STOCHASTIC ECONOMIC OPTIMIZATION OF ELECTRICAL SYSTEMS, AND RELATED SYSTEMS, APPARATUSES, AND METHODS

Abstract

Electrical system controllers, computer-readable storage media, and related methods for stochastic control of electrical systems. An electrical system controller includes one or more data storage devices and one or more processors. The one or more data storage devices are configured to store data corresponding to one or more random variables associated with operation of an electrical system. The one or more processors are configured to determine a set of control values for a set of control variables to effectuate a change to the electrical system toward meeting a controller objective for economical optimization of the electrical system. The set of control values are determined by the one or more processors utilizing an optimization algorithm to identify the set of control values as a function of the one or more random variables. The one or more processors are also configured to control the electrical system based on the control values.

Description

TECHNICAL FIELD

The present disclosure is directed to systems and methods for control of an electrical system, and more particularly to controllers and methods of controlling an electrical system.

BACKGROUND

Electricity supply and delivery costs continue to rise, especially in remote or congested areas. Moreover, load centers (e.g., population centers where electricity is consumed) increasingly demand more electricity. In the U.S. energy infrastructure is such that power is mostly produced by resources inland, and consumption of power is increasing at load centers along the coasts. Thus, transmission and distribution (T&D) systems are needed to move the power from where it's generated to where it's consumed at the load centers. As the load centers demand more electricity, additional T&D systems are needed, particularly to satisfy peak demand. However, a major reason construction of additional T&D systems is unwise and/or undesirable is because full utilization of this infrastructure is really only necessary during relatively few peak demand periods, and would otherwise be unutilized or underutilized. Justifying the significant costs of constructing additional T&D resources may make little sense when actual utilization may be relatively infrequent.

Distributed energy storage is increasingly seen as a viable means for minimizing rising costs by storing electricity at the load centers for use during the peak demand times. An energy storage system (ESS) can enable a consumer of energy to reduce or otherwise control a net consumption from an energy supplier. For example, if electricity supply and/or delivery costs are high at a particular time of day, an ESS, which may include one or more batteries or other storage devices, can generate/discharge electrical energy at that time when costs are high in order to reduce the net consumption from the supplier. Likewise, when electricity rates are low, the ESS may charge so as to have reserve energy to be utilized in a later scenario as above when supply and/or delivery costs are high.

Presently available automatic controllers of electrical systems utilize rule sets and iteration to find an operating command that in its simplest form can be a single scalar value that specifies the charge (or discharge) power setting of a battery. The main drawbacks of this existing approach are that it doesn't necessarily provide economically optimal control considering all costs and benefits, rule sets become complex quickly (which makes the algorithm difficult to build and maintain, even for just two value streams), and this approach is not easily scalable to new rate tariffs or other markets or value streams (because rule sets must be rewritten).

An economically optimizing automatic controller may be beneficial and may be desirable to enable intelligent actions to be taken to more effectively utilize controllable components of an electrical system, and without the aforementioned drawbacks.

BRIEF DESCRIPTION OF THE DRAWINGS

Additional aspects and advantages will be apparent from the following detailed description of preferred embodiments, which proceeds with reference to the accompanying drawings, in which:

FIG. 1 is a block diagram illustrating a system architecture of a controllable electrical system, according to one embodiment of the present disclosure.

FIG. 2 is a flow diagram of a method or process of controlling an electrical system, according to one embodiment of the present disclosure.

FIG. 3 is a graph illustrating an example of nonlinear continuous optimization to determine a minimum or maximum of an equation given specific constraints.

FIG. 4 is a contour plot illustrating an example of nonlinear continuous optimization of FIG. 3 to determine a minimum or maximum of an equation given specific constraints.

FIG. 5 is a block diagram illustrating a system architecture of a controllable electrical system, according to one embodiment of the present disclosure.

FIG. 6 is a flow diagram of a method or process of controlling an electrical system, according to one embodiment of the present disclosure.

FIG. 7 is a flow diagram of a method of predicting load and/or generation of an electrical system during an upcoming time domain, according to one embodiment of the present disclosure.

FIG. 8 is a graphical representation 800 of predicting load and/or generation of an electrical system during an upcoming time domain.

FIG. 9 is a graph illustrating one example of segmenting an upcoming time domain into multiple time segments.

FIG. 10 is a diagrammatic representation of a cost function evaluation module, according to one embodiment of the present disclosure.

FIG. 11 is a flow diagram of a method of preparing a cost function f_c(X), according to one embodiment of the present disclosure.

FIG. 12 is a flow diagram of a method of evaluating a cost function that is received from an external source or otherwise unprepared, according to one embodiment of the present disclosure.

FIG. 13 is a flow diagram of a method of evaluating a prepared cost function, according to one embodiment of the present disclosure.

FIG. 14 is a diagrammatic representation of an optimizer that utilizes an optimization algorithm to determine an optimal control parameter set.

FIG. 15 is a graph illustrating an example result from an economic optimizer (EO) for a battery ESS.

FIG. 16 is a method of a dynamic manager, according to one embodiment of the present disclosure.

FIG. 17 is a graph showing plots for an example of application of a particular four-parameter control set during a time segment.

FIG. 18 is a graph providing a plot of the wear rate vs. state of charge for a specific battery degradation model, according to one embodiment of the present disclosure.

FIG. 19 is a graph providing a plot showing a relationship between state of charge and aging rate (or an aging factor) for a specific battery degradation model, according to one embodiment of the present disclosure.

FIG. 20 is a pair of graphs that illustrate a battery's lifetime.

FIG. 21 is a diagram of an economic optimizer, according to one embodiment of the present disclosure.

FIG. 22 is a diagram of a dynamic manager, according to one embodiment of the present disclosure.

FIG. 23 is a graph illustrating how Time-of-Use (ToU) supply charges impact energy costs of a customer.

FIG. 24 is a graph illustrating how demand charges impact energy costs of a customer.

FIG. 25 is a graph illustrating the challenge of maximizing a customer's economic returns for a wide range of system configurations, building load profiles, and changing utility tariffs.

FIG. 26 is a simplified block diagram illustrating an electrical power system, according to one embodiment of the present disclosure.

FIG. 27 is a simplified flowchart illustrating a method of operating an electrical power system, according to one embodiment of the present disclosure.

FIG. 28 is a simplified block diagram of an electrical system controller, according to some embodiments.

FIGS. 29A-29G illustrate an example of generating random variables to account for uncertainty in a load, according to some embodiments.

FIG. 30 is a graphical representation illustrating an accounting for uncertainty in a cost of operating an electrical system during an upcoming time domain, according to some embodiments.

FIG. 31 is a simplified flowchart illustrating a method of operating an electrical system, according to some embodiments.

DETAILED DESCRIPTION

As electricity supply and delivery costs increase, especially in remote or congested areas, distributed energy storage is increasingly seen as a viable means for reducing those costs. The reasons are numerous, but primarily an energy storage system (ESS) gives a local generator or consumer the ability to control net consumption and delivery of electrical energy at a point of interconnection, such as a building's service entrance in example implementations where an ESS is utilized in an apartment building or office building. For example, if electricity supply and/or delivery costs (e.g., charges) are high at a particular time of day, an ESS can generate/discharge electrical energy from a storage system at that time to reduce the net consumption of a consumer (e.g., a building), and thus reduce costs to the consumer. Likewise, when electricity rates are low, the ESS may charge its storage system which may include one or more batteries or other storage devices; the lower-cost energy stored in the ESS can then be used to reduce net consumption and thus costs to the consumer at times when the supply and/or delivery costs are high. There are many ways an ESS can provide value.

One possible way in which ESSs can provide value (e.g., one or more value streams) is by reducing time-of-use (ToU) supply charges. ToU supply charges are typically pre-defined in a utility's tariff document by one or more supply rates and associated time windows. ToU supply charges may be calculated as the supply rate multiplied by the total energy consumed during the time window. ToU supply rates in the United States may be expressed in dollars per kilowatt-hour ($/kWh). The ToU supply rates and time windows may change from time to time, for example seasonally, monthly, daily, or hourly. Also, multiple ToU supply rates and associated time windows may exist and may overlap. ToU supply rates are time-varying which makes them different from “flat” supply rates that are constant regardless of time of use. An example of ToU charges and the impact on customer energy costs is illustrated in FIG. 24 and described more fully below with reference to the same. An automatic controller may be beneficial and may be desirable to enable intelligent actions to be taken as frequently as may be needed to utilize an ESS to reduce ToU supply charges.

Another possible way in which ESSs can provide value is by reducing demand charges. Demand charges are electric utility charges that are based on the rate of electrical energy consumption (also called “demand”) during particular time windows (which we will call “demand windows”). A precise definition of demand and the formula for demand charges may be defined in a utility's tariff document. For example, a tariff may specify that demand be calculated at given demand intervals (e.g., 15-minute intervals, 30-minute intervals, 40-minute intervals, 60-minute intervals, 120-minute intervals, etc.). The tariff may also define demand as being the average rate of electrical energy consumption over a previous period of time (e.g., the previous 15 minutes, 30 minutes, 40 minutes, etc.). The previous period of time may or may not coincide with the demand interval. Demand may be expressed in units of power such as kilowatts (kW) or megawatts (MW). The tariff may describe one or more demand rates, each with an associated demand window (e.g., a period of time during which a demand rate applies). The demand windows may be contiguous or noncontiguous and may span days, months, or any other total time interval per the tariff. Also, one or more demand window may overlap which means that, at a given time, more than one demand rate may be applicable. Demand charges for each demand window may be calculated as a demand rate multiplied by the maximum demand during the associated demand window. Demand rates in the United States may be expressed in dollars per peak demand ($/kW). An example of demand charges is shown in FIG. 24 and described more fully below with reference to the same. As can be appreciated, demand tariffs may change from time to time, or otherwise vary, for example annually, seasonally, monthly, or daily. An automatic controller may be beneficial and may be desirable to enable intelligent actions to be taken as frequently as may be needed to utilize an ESS to reduce demand charges.

Another possible way in which ESSs can provide value is through improving utilization of local generation by: (a) maximizing self-consumption of renewable energy, or (b) reducing fluctuations of a renewable generator such as during cloud passage on solar photovoltaic (PV) arrays. An automatic controller may be beneficial and may be desirable to enable intelligent actions to be taken to effectively and more efficiently utilize locally generated power with an ESS.

Another possible way in which ESSs can provide value is through leveraging local contracted or incentive maneuvers. For example New York presently has available a Demand Management Program (DMP) and a Demand Response Program (DRP). These programs, and similar programs, offer benefits (e.g., a statement credit) or other incentives for consumers to cooperate with the local utility(ies). An automatic controller may be beneficial and may be desirable to enable intelligent actions to be taken to utilize an ESS to effectively leverage these contracted or incentive maneuvers.

Still another possible way in which ESSs can provide value is through providing reserve battery capacity for backup power in case of loss of supply. An automatic controller may be beneficial and may be desirable to enable intelligent actions to be taken to build and maintain such reserve battery backup power with an ESS.

As can be appreciated, an automatic controller that can automatically operate an electrical system to take advantage of any one or more of these value streams using an ESS may be desirable and beneficial.

Controlling Electrical Systems

An electrical system, according to some embodiments, may include one or more electrical loads, generators, and ESSs. An electrical system may include all three of these components (loads, generators, ESSs), or may have varying numbers and combinations of these components. For example, an electrical system may have loads and an ESS, but no local generators (e.g., photovoltaic, wind). The electrical system may or may not be connected to an electrical utility distribution system (or “grid”). If not connected to an electrical utility distribution system, it may be termed “off-grid.”

An ESS of an electrical system may include one or more storage devices and any number of power conversion devices. The power conversion devices are able to transfer energy between an energy storage device and the main electrical power connections that in turn connect to the electrical system loads and, in some embodiments, to the grid. The energy storage devices may be different in different implementations of the ESS. A battery is a familiar example of a chemical energy storage device. For example, in one embodiment of the present disclosure, one or more electric vehicle batteries is connected to an electrical system and can be used to store energy for later use by the electrical system. A flywheel is an example of a mechanical energy storage device.

FIG. 1 is a control diagram of an electrical system 100, according to one embodiment of the present disclosure. Stated otherwise, FIG. 1 is a representative diagram of a system architecture of an electrical system 100 including a controller 110, according to one embodiment. The electrical system 100 comprises a building electrical system 102 that is controlled by the controller 110. The building electrical system 102 includes one or more loads 122, one or more generators 124, and an energy storage system (ESS) 126. The building electrical system 102 is coupled to an electrical utility distribution system 150, and therefore may be considered on-grid. Similar electrical systems exist for other applications such as a photovoltaic generator plant and an off-grid building.

In the control diagram of FIG. 1, the controller 110 is shown on the left-hand side and the building electrical system 102, sometimes called the “plant,” is on the right-hand side. The controller 110 may include electronic hardware and software in one embodiment. In one example arrangement, the controller 110 includes one or more processors and suitable storage media, which stores programming in the form of executable instructions which are executed by the processors to implement the control processes. In some embodiments, the building electrical system 102 is the combination of all local loads 122, local generators 124, and the ESS 126.

Loads are consumers of electrical energy within an electrical system. Examples of loads are air conditioning systems, motors, electric heaters, etc. The sum of the loads' electricity consumption rates can be measured in units of power (e.g. kW) and simply called “load” (e.g., a building load).

Generators may be devices, apparatuses, or other means for generating electrical energy within an electrical system. Examples are solar photovoltaic systems, wind generators, combined heat and power (CHP) systems, and diesel generators or “gen-sets.” The sum of electric energy generation rates of the generators 124 can be measured in units of power (e.g., kW) and simply referred to as “generation.”

As can be appreciated, loads may also generate at certain times. An example may be an elevator system that is capable of regenerative operation when the carriage travels down.

Unadjusted net power may refer herein to load minus generation in the absence of active control by a controller described herein. For example, if at a given moment a building has loads consuming 100 kW, and a solar photovoltaic system generating at 25 kW, the unadjusted net power is 75 kW. Similarly, if at a given moment a building has loads consuming 70 kW, and a solar photovoltaic system generating at 100 kW, the unadjusted net power is −30 kW. As a result, the unadjusted net power is positive when the load energy consumption exceeds generation, and negative when the generation exceeds the load energy consumption.

ESS power refers herein to a sum of a rate of electric energy consumption of an ESS. If ESS power is positive, an ESS is charging (consuming energy). If ESS power is negative, an ESS is generating (delivering energy).

Adjusted net power refers herein to unadjusted net power plus the power contribution of any controllable elements such as an ESS. Adjusted net power is therefore the net rate of consumption of electrical energy of the electrical system considering all loads, generators, and ESSs in the system, as controlled by a controller described herein.

Unadjusted demand is demand defined by the locally applicable tariff, but only based on the unadjusted net power. In other words, unadjusted demand does not consider the contribution of any ESS.

Adjusted demand or simply “demand” is demand as defined by the locally applicable tariff, based on the adjusted net power, which includes the contribution from any and all controllable elements such as ESSs. Adjusted demand is the demand that can be monitored by the utility and used in the demand charge calculation.

Referring again to FIG. 1, the building electrical system 102 may provide information to the controller 110, such as in a form of providing process variables. The process variables may provide information, or feedback, as to a status of the building electrical system 102 and/or one or more components (e.g., loads, generators, ESSs) therein. For example, the process variable may provide one or more measurements of a state of the electrical system. The controller 110 receives the process variables for determining values for control variables to be communicated to the building electrical system 102 to effectuate a change to the building electrical system 102 toward meeting a controller objective for the building electrical system 102. For example, the controller 110 may provide a control variable to adjust the load 122, to increase or decrease generation by the generator 124, and to utilize (e.g., charge or discharge) the ESS 126. The controller 110 may also receive a configuration (e.g., a set of configuration elements), which may specify one or more constraints of the electrical system 102. The controller 110 may also receive external inputs (e.g., weather reports, changing tariffs, fuel costs, event data), which may inform the determination of the values of the control variables. A set of external inputs may be received by the controller 110. The set of external inputs may provide indication of one or more conditions that are external to the controller and the electrical system.

As noted, the controller 110 may attempt to meet certain objectives by changing a value associated with one or more control variables, if necessary. The objectives may be predefined, and may also be dependent on time, on any external inputs, on any process variables that are obtained from the building electrical system 102, and/or on the control variables themselves. Some examples of controller objectives for different applications are:

• • Minimize demand (kW) over a prescribed time interval;
  • Minimize demand charges ($) over a prescribed time interval;
  • Minimize total electricity charges ($) from the grid;
  • Reduce demand (kW) from the grid by a prescribed amount during a prescribed time window; and
  • Maximize the life of the energy storage device.

Objectives can also be compound—that is, a controller objective can be comprised of multiple individual objectives. One example of a compound objective is to minimize demand charges while maximizing the life of the energy storage device. Other compound objectives including different combinations of the individual objectives are possible.

The inputs that the controller 110 may use to determine (or otherwise inform a determination of) the control variables can include configuration, external inputs, and process variables.

Process variables are typically measurements of the electrical system state and are used by the controller 110 to, among other things, determine how well its objectives are being met. These process variables may be read and used by the controller 110 to generate new control variable values. The rate at which process variables are read and used by the controller 110 depends upon the application but typically ranges from once per millisecond to once per hour. For battery energy storage system applications, the rate is often between 10 times per second and once per 15 minutes. Examples of process variables may include:

• • Unadjusted net power
  • Unadjusted demand
  • Adjusted net power
  • Demand
  • Load (e.g., load energy consumption for one or more loads)
  • Generation for one or more loads
  • Actual ESS charge or generation rate for one or more ESS
  • Frequency
  • Energy storage device state of charge (SoC) (%) for one or more ESS
  • Energy storage device temperature (deg. C) for one or more ESS
  • Electrical meter outputs such as kilowatt-hours (kWh) or demand.

A configuration received by the controller 110 (or input to the controller 110) may include or be received as one or more configuration elements (e.g., a set of configuration elements). The configuration elements may specify one or more constraints associated with operation of the electrical system. The configuration elements may define one or more cost elements associated with operation of the electrical system 102. Each configuration element may set a status, state, constant or other aspect of the operation of the electrical system 102. The configuration elements may be values that are typically constant during the operation of the controller 110 and the electrical system 102 at a particular location. The configuration elements may specify one or more constraints of the electrical system and/or specify one or more cost elements associated with operation of the electrical system.

Examples of configuration elements may include:

• • ESS type (for example if a battery: chemistry, manufacturer, and cell model)
  • ESS configuration (for example, if a battery: number of cells in series and parallel) and constraints (such as maximum charge and discharge powers)
  • ESS efficiency properties
  • ESS degradation properties (as a function of SoC, discharge or charge rate, and time)
  • Electricity supply tariff (including ToU supply rates and associated time windows)
  • Electricity demand tariff (including demand rates and associated time windows)
  • Electrical system constraints such as minimum power import
  • ESS constraints such as SoC limits or power limits
  • Historic data such as unadjusted net power or unadjusted demand, weather data, and occupancy
  • Operational constraints such as a requirement for an ESS to have a specified minimum amount of energy at a specified time of day.

External inputs are variables that may be used by the controller 110 and that may change during operation of the controller 110. Examples are weather forecasts (e.g., irradiance for solar generation and wind speeds for wind generation) and event data (e.g., occupancy predictions). In some embodiments, tariffs (e.g., demand rates defined therein) may change during the operation of the controller 110, and may therefore be treated as an external input.

The outputs of the controller 110 are the control variables that can affect the electrical system behavior. Examples of control variables are:

• • ESS power command (kW or %). For example, an ESS power command of 50 kW would command the ESS to charge at a rate of 50 kW, and an ESS power command of −20 kW would command the ESS to discharge at a rate of 20 kW.
  • Building or subsystem net power increase or reduction (kW or %)
  • Renewable energy increase or curtailment (kW or %). For example a photovoltaic (PV) system curtailment command of −100 kW would command a PV system to limit generation to no less than −100 kW. Again, the negative sign is indicative of the fact that that the value is generative (non-consumptive).In some embodiments, control variables that represent power levels may be signed, e.g., positive for consumptive or negative for generative.

In one illustrative example, consider that an objective of the controller 110 may be to reduce demand charges while preserving battery life. In this example, only the ESS may be controlled. To accomplish this objective, the controller should have knowledge of a configuration of the electrical system 102, such as the demand rates and associated time windows, the battery capacity, the battery type and arrangement, etc. Other external inputs may also be used to help the controller 110 meet its objectives, such as a forecast of upcoming load and/or forecast of upcoming weather (e.g., temperature, expected solar irradiance, wind). Process variables from the electrical system 102 that may be used may provide information concerning a net electrical system power or energy consumption, demand, a battery SoC, an unadjusted building load, and an actual battery charge or discharge power. In this one illustrative example, the control variable may be a commanded battery ESS's charge or discharge power. In order to more effectively meet the objective, the controller 110 may continuously track the peak net building demand (kW) over each applicable time window, and use the battery to charge or generate at appropriate times to limit the demand charges. In one specific example scenario, the ESS may be utilized to attempt to achieve substantially flat (or constant) demand from the electrical utility distribution system 150 (e.g., the grid) during an applicable time window when a demand charge applies.

FIG. 2 is a flow diagram of a method 200 or process of controlling an electrical system, according to one embodiment of the present disclosure. The method 200 may be implemented by a controller of an electrical system, such as the controller 110 of FIG. 1 controlling the building electrical system 102 of FIG. 1. The controller may read 202 or otherwise receive a configuration (e.g., a set of configuration elements) of the electrical system.

The controller may also read 204 or otherwise receive external inputs, such as weather reports (e.g., temperature, solar irradiance, wind speed), changing tariffs, event data (e.g., occupancy prediction, sizeable gathering of people at a location or venue), and the like.

The controller may also read 206 or otherwise receive process variables, which may be measurements of a state of the electrical system and indicate, among other things, how well objectives of the controller are being met. The process variables provide feedback to the controller as part of a feedback loop.

Using the configuration, the external inputs, and/or the process variables, the controller determines 208 new control variables to improve achievement of objectives of the controller. Stated differently, the controller determines 208 new values for each control variable to effectuate a change to the electrical system toward meeting one or more controller objectives for the electrical system. Once determined, the control variables (or values thereof) are transmitted 210 to the electrical system or components of the electrical system. The transmission 210 of the control variables to the electrical system allows the electrical system to process the control variables to determine how to adjust and change state, which thereby can effectuate the objective(s) of the controller for the electrical system.

Optimization

In some embodiments, the controller uses an algorithm (e.g., an optimization algorithm) to determine the control variables, for example, to improve performance of the electrical system. Optimization can be a process of finding a variable or variables at which a function f(x) is minimized or maximized. An optimization may be made with reference to such global extrema (e.g., global maximums and/or minimums), or even local extrema (e.g., local maximums and/or minimums). Given that an algorithm that finds a minimum of a function can generally also find a maximum of the same function by negating it, this disclosure will sometimes use the terms “minimization,” “maximization,” and “optimization,” interchangeably.

An objective of optimization may be economic optimization, or determining economically optimal control variables to effectuate one or more changes to the electrical system to achieve economic efficiency (e.g., to operate the electrical system at as low a cost as may be possible, given the circumstances). As can be appreciated, other objectives may be possible as well (e.g., prolong equipment life, system reliability, system availability, fuel consumption, etc.).

The present disclosure includes embodiments of controllers that optimize a single parameterized cost function (or objective function) for effectively utilizing controllable components of an electrical system in an economically optimized manner. Various forms of optimization may be utilized to economically optimize an electrical system.

Continuous Optimization

A controller according to some embodiments of the present disclosure may use continuous optimization to determine the control variables. More specifically, the controller may utilize a continuous optimization algorithm, for example, to find economically optimal control variables to effectuate one or more changes to the electrical system to achieve economic efficiency (e.g., to operate the electrical system at as low a cost as may be possible, given the circumstances). The controller, in one embodiment, may operate on a single objective: optimize overall system economics. Since this approach has only one objective, there can be no conflict between objectives. And by specifying system economics appropriately in the cost function (or objective function), all objectives and value streams can be considered simultaneously based on their relative impact on a single value metric. The cost function may be continuous in its independent variables x, and optimization can be executed with a continuous optimization algorithm that is effective for continuous functions. Continuous optimization differs from discrete optimization, which involves finding the optimum value from a finite set of possible values or from a finite set of functions.

As can be appreciated, in another embodiment, the cost function may be discontinuous in x (e.g., discrete or finite) or piecewise continuous in x, and optimization can be executed with an optimization algorithm that is effective for discontinuous or piecewise continuous functions.

Constrained Optimization

In some embodiments, the controller utilizes a constrained optimization to determine the control variables. In certain embodiments, the controller may utilize a constrained continuous optimization to find a variable or variables x_opt at which a continuous function f(x) is minimized or maximized subject to constraints on the allowable x.

FIGS. 3 and 4 show a graph 300 and a contour plot 400 illustrating an example of a constrained continuous optimization to determine a minimum or maximum of an equation given specific constraints. Possible constraints may be an equation or inequality. FIGS. 3 and 4 consider an equation:

f(x)=100(x₂−x₁²)²+(1−x₁)².

The set x includes the independent variables x₁, x₂. Constraints are defined by the equation:

x ₂ ² +x ₁ ²≤1.

FIG. 3 illustrates a curve 301 of ln (1+f(x)) vs. x₁ and x₂ and illustrates the constraint within an outlined unit disk 303. A minimum 305 is at (0.7864, 0.6177).

FIG. 4 illustrates a contour plot 401 of log₁₀(f(x)), which also shows the constraints within the outlined unit disk 403 and a minimum 405 is at (0.7864, 0.6177).

Constrained continuous optimization algorithms are useful in many areas of science and engineering to find a “best” or “optimal” set of values that affect a governing of a process. They are particularly useful in cases where a single metric is to be optimized, but the relationship between that metric and the independent (x) variables is so complex that a “best” set of x values cannot easily be found symbolically in closed form. For example, consider a malignant tumor whose growth rate over time is dependent upon pH and on the concentration of a particular drug during various phases of growth. The equation describing growth rate as a function of the pH and drug concentration is known and can be written down but may be complex and nonlinear. It might be very difficult or impossible to solve the equation in closed form for the best pH and drug concentration at various stages of growth. It may also depend on external factors such as temperature. To solve this problem, pH and drug concentration at each stage of growth can be combined into an x vector with two elements. Since the drug concentration and pH may have practical limits, constraints on x can be defined. Then the function can be minimized using constrained continuous optimization. The resulting x where the growth rate is minimized contains the “best” pH and drug concentration to minimize growth rate. Note this approach can find the optimum pH and drug concentration (to machine precision) from a continuum of infinite possibilities of pH and drug concentration, not just from a predefined finite set of possibilities.

Generalized Optimization

A controller according to some embodiments of the present disclosure may use generalized optimization to determine the control variables. More specifically, the controller may utilize a generalized optimization algorithm, for example, to find economically optimal control variables to effectuate one or more changes to the electrical system to achieve economic efficiency (e.g., to operate the electrical system at as low a cost as may be possible, given the circumstances).

An algorithm that can perform optimization for an arbitrary or general real function f(x) of any form may be called a generalized optimization algorithm. An algorithm that can perform optimization for a general continuous real function f(x) of a wide range of possible forms may be called a generalized continuous optimization algorithm. Some generalized optimization algorithms may be able to find optimums for functions that may not be continuous everywhere, or may not be differentiable everywhere. Some generalized optimization algorithms are available as pre-written software in many languages including Java®, C++, and MATLAB®. They often use established and well-documented iterative approaches to find a function's minimum.

As can be appreciated, a generalized optimization algorithm may also account for constraints, and therefore be a generalized constrained optimization algorithm.

Nonlinear Optimization

A controller according to some embodiments of the present disclosure may use nonlinear optimization to determine the control variables. More specifically, the controller may utilize a nonlinear optimization algorithm, for example, to find economically optimal control variables to effectuate one or more changes to the electrical system to achieve economic efficiency (e.g., to operate the electrical system at as low a cost as may be possible, given the circumstances).

Nonlinear continuous optimization or nonlinear programming is similar to generalized continuous optimization and describes methods for optimizing continuous functions that may be nonlinear, or where the constraints may be nonlinear.

Multi-Variable Optimization

A controller according to some embodiments of the present disclosure may use multi-variable optimization to determine the control variables. More specifically, the controller may utilize a multivariable optimization algorithm, for example, to find economically optimal control variables to effectuate one or more changes to the electrical system to achieve economic efficiency (e.g., to operate the electrical system at as low a cost as may be possible, given the circumstances).

In the examples of FIGS. 3 and 4, the considered equation

f(x)=100(x₂−x₁²)²+(1−x₁)²

is a multi-variable equation. In other words, x is a set comprised of more than one element. Therefore, the optimization algorithm is “multivariable.” A subclass of optimization algorithms is the multivariable optimization algorithm that can find the minimum of f(x) when x has more than one element. Thus, the example of FIGS. 3 and 4 may illustrate solving for a generalized constrained continuous multi-variable optimization problem.

Economically Optimizing Electrical System Controller

A controller, according to one embodiment of the present disclosure, will now be described to provide an example of using optimization to control an electrical system. An objective of using optimization may be to minimize the total electrical system operating cost during a period of time. For example, the approach of the controller may be to minimize the operating cost during an upcoming time domain, or future time domain, which may extend from the present time by some number of hours (e.g., integer numbers of hours, fractions of hours, or combinations thereof). As another example, the upcoming time domain, or future time domain, may extend from a future time by some number of hours. Costs included in the total electrical system operating cost may include electricity supply charges, electricity demand charges, a battery degradation cost, equipment degradation cost, efficiency losses, etc. Benefits, such as incentive payments, which may reduce the electrical system operating cost, may be incorporated (e.g., as negative numbers or values) or otherwise considered. Other cost may be associated with a change in energy in the ESS such that adding energy between the beginning and the end of the future time domain is valued. Other costs may be related to reserve energy in an ESS such as for backup power purposes. All of the costs and benefits can be summed into a net cost function, which may be referred to as simply the “cost function.”

In certain embodiments, a control parameter set X can be defined (in conjunction with a control law) that is to be applied to the electrical system, how they should behave, and at what times in the future time domain they should be applied. In some embodiments, the cost function can be evaluated by performing a simulation of electrical system operation with a provided set X of control parameters. The control laws specify how to use X and the process variables to determine the control variables. The cost function can then be prepared or otherwise developed to consider the control parameter set X.

For example, a cost f_c(X) may consider the control parameter values in X and return the scalar net cost of operating the electrical system with those control parameter values. All or part of the control parameter set X can be treated as a variable set X_x (e.g., x as described above) in an optimization problem. The remaining part of X, X_logic, may be determined by other means such as logic (for example logic based on constraints, inputs, other control parameters, mathematical formulas, etc.). Any constraints involving X_x can be defined, if so desired. Then, an optimization algorithm can be executed to solve for the optimal X_x. We can denote X_opt as the combined X_x and X_logic values that minimize the cost function subject to the constraints, if any. Since X_opt represents the control parameters, this example process fully specifies the control that will provide minimum cost (e.g., optimal) operation during the future time domain. Furthermore, to the limits of computing capability, this optimization can consider the continuous domain of possible X_x values, not just a finite set of discrete possibilities. This example method continuously can “tune” possible control sets until an optimal set is found. As shorthand notation, we may refer to these certain example embodiments of an economically optimizing electrical system controller (EOESC).

Some of the many advantages of using an EOESC, according to certain embodiments, compared to other electrical system controllers are significant:

1) Any number of value streams may be represented in the cost function, giving the EOESC an ability to optimize on all possible value streams and costs simultaneously. As an example, generalized continuous optimization can be used to effectively determine the best control given both ToU supply charge reduction and demand charge reduction simultaneously, all while still considering battery degradation cost.

2) With a sufficiently robust optimization algorithm, only the cost function, control law, and control parameter definitions need be developed. Once these three components are developed, they can be relatively easily maintained and expanded upon.

3) An EOESC can yield a true economically optimum control solution to machine or processor precision limited only by the cost function, control laws, and control parameter definitions.

4) An EOESC may yield not only a control to be applied at the present time, but also the planned sequence of future controls. This means one execution of an EOESC can generate a lasting set of controls that can be used into the future rather than a single control to be applied at the present. This can be useful in case a) the optimization algorithm takes a significant amount of time to execute, or b) there is a communication interruption between the processor calculating the control parameter values and the processor interpreting the control parameters and sending control variables to the electrical system.

FIG. 5 is a control diagram of an electrical system 500, according to one embodiment of the present disclosure, including an EOESC 510. Stated otherwise, FIG. 5 is a diagram of a system architecture of the electrical system 500 including the EOESC 510, according to one embodiment. The electrical system 500 comprises a building electrical system 502 that is controlled by the EOESC 510. The building electrical system 502 includes one or more loads 522, one or more generators 524, an energy storage system (ESS) 526, and one or more sensors 528 (e.g., meters) to provide measurements or other indication(s) of a state of the building electrical system 502. The building electrical system 502 is coupled to an electrical utility distribution system 550, and therefore may be considered on-grid. Similar diagrams can be drawn for other applications such as a photovoltaic generator plant and an off-grid building.

The EOESC 510 receives or otherwise obtains a configuration of the electrical system, external inputs, and process variables and produces control variables to be sent to the electrical system 502 to effectuate a change to the electrical system toward meeting a controller objective for economical optimization of the electrical system, for example during an upcoming time domain. The EOESC 510 may include electronic hardware and software to process the inputs (e.g., the configuration of the electrical system, external inputs, and process variables) to determine values for each of the control variables. The EOESC 510 may include one or more processors and suitable storage media which stores programming in the form of executable instructions which are executed by the processors to implement the control processes.

In the embodiment of FIG. 5, the EOESC 510 includes an economic optimizer (EO) 530 and a dynamic manager 540 (or high speed controller (HSC)). The EO 530 according to some embodiments is presumed to have ability to measure or obtain a current date and time. The EO 530 may determine a set of values for a control parameter set X and provide the set of values and/or the control parameter set X to the HSC 540. The EO 530 uses a generalized optimization algorithm to determine an optimal set of values for the control parameter set X_opt. The HSC 540 utilizes the set of values for the control parameter set X (e.g., an optimal control parameter set X_opt) to determine the control variables to communicate to the electrical system 502. The HSC 540 in some embodiments is also presumed to have ability to measure or obtain a current date and time. The two part approach of the EOESC 510, namely the EO 530 determining control parameters and then the HSC 540 determining the control variables, enables generation of a lasting set of controls, or a control solution (or plan) that can be used into the future rather than a single control to be applied at the present. Preparing a lasting control solution can be useful if the optimization algorithm takes a significant amount of time to execute. Preparing a lasting control solution can also be useful if there is a communication interruption between the calculating of the control parameter values and the processor interpreting the control parameters and sending control variables to the electrical system 502. The two part approach of the EOESC 510 also enables the EO 530 to be disposed or positioned at a different location from the HSC 540. In this way, intensive computing operations that optimization may require can be performed by resources with higher processing capability that may be located remote from the building electrical system 502. These intensive computing operations may be performed, for example, at a data center or server center (e.g., in the cloud).

In some embodiments, a future time domain begins at the time the EO 530 executes and can theoretically extend any amount of time. In certain embodiments, analysis and experimentation suggest that a future time domain extent of 24 to 48 hours generates sufficiently optimal solutions in most cases.

As can be appreciated, the EOESC 510 of FIG. 5 may be arranged and configured differently from that shown in FIG. 5, in other embodiments. For example, instead of the EO 530 passing the control parameter set X_opt (the full set of control parameters found by a generalized optimization algorithm of the EO 530) to the HSC 540, the EO 530 can pass a subset of X_opt to the HSC 540. Similarly, the EO 530 can pass X_opt and additional control parameters to the HSC 540 that are not contained in X_opt. Likewise, the EO 530 can pass modified elements of X_opt to the HSC 540. In one embodiment, the EO 530 finds a subset X_x of the optimal X, but then determines additional control parameters X_logic, and passes X_logic together with X_x to the HSC 540. In other words, in this example, the X_x values are to be determined through an optimization process of the EO 530 and the X_logic values can be determined from logic. An objective of the EO 530 is to determine the values for each control parameter whether using optimization and/or logic.

For brevity in this disclosure, keeping in mind embodiments where X consists of independent (X_x) parameters and dependent (X_logic) parameters, when describing optimization of a cost function versus X, what is meant is variation of the independent variables X_x until an optimum (e.g., minimum) cost function value is determined. In this case, the resulting X_opt will consist of the combined optimum X_x parameters and associated X_logic parameters.

In one embodiment, the EOESC 510 and one or more of its components are executed as software or firmware (for example stored on non-transitory media, such as appropriate memory) by one or more processors. For example, the EO 530 may comprise one or more processors to process the inputs and generate the set of values for the control parameter set X. Similarly, the HSC 540 may comprise one or more processors to process the control parameter set X and the process variables and generate the control variables. The processors may be computers, microcontrollers, CPUs, logic devices, or any other digital or analog device that can operate on pre-programmed instructions. If more than one processor is used, they can be connected electrically, wirelessly, or optically to pass signals between one another. In addition, the control variables can be communicated to the electrical system components electrically, wirelessly, or optically or by any other means. The processor has the ability to store or remember values, arrays, and matrices, which can be viewed as multi-dimensional arrays, in some embodiments. This storage may be performed using one or more memory devices, such as read access memory (RAM, disk drives, etc.).

FIG. 6 is a flow diagram of a method 600 of controlling an electrical system, according to one embodiment of the present disclosure. The method 600 includes two separate processes, namely an economic optimizer (EO) process 601a and a high speed controller (HSC) process 601b. The HSC process 601b may also be referred to herein as a dynamic manager process 601b. The HSC process 601b may utilize a control parameter set X determined by the EO process 601a. Nevertheless, the HSC process 601b may execute separate from, or even independent from the EO process 601a, based on a control parameter set X determined at an earlier time by the EO process 601a. Because the EO process 601a can run separate and distinct from the HSC process 601b, the execution of these processes 601a, 601b may be collocated on a single system or isolated on remote systems.

The EO process 601a may be a computer-implemented process executed by one or more computing devices, such as the EO 530 of FIG. 5. The EO process 601a may receive 602 a configuration, or a set of configuration elements, of the electrical system. The configuration may specify one or more constraints of the electrical system. The configuration may specify one or more constants of the electrical system. The configuration may specify one or more cost elements associated with operation of the electrical system. The cost elements may include one or more of an electricity cost (e.g., an electricity supply charge, an electricity demand charge), a battery degradation cost, equipment degradation cost, a tariff definition (e.g., an electricity supply tariff providing ToU supply rates and associated time windows, or an electricity demand tariff providing demand rates and associated time windows), a cost of local generation, penalties associated with deviation from an operating plan (e.g., a prescribed operating plan, a contracted operating plan), costs or benefits associated with a change in energy in the ESS such that adding energy between the beginning and the end of the future time domain is valued, costs or benefits (e.g., a payment) for contracted maneuvers, costs or benefits associated with the amount of energy stored in an ESS as a function of time, a value of comfort that may be a function of other process variables such as building temperature.

In certain embodiments, the set of configuration elements define the one or more cost elements by specifying how to calculate an amount for each of the one or more cost elements. For example, the definition of a cost element may include a formula for calculating the cost element.

In certain embodiments, the cost elements specified by the configuration elements may include one or more incentives associated with operation of the electrical system. An incentive may be considered as a negative cost. The one or more incentives may include one or more of an incentive revenue, a demand response revenue, a value of reserve energy or battery capacity (e.g., for backup power as a function of time), a contracted maneuver, revenue for demand response opportunities, revenue for ancillary services, and revenue associated with deviation from an operating plan (e.g., a prescribed operating plan, a contracted operating plan).

In other embodiments, the configuration elements may specify how to calculate an amount for one or more of the cost elements. For example, a formula may be provided that indicates how to calculate a given cost element.

External inputs may also be received 604. The external inputs may provide indication of one or more conditions that are external to the controller and/or the electrical system. For example, the external inputs may provide indication of the temperature, weather conditions (e.g., patterns, forecasts), and the like.

Process variables are received 606. The process variables provide one or more measurements of a current state of the electrical system. The set of process variables can be used to determine progress toward meeting an objective for economical optimization of the electrical system. The process variables may be feedback in a control loop for controlling the electrical system.

The EO process 601a may include predicting 608 a local load and/or generation during an upcoming time domain. The predicted local load and/or local generation may be stored for later consideration. For example, the predicted load and/or generation may be used in a later process of evaluating the cost function during a minimization of the cost function.

A control parameter set X may be defined 610 to be applied during an upcoming time domain. In defining the control parameter set X, the meaning of each element of X is established. A first aspect in defining 610 the control parameter set X may include selecting a control law. Then, for example, X may be defined 610 as a matrix of values such that each column of X represents a set of control parameters for the selected control law to be applied during a particular time segment of the future time domain. In this example, the rows of X represent individual control parameters to be used by the control law. Further to this example, the first row of X can represent the nominal ESS power during a specific time segment of the future time domain. Likewise, X may be further defined such that the second row of X is the maximum demand limit (e.g., a maximum demand setpoint). A second aspect in defining 610 may include splitting the upcoming time domain into sensible segments and selecting the meaning of the control parameters to use during each segment. The upcoming future time domain may be split into different numbers of segments depending on what events are coming up during the future time domain. For example if there are no supply charges, and there is only one demand period, the upcoming time domain may be split into a few segments. But if there is a complicated scenario with many changing rates and constraints, the upcoming time domain may be split into many segments. Lastly, in defining 610 the control parameters X, some control parameters X_x may be marked for determination using optimization, and others X_logic may be marked for determination using logic (for example logic based on constraints, inputs, other control parameters, mathematical formulas, etc.).

The EO process 601a may also prepare 612 or obtain a cost function. Preparing 612 the cost function may be optional and can increase execution efficiency by pre-calculating certain values that will be needed each time the cost function is evaluated. The cost function may be prepared 612 (or configured) to include or account for any constraints on the electrical system.

With the control parameter set X defined 610 and the cost function prepared 612, the EO process 601a can execute 614 a minimization or optimization of the cost function resulting in the optimal control parameter set X_opt. For example, a continuous optimization algorithm may be used to identify an optimal set of values for the control parameter set X_opt (e.g., to minimize the cost function) in accordance with the one or more constraints and the one or more cost elements. The continuous optimization algorithm may be one of many types. For example, it may be a generalized continuous optimization algorithm. The continuous optimization algorithm may be a multivariable continuous optimization algorithm. The continuous optimization algorithm may be a constrained continuous optimization algorithm. The continuous optimization algorithm may be a Newton-type algorithm. It may be a stochastic-type algorithm such as Covariance Matrix Adaption Evolution Strategy (CMAES). Other algorithms that can be used are BOBYQA (Bound Optimization by Quadratic Approximation) and COBYLA (Constrained Optimization by Linear Approximation).

To execute the optimization of the cost function, the cost function may be evaluated many times. Each time, the evaluation may include performing a simulation of the electrical system operating during the future time domain with a provided control parameter set X, and then calculating the cost associated with that resulting simulated operation. The cost function may include or otherwise account for the one or more cost elements received 602 in the configuration. For example, the cost function may be a summation of the one or more cost elements (including any negative costs, such as incentive, revenues, and the like). In this example, the optimization step 614 would find X_opt that minimizes the cost function. The cost function may also include or otherwise account for the one or more constraints on the electrical system. The cost function may include or otherwise account for any values associated with the electrical system that may be received 602 in the configuration.

The cost function may also evaluate another economic metric such as payback period, internal rate of return (IRR), return on investment (ROI), net present value (NPV), or carbon emission. In these examples, the function to minimize or maximize would be more appropriately termed an “objective function.” In case the objective function represents a value that should be maximized such as IRR, ROI, or NPV, the optimizer should be set up to maximize the objective function when executing 614, or the objective function could be multiplied by −1 before minimization. Therefore, as can be appreciated, elsewhere in this disclosure, “minimizing” the “cost function” may also be more generally considered for other embodiments as “optimizing” an “objective function.”

The continuous optimization algorithm may execute the cost function (e.g., simulate the upcoming time domain) a plurality of times with various parameter sets X to identify an optimal set of values for the control parameter set X_opt to minimize the cost function. The cost function may include a summation of the one or more cost elements and evaluating the cost function may include returning a summation of the one or more cost elements incurred during the simulated operation of the control system over the upcoming time domain.

The optimal control parameter set X_opt is then output 616. In some embodiments, the output 616 of the optimal control parameter set X_opt may be stored locally, such as to memory, storage, circuitry, and/or a processor disposed local to the EO process 601a. In some embodiments, the outputting 616 may include transmission of the optimal control parameter set X_opt over a communication network to a remote computing device, such as the HSC 540 of FIG. 5.

The EO process 601a repeats for a next upcoming time domain (a new upcoming time domain). A determination 618 is made whether a new configuration is available. If yes, then the EO process 601a receives 602 the new configuration. If no, then the EO process 601a may skip receiving 602 the configuration and simply receive 604 the external inputs.

As can be appreciated, in other embodiments an EO process may be configured differently, to perform operations in a differing order, or to perform additional and/or different operations. In certain embodiments, an EO process may determine values for a set of control variables to provide to the electrical system to effectuate a change to the electrical system toward meeting the controller objective for economical optimization of the electrical system during an upcoming time domain, rather than determining values for a set of control parameters to be communicated to a HSC process. The EO process may provide the control variables directly to the electrical system, or to an HSC process for timely communication to the electrical system at, before, or during the upcoming time domain.

The HSC process 601b may be a computer-implemented process executed by one or more computing devices, such as the HSC 540 of FIG. 5. The HSC process 601b may receive 622 a control parameter set X, such as the optimal control parameter set X_opt output 616 by the EO process 601a. Process variables are also received 624 from the electrical system. The process variables include information, or feedback, about a current state or status of the electrical system and/or one or more components therein.

The HSC process 601b determines 626 values for a set of control variables for controlling one or more components of the electrical system at the current time. The HSC process 601b determines 626 the values for the control variables by using the optimal control parameter set X_opt in conjunction with a control law. The control laws specify how to determine the control variables from X (or X_opt) and the process variables. Stated another way, the control law enforces the definition of X. For example, for a control parameter set X defined such that a particular element, X_i, is an upper bound on demand to be applied at the present time, the control law may compare process variables such as the unadjusted demand to X_i. If unadjusted building demand exceeds X_i, the control law may respond with a command (in the form of a control variable) to instruct the ESS to discharge at a rate that will make the adjusted demand equal to or less than X_i.

The control variables (including any newly determined values) are then output 628 from the HSC process 601b. The control variables are communicated to the electrical system and/or one or more components therein. Outputting 628 the control variables may include timely delivery of the control variables to the electrical system at, before, or during the upcoming time domain and/or applicable time segment thereof. The timely delivery of the control variables may include an objective to effectuate a desired change or adjustment to the electrical system during the upcoming time domain.

A determination 630 is then made whether a new control parameter set X (and/or values thereof) is available. If yes, then the new control parameter set X (or simply the values thereof) is received 622 and HSC process 601b repeats. If no, then the HSC process 601b repeats without receiving 622 a new control parameter set X, such as a new optimal control parameter set X_opt.

As can be appreciated, in other embodiments an HSC process may be configured differently, to perform operations in a differing order, or to perform additional and/or different operations. For example, in certain embodiments, an HSC process may simply receive values for the set of control variables and coordinate timely delivery of appropriate control variables to effectuate a change to the electrical system at a corresponding time segment of the upcoming time domain.

The example embodiment of a control 510 in FIG. 5 and a control method 600 in FIG. 6 illustrate a two-piece or staged controller, which split a control problem into two pieces (e.g., a low speed optimizer and a high speed dynamic manager (or high speed controller (HSC)). The two stages or pieces of the controller, namely an optimizer and a dynamic manager, are described more fully the sections below. Nevertheless, as can be appreciated, in certain embodiments a single stage approach to a control problem may be utilized to determine optimal control values to command an electrical system.

Economic Optimizer (EO)

Greater detail will now be provided about some elements of an EO, according to some embodiments of the present disclosure.

Predicting a Load/Generation of an Upcoming Time Domain

In many electrical system control applications, a load of the electrical system (e.g., a building load) changes over time. Load can be measured as power or as energy change over some specified time period, and is often measured in units of kW. As noted above with reference to FIG. 6, an EO process 601a may predict 608 a local load and/or generation during an upcoming time domain.

FIG. 7 is a flow diagram of a method 700 of predicting load and/or generation of an electrical system during an upcoming time domain. A controller, according to some embodiments of the present disclosure, may have the ability to predict the changing load that may be realized during an upcoming time domain. These load and generation predictions may be used when the cost function is evaluated. To account for and reap a benefit from some types of value streams such as demand charge reduction, an accurate estimate of the upcoming load can be important. An accurate projection of a load during an upcoming time domain enables an EO to make better control decisions to capitalize on value streams such as demand charge reduction.

A method of predicting load, according to one embodiment of the present disclosure, may perform a load prediction considering historic periodic trends or shapes such as a daily trend or shape. The load prediction can execute every time an EO executes an EO process, or it can execute more or less frequently. The load prediction may be executed by performing a regression of a parameterized historic load shape against historic load data (typically less than or equal to 24 hours) in one embodiment. Regression algorithms such as least squares may be used. A compilation of historic trends may be recorded as a historic average (or typical) profile or an average load shape. The historic average profile or average load shape may be a daily (24-hour) historic average profile that represents a typical day. The compilation of historic observations and/or historic average profile may be received from another system, or may be gathered and compiled (or learned) as part of the method of predicting load, as will be explained below with reference to FIG. 8.

Referring to FIG. 7, historic observations of load are recorded 702. For example, the last h hours of historic observations of load may be continuously recorded and stored in memory, each measurement having a corresponding time of day at which time it was measured in an array pair historic_load_observed and historic_load_observed_time_of_day. The last h hours can be any amount of time, but in one embodiment, it is between 3 and 18 hours.

Assume for now a daily average load shape array or vector is in memory named avg_load_shape, each with a corresponding array avg_load_shape_time_of_day of the same length. The avg_load_shape and avg_load_shape_time_of_day represents a historic average profile and/or historic trends. The time domain of avg_load_shape_time_of_day is 24 hours, and the time interval of discretization of avg_load_shape_time_of_day could be any value. Between 5 and 120 minutes may be used, depending on the application, in some embodiments. As an example, if the interval of discretization is chosen to be 30 minutes, there will be 48 values comprising avg_load_shape and 48 values comprising avg_load_shape_time_of_day.

An interpolation is performed 704 to find the avg_load_shape values at each of the times in historic_load_observed_time_of_day. Call this new interpolated array avg_load_shape_interpolated. Consider mathematically avg_load_shape_interpolated with a scale and offset defined as: average_load_shape_interpolated_p=avg_load_shape_interpolated*scale+offset. In some embodiments, the interpolation is a linear interpolation. In other embodiments, the interpolation is a nonlinear interpolation.

A scale and offset are determined 706. For example, the method 700 may perform a least squares regression to determine 706 scale and offset that minimize the sum of the squares of the error between average_load_shape_interpolated_p and historic_load_observed. Call these resulting scale and offset values scale fit and offset fit. In some embodiments, the determining 706 of scale and offset can utilize weighted least squares techniques that favor more recent observations.

A corrected daily average load shape is generated 708 based on the scale and/or offset. For example a corrected load shape may be generated 708 for a full day as avg_load_shape_fit=avg_load_shape*scale_fit+offset_fit.

The future load values can then be estimated 710, such as by interpolating. A future load value at any time of day in the future time domain can now be estimated by interpolating 710 to that time of day from the pair of arrays avg_load_shape_fit and avg_load_shape_time_of_day.

FIG. 8 provides a graphical representation 800 of predicting load and/or generation of an electrical system during an upcoming time domain, according to one embodiment. The graphical representation may track the method 700 of FIG. 7. A historic average daily load shape 802 is generated or learned. Historic observations of load 804 are generated. A fitted current day historic load shape 806 is generated. A predicted load shape 808 can then be generated. The predicted load shape 808 is fairly accurate based on an actual load shape observed 810.

One advantage of a method of predicting load and/or generation of an electrical system during an upcoming time domain, such as previously described, compared to other methods, is that such method can adapt to changes in load scale and offset while still conforming to the general expected average daily load shape.

In other embodiments, the predicted load can be further modified after it has been calculated with the above method. For example, the predicted load may be modified to bring it closer to the daily average historic load as the prediction time becomes farther away from the time the prediction was made.

As mentioned previously, in certain embodiments, a method of predicting load may include compiling or otherwise gathering historic trends to determine a historic average profile or an average load shape on a typical day. One possible method for load learning (e.g., determining and updating avg_load_shape) is as follows:

1) Create arrays avg_load_shape and avg_load_shape_time_of_day which are defined as above. Initialize avg_load_shape to some reasonable value such as a constant load equal to the current load, or to an initial load shape provided in the configuration information.

2) Begin recording load observations and storing each along with its associated time of day in historic_load_observed_2 and historic_load_observed_time_of_day_2.

3) After at least one full day of load observations has been stored in historic_load_observed_2 and historic_load_observed_time_of_day_2, assign a last 24 hours of load data in historic_load_observed_2 to a temporary array in memory named avg_load_shape_last_24_hr which has the same number of elements as avg_load_shape, and whose associated time of day vector is also avg_load_shape_time_of_day. To perform this operation, a number of well-known approaches can be used including regression and interpolation, linear weighted averaging, and nearest neighbor.

4) Assign avg_load_shape_last_24_hr to avg_load_shape.

5) Wait for a new 24-hour period of data to be recorded in historic_load_observed_2 and historic_load_observed_time_of_day_2.

6) Again assign the last 24 hours of load data in historic_load_observed_2 to a temporary array in memory named avg_load_shape_last_24_hr which has the same number of elements as avg_load_shape, and whose associated time of day vector is also avg_load_shape_time_of_day. Again to perform this operation, a number of well-known approaches can be used including regression and interpolation, linear weighted averaging, and nearest neighbor.

7) Update each element k of avg_load_shape by performing a digital filter operation with it and avg_load_shape_last_24_hr. In one embodiment, this digital filter operation is performed as a first order infinite impulse response (IIR) filter with the inputs being elements k of avg_load_shape_last_24_hr and the original avg_load_shape, and the output being a modified element k of avg_load_shape. In one embodiment, the time constant of the first-order IIR filter is set to between 2 and 60 days. Other types of digital filters including low pass digital filters may be used.

8) Return to 5 above.

Some unique advantages of this embodiment of learning and predicting load and/or generation are obtained. For example, previous load information to construct an average daily load shape is not required. It learns the average daily load shape day-by-day as it observes the actual load. It requires very little memory: only enough for 24-hours of observed load, the load shape itself (24-hours), and supporting arrays and scalar values. Due to the filtering described above, it allows the load shape to change seasonally as seasonal changes occur and are observed. In other words, it is adaptive.

The method of FIG. 7 and illustration of FIG. 8 describe one embodiment of a method for predicting load. If a local generator is present in an electrical system, the same or a similar method can be applied for predicting generation. Instead of a “load shape,” a “generation shape” can be stored in memory. For generators where the generation is known at a particular time (such as a photovoltaic generator which would be expected to have nearly zero generation at nighttime), the prediction and generation shape can be constrained to specific values at specific times of the day. In this case, instead of using regression to determine both scale and offset, perhaps only scale may be needed.

Another aspect of this embodiment of a method to predict load and/or generation is the ability to incorporate external inputs to modify the prediction of load or generation. In one embodiment, the prediction is made as already described, then the prediction is modified with the use of external information such as a weather forecast or building occupancy forecast.

By having a pre-determined differential relationship for load (or generation) vs. input data, the prediction can be modified in one example as follows:

1) An external input is read which contains a forecasted variable x_{input, forecast}.

2) From configuration information, a value of the differential

${\left[\frac{d\left(\mathrm{load}\right)}{d\left(x_{\mathrm{input}}\right)}\right]}_{x_{\mathrm{nom}}}$

is available which is valid near some nominal x_input value of x_nom.

3) The predicted load can be modified to account for the difference between the input x_input and x_nom:

${\mathrm{load}}_{\mathrm{predicted},\mathrm{modified}}={\mathrm{load}}_{\mathrm{predicted}}+\left(x_{\mathrm{input},\mathrm{forecast}}-x_{\mathrm{nom}}\right)\times {\left[\frac{d\left(\mathrm{load}\right)}{d\left(x_{\mathrm{input}}\right)}\right]}_{x_{\mathrm{nom}}}$

The same approach can be used for modifying a generation prediction by replacing “load” with “generation” in the formula above.

Define the Control Parameter Set X

Defining the Control Parameter Set X involves defining or otherwise specifying times at which each control parameter is to be applied during a future time domain, and the control law(s) that are to be applied at each time in the future time domain.

An EO, according to certain embodiments of the present disclosure, is configured to define the control parameter set X. While there are many ways to define a control parameter set X, three possible approaches are:

1. a single set of parameters of a control law to be applied during the entire upcoming time domain;

2. a sequence of parameter sets that are each to be applied to a single control law at different contiguous sequential time intervals throughout the upcoming time domain; and

3. a sequence of parameters that specifies different control laws to be applied at different contiguous sequential time intervals throughout the future time domain.

An example of Approach 1 above of a single set of parameters of the control parameter set X (and example values) for a four-parameter control law is shown in Table 1.

TABLE 1
Pa-		Example
rameter	Description	Value
Pnom	Nominal ESS power (or discharge power if	−40	W
	negative) to be applied in the absence of other
	constraints or rules (such as those related to
	UB, UB0, or LB below).
UB	Upper bound on adjusted demand (e.g., an upper	100	kW
	setpoint). Not to be exceeded unless the ESS is
	incapable of discharging at sufficient power.
UB0	Upper bound on electrical system adjusted	80	kW
	demand (e.g., an upper setpoint) not to be
	actively exceeded (e.g., electrical system
	adjusted demand may exceed this value only
	with ESS power less than or equal to 0).
LB	Lower bound on adjusted net power (e.g., a lower	0	kW
	setpoint). Sometimes referred to as “minimum
	import,” or, if 0, “zero export.” Adjusted net
	power will be kept above this value unless the
	ESS is incapable of charging at sufficient power
	and generators cannot be throttled sufficiently.

Approaches 2 and 3 above utilize segmentation of the future time domain.

FIG. 9 is a graph 900 illustrating one example of segmenting an upcoming time domain into a plurality of time segments 902. A plot 904 of predicted unadjusted net power (kW) versus future time (e.g., of an upcoming time domain) is provided. A plot 906 of energy supply rate ($/kWh) versus future time is also provided. A plot 908 of a demand rate ($/kW) versus future time is also provided. A 25-hour future time domain is segmented into nine discrete sequential time segments 902 (e.g., i=1, 2, 3, 4, 5, 6, 7, 8, 9). Each segment 902 will be assigned a single set of one or more parameters from the control parameter set X to be applied during that time segment.

Segmentation of the future time domain can be done in many ways. In one embodiment, segmentation is performed such that:

i. the electric rates (both supply and demand) are constant within each time segment,

ii. the number of segments is minimized but large enough to provide a different segment for each region of the future time domain that is expected to have significantly different operating behavior or conditions, and

iii. the segment length does not exceed a prescribed maximum segment length.

In cases where rates are changing very frequently (every hour for example), some minimum time segment length can be specified (every four hours for example) to reduce the number of time segments while still maintaining acceptable computational fidelity. Likewise, a maximum segment length (for example six hours) may also be prescribed to increase computational fidelity.

Smaller numbers of segments are less burdensome on the EO processor computationally, while large numbers of segments provide higher fidelity in the final optimized solution. A desirable segment length of between 0.5 and 6 hours in some embodiments has been found to provide a good balance between these criteria.

The time segments of the upcoming time domain may be defined such that one or more of supply rate cost elements and delivery rate cost elements are constant during each time segment. The time segments of the upcoming time domain may be defined such that one or more of contracted maneuvers, demand response maneuvers, and ancillary service maneuvers are continuous during each time segment.

FIG. 9 also illustrates a representation 910 of an example of control parameter set X that includes multiple sets of parameters. The control parameter set X is for a three-parameter control law, which may be defined similar to the set illustrated above in Table 1, but without UB₀. The values for the parameters are not initialized, but the cells of the table X in FIG. 9 represent a parameter for which a value may be associated. In this example, the un-shaded values (X_x) are to be determined through an optimization process of the EO and the shaded values (X_logic) can be determined from logic. An objective of the EO is to fill in the values for each control parameter that minimizes the cost of operating the electrical system during the future time domain.

In some instances, it may make sense for an EO (or an EOESC) to operate with a single control parameter (e.g., a single set with a single element in X, such as P_nom) or with multiple control parameters (a single set of multiple elements in X, such as P_nom, UB, and LB) to be applied during the entire future time domain. In these two cases, the future time domain would be segmented into only one time segment 902. Correspondingly, the EO would only consider control parameters that are constant over the whole future time domain in this example.

Prepare the Cost Function

An EO, according to certain embodiments of the present disclosure, prepares or otherwise obtains a cost function. As already mentioned, the cost function f_c(X) is a function that considers particular control parameters (e.g., control parameter set X) and returns the scalar net cost of operating the electrical system with X during the future time domain.

FIG. 10 is a diagrammatic representation of a cost function evaluation module 1000 (or cost function evaluator) that implements a cost function f_c(X) 1002 that includes models 1004 for one or more electrical system components (e.g., loads, generators, ESSs). The cost function f_c(X) 1002 receives as inputs initialization information 1006 and control parameters 1008 (e.g., a control parameter set X). The cost function f_c(X) 1002 provides as an output a scalar value 1010 representing a cost of operating the electrical system during the future time domain.

The scalar value 1010 representing the cost, which is the output of the cost function f_c(X) 1002, can have a variety of different units in different examples. For example, it can have units of any currency. Alternately, the cost can have units of anything with an associated cost or value such as electrical energy or energy credits. The cost can also be an absolute cost, cost per future time domain, or a cost per unit time such as cost per day. In one embodiment, the units of cost are U.S. dollars per day.

Prior to using the cost function, several elements of it can be initialized. The initialization information that is provided in one embodiment is:

Date and time. For determining the applicable electric utility rates.

Future time domain extent. For defining the time extent of the cost calculation.

Electric utility tariff definition. This is a set of parameters that defines how the electrical utility calculates charges.

Electrical system configuration. These configuration elements specify the sizes and configuration of the components of the electrical system. An example for a battery energy storage system is the energy capacity of the energy storage device.

Electrical system component model parameters. These model parameters work in conjunction with analytic or numerical models to describe the physical and electrical behavior and relationships governing the operation of electrical components in the electrical system. For battery energy storage systems, a “battery model” is a component, and these parameters specify the properties of the battery such as its Ohmic efficiency, Coulombic efficiency, and degradation rate as a function of its usage.

States of the electrical system. This is information that specifies the state of components in the electrical system that are important to the economic optimization. For battery energy storage systems, one example state is the SoC of the energy storage device.

Operational constraints. This information specifies any additional operational constraints on the electrical system such as minimum import power.

Control law(s). The control law(s) associated with the definition of X.

Definition of control parameter set X. The definition of the control parameter set X may indicate the times at which each control parameter is to be applied during a future time domain. The definition of the control parameter set X may indicate which control law(s) are to be applied at each time in the future time domain.

Net load (or power) prediction. Predicted unadjusted net load (or predicted unadjusted net power) during the future time domain.

Pre-calculated values. While segments are defined, many values may be calculated that the cost function can use to increase execution efficiency (help it “evaluate” faster). Pre-calculation of these values may be a desirable aspect of preparing the cost function f_c(X) 1002 to enable the cost function to be evaluated more efficiently (e.g., faster, with fewer resources).

Preparing the cost function f_c(X) 1002 can increase execution efficiency of the EO because values that would otherwise be re-calculated each time the cost function is evaluated (possible thousands of times per EO iteration) are pre-calculated a single time.

FIG. 11 is a flow diagram of a process 1100 of preparing a cost function f_c(X), according to one embodiment of the present disclosure. Cost function initialization information may be received 1102. A simulation of electrical system operation is initialized 1104 with the received 1102 cost function initialization information. Cost function values may be pre-calculated 1106. The pre-calculated values may be stored 1100 for later use during evaluation of the cost function.

In certain embodiments, defining a control parameter set X and preparing a cost function f_c(X) may be accomplished in parallel.

Evaluation of the Cost Function

During execution of an EO, according to some embodiments of the present disclosure, the cost function is evaluated. During evaluation of the cost function, operation of the electrical system with the control parameter set X is simulated. The simulation may be an aspect of evaluating the cost function. Stated otherwise, one part of evaluating the cost function for a given control parameter set X may be simulating operation of the electrical system with that given control parameter set X. In the simulation, the previously predicted load and generation are applied. The simulation takes place on the future time domain. As time advances through the future time domain in the simulation, costs and benefits (as negative costs) can be accumulated. What is finally returned by the simulation is a representation of how the electrical system state may evolve during the future time domain with control X, and what costs may be incurred during that time.

In some embodiments, the cost function, when evaluated, returns the cost of operating the electrical system with some specific control parameter set X. As can be appreciated, the cost of operating an electrical system may be very different, depending on X. So evaluation of the cost function includes a simulated operation of the electrical system with X first. The result of the simulation can be used to estimate the cost associated with that scenario (e.g., the control parameter set X).

As noted previously, some of the costs considered by the cost function in one embodiment are:

1. Electricity supply charges (both flat rates and ToU rates)

2. Electricity demand charges

3. Battery degradation cost

4. Reduction of energy stored in the energy storage system

5. Incentive maneuver benefits (as a negative number)

Electricity supply and demand charges have already been described. For monthly demand charges, the charge may be calculated as an equivalent daily charge by dividing the charge by approximately 30 days, or by dividing by some other number of days, depending on how many days are remaining in the billing cycle. Battery degradation cost is described in a later section. Reduction in energy stored in an ESS accounts for the difference in value of the storage energy at the beginning of the future time domain compared to the end. Incentive maneuver benefits such as demand response can be calculated as the benefit on a per day basis, but as a negative number.

During the cost function's electrical system simulation, several variables can be tracked and stored in memory. These include control variables, electrical power consumed by or supplied from various electrical systems, and the states of charge of any energy storage systems. Other variables can also be tracked and stored to memory. Any of the variables stored to memory can be output by the cost function.

FIG. 12 is a flow diagram of a method 1200 of evaluating a cost function that is received from an external source or otherwise unprepared, according to one embodiment of the present disclosure. Cost function initialization information may be received 1202. A simulation of electrical system operation is initialized 1204 with the received 1202 cost function initialization information. The simulation is performed 1206 of the electrical system operation with X over the future time domain. A calculation 1208 of the cost components of operating the electrical system with X is performed. The cost components are summed 1210 to yield a net cost of operating the electrical system with X. The net cost of operating the electrical system with X is returned 1212 or otherwise output.

FIG. 13 is a flow diagram of a method 1300 of evaluating a prepared cost function, according to one embodiment of the present disclosure. The cost function may be prepared according to the method of FIG. 11. Pre-calculated values are received 1302 as inputs to the method 1300. The values may be pre-calculated during an operation to prepare the cost function, such as the process of FIG. 11. A simulation is performed 1304 of the electrical system operating with X over the future time domain. A calculation 1306 of the cost components of operating the electrical system with X is performed. The cost components are summed 1308 to yield a net cost of operating the electrical system with X. The net cost of operating the electrical system with X is returned 1310 or otherwise output.

In some embodiments, rather than returning the net cost of operating the electrical system with X during the future time domain, what is returned is the net cost of operating the electrical system with X as a cost per unit time (such as an operating cost in dollars per day). Returning a per day cost can provide better normalization between the different cost elements that comprise the cost function. The cost per day for example can be determined by multiplying the cost of operating during the future time domain by 24 hours and dividing by the length (in hours) of the future time domain.

Execute Continuous Minimization of the Cost Function

With a prediction of load and generation made, the control parameter set X defined, and the cost function obtained and initialized and/or prepared, minimization of cost can be performed.

Minimization of the cost function may be performed by an optimization process or module that is based on an optimization algorithm. Minimization (or optimization) may include evaluating the cost function iteratively with different sets of values for the control parameter set X (e.g., trying different permutations from an initial value) until a minimum cost (e.g., a minimum value of the cost function) is determined. In other words, the algorithm may iteratively update or otherwise change values for the control parameter set X until the cost function value (e.g. result) converges at a minimum (e.g., within a prescribed tolerance). The iterative updating or changing of the values may include perturbing or varying one or more values based on prior one or more values.

Termination criteria (e.g., a prescribed tolerance, a delta from a prior value, a prescribed number of iterations) may aid in determining when convergence at a minimum is achieved and stopping the iterations in a finite and reasonable amount of time. The number of iterations that may be performed to determine a minimum could vary from one optimization cycle to a next optimization cycle. The set of values of the control parameter set X that results in the cost function returning the lowest value may be determined to be the optimal control parameter set X_opt.

In one embodiment, a numerical or computational generalized constrained nonlinear continuous optimization (or minimization) algorithm is called (e.g., executed) by a computing device.

FIG. 14 is a diagrammatic representation of an optimization subsystem 1400 that utilizes or otherwise implements an optimization algorithm 1401 to determine an optimal control parameter set X₀ 1410, which minimizes the cost function f_c(X). In the embodiment of FIG. 14, the optimization algorithm 1401 utilized by the optimization subsystem 1400 may be a generalized constrained multivariable continuous optimization (or minimization) algorithm. A reference 1402 is provided for the cost function f_c(X).

The optimization algorithm can be implemented in software, hardware, firmware, or any combination of these. The optimization algorithm may be implemented based on any approach from descriptions in literature, pre-written code, or developed from first principles. The optimization algorithm implementation can also be tailored to the specific problem of electrical system economic optimization, as appropriate in some embodiments.

Some algorithms for generalized constrained multivariable continuous optimization include:

• • Trust-region reflective

Active set

SQP

Interior Point

Covariance Matrix Adaption Evolution Strategy (CMAES)

Bound Optimization by Quadratic Approximation (BOBYQA)

Constrained Optimization by Linear Approximation (COBYLA)

The optimization algorithm may also be a hybrid of more than one optimization algorithm. For example, the optimization algorithm may use CMAES to find a rough solution, then Interior Point to converge tightly to a minimum cost. Such hybrid methods may produce robust convergence to an optimum solution in less time than single-algorithm methods.

Regardless of the algorithm chosen, it may be useful to make an initial guess of the control parameter set X 1404. This initial guess enables an iterative algorithm such as those listed above to more quickly find a minimum. In one embodiment, the initial guess is derived from the previous EO execution results.

Any constraints 1406 on X can also be defined or otherwise provided. Example constraints include any minimum or maximum control parameters for the electrical system.

An Example EO Result

FIG. 15 is a graph 1500 illustrating an example result from an EO for a small battery energy storage system, using the same example upcoming time domain, segmentation of the upcoming time domain into a plurality of segments 902, predicted unadjusted net power plot 904, supply rate plot 906, daily demand rate plot 908, and representation 910 of the control parameter set X as in FIG. 9.

The graph 1500 also includes plots for UB (kW) 1522, LB (kW) 1524, Pnom (kW) 1526, ESS power (kW) 1528, adjusted net power (kW) 1530, and battery SoC 1532.

In FIG. 15, as in FIG. 9, the future time domain is split into nine segments 902, and nine optimal sets of parameters 1502 were determined (e.g., a control parameter set X_opt 910 that includes values for nine optimal sets of parameters, one optimal set of parameters for each segment 902). Daily demand charges are applicable and a net export of energy (e.g., to the grid) is not allowed in the illustrated example. An objective of the controller is to find an optimal sequence of electrical system control parameters.

The control parameter set X in this case is defined to include three parameters: Pnom, UB, and LB as described above. In this example, during execution of the optimization algorithm, the optimal values in the unshaded boxes (X_x) of the representation 910 of X are determined, P_nom 1502 which is the battery inverter power (where charge values are positive and generation/discharge values are negative) during each time segment 902, and UB 1502 which is the upper limit on demand during each time segment 902). The date and time to apply each specific control parameter is part of the definition of X. The shaded values (X_logic, which includes LB and some UB values) in the representation 910 of X are determined by logic. For example, when no demand charge is applicable, the UB can be set to infinity. And since net export of power is not permitted in this example, LB can be set to zero. There is no need to determine optimal values for these shaded parameters when executing the optimization because their values are dictated by constraints and logic.

Applying the optimal values of X, the expected cost per day of operating the electrical system in the example of FIG. 15 is $209.42 per day. This total cost is the sum of the ToU supply cost ($248.52), the daily demand cost ($61.52), the cost of battery energy change ($−115.93), and the cost of battery degradation ($15.32).

As can be appreciated, in other embodiments, the EO may determine a set of control values for a set of control variables, instead of a control parameter set X. The EO may determine the set of control values to effectuate a change to the electrical system toward meeting a controller objective for economical optimization of the electrical system. The EO may then output the control values or the set of control variables for delivery directly to the electrical system. In such embodiment, the EO may be a primary component of the controller and the controller may not include a dynamic manager (e.g., a high speed controller).

Dynamic Manager or High Speed Controller (HSC)

Greater detail will now be provided about some elements of a dynamic manager, or an HSC, according to some embodiments of the present disclosure. Because the control parameter set X is passed to the high speed controller, the definition of the control parameter set X may be tightly linked to the HSC's control law. The interaction between an example HSC and control parameter set X is described below.

Storing a Control Plan

As already mentioned, the control parameter set X can contain multiple sets of parameters and dates and times that those sets of parameters are meant to be applied by the HSC. One embodiment of the present disclosure takes this approach. Multiple sets of parameters are included in X, each set of parameters with a date and time the set is intended to be applied to the electrical system being controlled. Furthermore, each controllable system within the electrical system can have a separate set of controls and date and time on which the set of controls is intended to be applied. The HSC commits the full control parameter set X to memory and applies each set of parameters therein to generate control variables to deliver to, and potentially effectuate a change to, the electrical system at the specified times. Stated differently, the HSC stores and schedules a sequence of optimal sets of parameters, each to be applied at an appropriate time. In other words, the HSC stores a control plan. This first task of storing and scheduling a sequence of optimal control parameter sets (e.g., a control plan) by the high speed controller provides distinct advantages over other control architectures.

For example, storing of a control plan by the HSC reduces the frequency that the computationally intensive (EO) portion of the controller is executed. This is because even if the first sequential time interval expires before the EO executes again, the HSC will switch to the next sequential control set at the appropriate time. In other words, the EO does not have to execute again before the first sequential time interval expires since multiple optimal control sets can be queued up in sequence.

As another example, storing of a control plan by the HSC enables operation (e.g., control of the electrical system) for significant periods of time without additional EO output. This may be important for example if the EO is executing in a remote processor such as a cloud computing environment and the HSC is running on a processor local to a building. If communication is lost for a period of time less than the future time domain, the HSC can continue to use the already-calculated optimal control parameter sets at the appropriate times. Although operation in such a manner during outage may not be optimal (because fresh EO executions are desirable as they take into account the latest data), this approach may be favored compared with use of a single invariant control set or shutting down.

Application of Presently Applicable Control Parameters

A second task of the HSC, according to one embodiment, is to control some or all of the electrical system components within the electrical system based on the presently applicable control parameter set. In other words, the HSC applies each set of parameters of a control parameter set X in conjunction with a control law to generate control variables to deliver to, and potentially effectuate a change to, the electrical system at appropriate times.

For an electrical system with a controllable battery ESS, this second task of the HSC may utilize four parameters for each time segment. Each of the four parameters may be defined as in Table 1 above. In one embodiment, these parameters are used by the HSC to control the battery inverter to charge or discharge the energy storage device. For a battery ESS, the typical rate at which the process variables are read and used by the HSC and new control variables are generated may be from 10 times per second to once per 15 minutes. The control variables (or the set of values for the set of control variables) for a given corresponding time segment may be provided to the electrical system at (e.g., before or during) the given corresponding time segment of the upcoming time domain.

As can be appreciated, in other embodiments, an entire control plan (e.g., a control parameter set X comprising a set of sets) may be processed by the HSC to determine a plurality of sets of control variables, each set of control variables for a corresponding time segment. The plurality of sets of control variables may be provided at once (e.g., before the upcoming time domain or no later than during a first time segment of the upcoming time domain). Or, each set of the plurality of sets may be provided individually to the electrical system at (e.g., before or during) the given corresponding time segment.

Another aspect of the HSC, according to one embodiment, is that the HSC can also be used to curtail a generator (such as a photovoltaic generator) if necessary to maintain the lower bound on electrical system power consumption specified by LB.

FIG. 16 is a method 1600 of a dynamic manager, or HSC, according to one embodiment of the present disclosure, to use a set of optimal control parameters X_opt in conjunction with a control law to determine values of a set of control variables to command the electrical system. A set of optimal control parameters (X_opt), a measurement of unadjusted building load (Load), and PV maximum power (PV_max_power) are received or otherwise available as inputs to the method 1600. The dynamic manager processes X_opt to determine a set of control values to effectuate a change to the electrical system toward meeting an objective for economical optimization of the electrical system during an upcoming time domain. The output control variables are the ESS power command (ESS command) and the photovoltaic limit (PV_limit), which are output to the building electrical system to command an ESS and a photovoltaic subsystem.

The presently applicable P_nom, UB, UB₀, and LB are extracted 1602 from) X_opt. The ESS power command, ESS command, is set 1604 equal to P_nom. The photovoltaic limit, PV_limit, is set 1606 equal to PV maximum power, PV_max_power. The building power, P_building, is calculated 1608 as a summation of the unadjusted building load, the photovoltaic limit, and the ESS power command (P_building=Load+PV_limit+ESS_command).

A determination 1610 is made whether the building power is greater than UB₀ (P_building>UB₀) and whether the ESS command is greater than zero (ESS_command>0). If yes, then variables are set 1612 as:

• • ESS_command=UB₀−Load−PV_limit
  • P_building=Load+PV_limit+ESS_command.

A determination 1614 is made whether building power is greater than UB (P_building>UB). If yes, then variables are set 1616 as:

• • ESS_command=UB−Load−PV_limit
  • P_building=Load+PV_limit+ESS_command.

A determination 1618 is made whether building power is less than LB (P_building<LB). If yes, then variables are set 1620 as:

• • ESS_command=LB−Load−PV_limit
  • P_building=Load+PV_limit+ESS_command,and another determination 1622 is made whether building power remains less than LB (P_building<LB). If yes, then the photovoltaic limit PV_limit is set 1624 as:
  • PV_limit+(LB−P_building).Then the control variables ESS_command and PV_limit are output 1630 to the electrical system.

An Example HSC Result

FIG. 17 is a graph 1700 showing plots for an example of application of a particular four-parameter control set during a time segment. The graph 1700 shows a value for each of UB, UB₀, LB, and P_nom, which are defined above in Table 1. A vertical axis is the power consumption (or rate of energy consumed), with negative values being generative. A first plot 1702 provides unadjusted values of power consumption (kW) for the electrical system load plus renewable (photovoltaic) generation and excluding battery operation, over the time segment. In other words, the first plot 1702 shows operation of the electrical system without benefit of a controllable ESS (battery) that is controlled by a controller, according to the present disclosure. A second plot 1704 provides values of power consumption (kW) for battery operation over the time segment. The second plot 1704 may reflect operation of an ESS as commanded by the controller. In other words, the second plot 1704 is the control variable for the ESS. The battery operation value may be the value of the control variable to be provided by the HSC to command operation of the ESS. A third plot 1706 provides values of power consumption (kW) for the electrical system load plus renewable (photovoltaic) generation and including battery operation, over the time segment. The third plot 1706 illustrates how the controlled ESS (or battery) affects the power consumption of the electrical system from the grid. Specifically, the battery in this example is controlled (e.g., by the battery operation value) to discharge to reduce the load of the electrical system on the grid and limit peak demand to the UB value when desired. Furthermore, this example shows LB being enforced by commanding the ESS to charge by an amount that limits the adjusted net power to be no less than LB when necessary. Furthermore, this example shows that the nominal ESS power (Pnom) is commanded to the extent possible while still meeting the requirements of UB, UB₀, and LB.

In other embodiments, the control parameter set X may have fewer or more parameters than the four described for the example embodiment above. For example, the control parameter set X may be comprised of only three parameters: Pnom, UB, and LB. Alternately, the control parameter set X may be comprised of only two parameters: Pnom and UB. Alternately, the control parameter set X may include only of UB or only of Pnom. Or, it may include any other combination of four or fewer parameters from the above list.

Battery Models

In a battery ESS, battery cost can be a significant fraction of the overall system cost and in many instances can be greater than 60% of the cost of the system. The cost of the battery per year is roughly proportional to the initial cost of the battery and inversely proportional to the lifetime of the battery. Also, any estimated costs of system downtime during replacement of a spent battery may be taken into account. A battery's condition, lifetime, and/or state of health (SoH) may be modeled and/or determined by its degradation rate (or rate of reduction of capacity and its capacity at end of life). A battery's degradation rate can be dependent upon many factors, including time, SoC, discharge or charge rate, energy throughput, and temperature of the battery. The degradation rate may consider capacity of the battery (or loss thereof). Other ways that a battery's condition, lifetime, and/or SoH may be evaluated may be based on a maximum discharge current of the battery or the series resistance of the battery.

Described herein are battery models based on battery degradation as a function of battery capacity as compared to initial capacity or capacity at the beginning of life of the battery. Stated otherwise, the disclosed battery models consider battery condition or state of health according to the battery capacity lost from the capacity at the beginning of life of the battery. As can be appreciated, other battery models may model battery condition according to another way, such as maximum discharge current of the battery, the series resistance of the battery, or the like.

In one embodiment, the battery degradation and its associated cost is included as a cost element in the cost function. By including battery degradation cost in the cost function, as the EO executes to find the minimum cost, the EO can effectively consider the contribution of battery degradation cost for each possible control parameter set X. In other words, the EO can take into account a battery degradation cost when determining (e.g., from a continuum of infinite control possibilities) an optimal control parameter set X_opt. To accomplish this, a parameterized model of battery performance, especially its degradation rate, can be developed and used in the cost function during the simulation of potential control solutions (e.g., sets of control parameters X). The battery parameters (or constants) for any battery type can be determined that provide a closest fit (or sufficiently close fit within a prescribed tolerance) between the model and the actual battery performance or degradation. Once the parameters are determined, the cost function can be initialized with configuration information containing those parameters so that it is able to use the model in its control simulation in some implementations.

In one embodiment, battery degradation is written in the form of a time or SoC derivative that can be integrated numerically as part of the cost function control simulation to yield battery degradation during the future time domain. In one embodiment, this degradation derivative can be comprised of two components: a wear component (or throughput component) and an aging component. The components can be numerically integrated vs. time using an estimate of the battery SoC at each time step in one embodiment.

Examples of components of a battery degradation model, according to one embodiment, that meet these criteria are illustrated by FIGS. 18 and 19. FIG. 18 shows the relationship of wear rate versus SoC for a lead acid battery, based on a battery degradation model that includes wear. FIG. 19 shows a relationship between SoC and aging rate (or aging factor) for a lead acid battery, based on a battery model that includes aging. Battery models that combine both wear and aging can then be fit to match a specific battery's cycle life similar to FIG. 20.

Formulating the battery degradation model as a time or SoC derivative is also beneficial because the model can be used to calculate battery degradation for any arbitrary battery operation profile. (A battery operation profile is a battery's SoC vs. time.) Calculating battery degradation is useful in simulation of the performance (both physical and economic) of a battery ESS. After a simulation produces a battery operation profile, the derivative-based degradation model can be integrated numerically over that profile to produce an accurate estimate of the battery degradation in one embodiment.

Other common degradation or “lifetime” models only provide degradation based on the number of cycles in the profile. With those models, the definition of a “cycle” is problematic, inconsistent, and difficult to use computationally for an arbitrary battery operation profile.

Note that other embodiments of wear and aging models, and their combinations, can be used in addition to those shown and described herein. For those other models, if the models are expressible as derivatives (and/or partial derivatives) with respect to time or with respect to battery SoC, they will also be afforded the advantages already mentioned and will be readily usable as a cost element in a cost function of an EO.

The graphs of FIGS. 18 and 19 illustrate different components of a battery degradation model, according to one embodiment of the present disclosure. The battery degradation model of the illustrated example may model a lead-acid battery and may include a wear component and an aging component. The wear component may include a function of the rate of charge or discharge of the battery, the SoC of the battery, and the temperature of the battery. The aging component may include a function of the battery state including the SoC of the battery and a temperature of the battery.

Consider one measure of a battery condition, a measure of battery degradation C_f, which is the maximum battery capacity divided by the maximum battery capacity at beginning of life (BoL). At BoL, C_f=1.0. As the battery degrades, C_f decreases. At end of life (EoL), C_f=C_f,EoL. C_f,EoL is typically between 0.5 and 0.9 and often around 0.8. In some embodiments of present disclosure, changes in C_f can be used to estimate the cost of operating the battery (e.g., battery degradation cost) during that future time domain as:

BatteryCost_{t1. . . t2}=BatteryCost_total*(C_f,t1−C_f,t2)/(1−C_f,EOL)

where BatteryCost_total is a total battery cost (for example an initial or net present cost), and C_f,t1−C_f,t2 is the change in C_f between time t₁ and t₂. In other words, C_f,t1−C_f,t2 is a measure of degradation of the battery between times t₁ and t₂. The battery lifetime is that point at which C_f reaches C_f,EoL, which is the manufacturer's failure limit (usually 0.8 or 80% of initial maximum capacity).

Determining the battery degradation cost for a time period may include multiplying the change in C_f by a cost factor. The cost factor may be the total cost of the battery divided by the total decrease in C_f at end of life, BatteryCost_total/(1−C_f,EoL). In other words, the cost factor may be determined as a lifetime cost of the battery divided by the amount of battery degradation resulting in end of life of the battery.

To determine the change in C_f between times t₁ and t₂, two components (wear and aging) can be considered as two partial derivatives of C_f with respect to SoC and t respectively. A battery's capacity change during some future time domain from t=t₁ to t=t₂ can be determined using calculus as:

$C_{f,t1}-C_{f,t2}=-{\int }_{t=t_1}^{t=t_2}\left[\frac{\partial C_f}{\partial \mathrm{SoC}}\frac{\mathrm{dSoC}}{\mathrm{dt}}+\frac{\partial C_f}{\partial t}\right]\mathrm{dt},$

where the rate of change of C_f due to wear (throughput), the “wear rate,” is denoted

$\frac{\partial C_f}{\partial \mathrm{SoC}},$

the rate or change or C_f due to aging, the “aging rate,” is denoted

$\frac{\partial C_f}{\partial t},$

and the rate of change of state of charge (denoted “SoC” and ranging from 0 to 1) versus time is denoted

$\frac{\mathrm{dSoC}}{\mathrm{dt}}.$

This equation may be integrated numerically using many commonly known methods including the trapezoidal rule. The derivative

$\frac{\mathrm{dSoC}}{\mathrm{dt}}$

can be obtained from the simulation performed by the cost function, and can be calculated as a discretized value

$\frac{\Delta \mathrm{SoC}}{\Delta t}.$

Regarding the rate of change of C_f due to wear, an exponential model can be used that depends upon whether the battery is discharging,

$\frac{\mathrm{dSoC}}{\mathrm{dt}}<0,$

or charging,

$\frac{\mathrm{dSoC}}{\mathrm{dt}}\ge 0.$

For example, the wear rate can be expressed as,

$\frac{\partial C_f}{\partial \mathrm{SoC}}=\{\begin{array}{cc}-A*e^{-\mathrm{BSoC}}-E& \frac{\mathrm{dSoC}}{\mathrm{dt}}<0\\ -C*e^{D\left(\mathrm{SoC}-1\right)}-F& \frac{\mathrm{dSoC}}{\mathrm{dt}}\ge 0\end{array},$

where A and B specify the rate of increase in degradation during discharging, E represents the baseline degradation during discharging, C and D specify the rate of increase in degradation during charging, and F represents the baseline degradation during charging.

FIG. 18 is a graph 1800 of the exponential battery wear model for a specific battery degradation model, according to one embodiment of the present disclosure. FIG. 18 provides plots 1802, 1804 of the negative of the “wear rate,”

$\frac{\partial C_{f,\mathrm{loss}}}{\partial \mathrm{SoC}}=-\frac{\partial C_f}{\partial \mathrm{SoC}},$

versus SoC. Put another way, plots 1802 and 1804 represent the rate of battery capacity loss versus a change in the state of charge of the battery. A vertical axis of the graph 1800 shows the negative of the wear rate in dimensionless units. A horizontal axis of the graph 1800 shows a battery SoC, where 1.0=100% (fully charged state). One plot 1802 shows the negative of the wear rate during charging, and a second plot 1804 shows the negative of the wear rate during discharging. The plots 1802, 1804 are computed using the above corresponding equations with parameters (A through F) selected specifically to match a type of lead-acid battery. The parameters in this case are A=4e-3, B=5.63, C=9e-4, D=27.4, E=3e-5, and F=3e-5.

Regarding the rate of change of C_f due to aging, an exponential model for the rate of change in fractional capacity versus time can be used. For example, the aging rate can be expressed as,

$\frac{\partial C_f}{\partial t}=-G*\left[\mathrm{Aging}\mathrm{Factor}\right]=-G*\left[\left(1-\left(1-H\right)e^{-\frac{\mathrm{SoC}}{I}}\right)*\left(1-\left(1-J\right)e^{\frac{\left(\mathrm{SoC}-1\right)}{K}}\right)\right],$

where G represents the nominal aging rate in units of fractional capacity lost (e.g., 2% per year) versus time. The Aging Factor, when multiplied by the nominal aging rate G and by −1, gives the aging rate. A 1.0 aging factor indicates the aging rate is at −G. Also, H and I define the rate of increase in aging rate as SoC approaches 0, and J and K define the rate of increase in aging rate as the SoC approaches 1.

FIG. 19 is a graph 1900 providing a plot 1902 showing a relationship between Aging Factor and SoC for a specific battery degradation model, according to one embodiment of the present disclosure. The vertical axis of the graph 1900 shows an Aging Factor. The horizontal axis of the graph 1900 shows the SoC of the battery being modeled. The plot 1902 reflects the values for the Aging Factor

$\left(1-\left(1-H\right)e^{-\frac{\mathrm{SoC}}{I}}\right)*\left(1-\left(1-J\right)e^{\frac{\left(\mathrm{SoC}-1\right)}{K}}\right),$

where in this example the aging parameters are H=15.0, 1=0.2, J=2.5, and K=0.02.

As noted above, a cost function, according to one embodiment, may sum multiple cost elements for operation of an electrical system, including the cost element BatteryCost_{t1 . . . t2}, or BatteryCost_{t1 . . . t2}*24 hr/(t₂−t₁) in embodiments where the cost function determines a cost per day and t₁ and t₂ have units of hours.

The model explained above can also be described in terms of capacity lost C_f,lost. The model includes both capacity lost due to battery wear (throughput), C_f,lost,wear, and capacity lost due to battery aging, C_r,lost,aging. In other words, capacity lost can be expressed as:

C _f,lost =C _f,lost,wear +C _f,lost,aging.

The battery end of life is that point at which C_f,lost reaches 1 minus the manufacturer's failure limit (usually 0.2 or 20% of initial capacity, which again sets a failure point at 80% of original capacity).

Capacity lost due to battery wear (throughput), C_f,lost,wear, can be modeled with an exponential model for the rate of change in C_f,lost versus SoC. For example, a discharge formulation of the loss of fractional capacity per unit change in fractional SoC applicable during a decreasing SoC can be expressed as,

$\frac{\partial C_{f,\mathrm{lost}}}{\partial {\mathrm{SoC}}_{\mathrm{discharge}}}=A*e^{-\mathrm{BSoC}}+E$

and a charge formulation of the loss of fractional capacity per unit change in fractional SoC applicable during an increasing SoC can be expressed as,

$\frac{\partial C_{f,\mathrm{lost}}}{\partial {\mathrm{SoC}}_{\mathrm{charge}}}=C*e^{D\left(\mathrm{SoC}-1\right)}+F.$

As before, SoC is the state of charge of the battery, A and B specify the rate of increase in degradation during discharging, E represents the baseline degradation during discharging, C and D specify the rate of increase in degradation during charging, and F represents the baseline degradation during charging. As noted previously, the graph 1800 of FIG. 18 provides plots 1802, 1804 of

$\frac{\partial C_{f,\mathrm{lost}}}{\partial \mathrm{SoC}}\mathrm{vs}.\mathrm{SoC}$

for this specific battery degradation model.

For a given battery SoC profile, the total capacity loss due to wear, C_f,lost,wear, between times t₁ and t₂ can be calculated with:

$\frac{\partial C_{f,\mathrm{lost}}}{\partial \mathrm{SoC}}\frac{d\mathrm{SoC}}{dt}=\{\begin{array}{cc}-\frac{\partial C_{f,\mathrm{lost}}}{\partial {\mathrm{SoC}}_{\mathrm{discharge}}}\frac{d\mathrm{SoC}}{\mathrm{dt}}& \frac{d\mathrm{SoC}}{\mathrm{dt}}<0\\ \frac{\partial C_{f,\mathrm{lost}}}{\partial {\mathrm{SoC}}_{\mathrm{charge}}}\frac{d\mathrm{SoC}}{\mathrm{dt}}& \frac{d\mathrm{SoC}}{\mathrm{dt}}\ge 0\end{array}C_{f,\mathrm{lost},\mathrm{wear}}={\int }_{t=t_1}^{t=t_2}\frac{\partial C_{f,\mathrm{lost}}}{\partial \mathrm{SoC}}\frac{d\mathrm{SoC}}{\mathrm{dt}}\mathrm{dt}$

Capacity lost due to battery aging, C_f,lost,aging, can be represented differentially by defining the rate of battery capacity loss versus time, or more specifically in one embodiment, the rate of change in fractional capacity lost versus time. In one example, this differential representation of battery capacity loss can take an exponential form as,

$\frac{\partial C_{f,\mathrm{lost}}}{\partial t}=G*\left[\mathrm{Aging}\mathrm{Factor}\right]=G*\left[\left(1-\left(1-H\right)e^{-\frac{\mathrm{SoC}}{I}}\right)*\left(1-\left(1-J\right)e^{\frac{\left(\mathrm{SoC}-1\right)}{K}}\right)\right],$

where, as before, G represents the nominal aging rate in units of fractional capacity lost (e.g., 2% per year) versus time. The Aging Factor, when multiplied by the nominal aging rate G and by −1, gives the aging rate. A 1.0 aging factor indicates the aging rate is at −G. Also, H and I define the rate of increase in aging rate as SoC approaches 0, and J and K define the rate of increase in aging rate as the SoC approaches 1.

For a given battery SoC profile, the total battery capacity loss due to aging, C_f,lost,aging, between times t₁ and t₂ can be calculated with:

$C_{f,\mathrm{lost},\mathrm{aging}}={\int }_{t=t_1}^{t=t_2}\frac{\partial C_{f,\mathrm{lost}}}{\partial t}\mathrm{dt}$

As noted previously, the graph 1900 of FIG. 19 provides a plot 1902 showing a relationship between SoC and aging rate (or an aging factor) for this specific battery degradation model.

Combining the two components (wear and aging) from the previous examples, a battery's capacity lost during some future time domain from t=t₁ to t₂ can be determined as:

$C_{f,\mathrm{lost}}=C_{f,\mathrm{lost},\mathrm{wear}}+C_{f,\mathrm{lost},\mathrm{aging}}={\int }_{t=t_1}^{t=t_2}\frac{\partial C_{f,\mathrm{lost}}}{\partial \mathrm{SoC}}\frac{d\mathrm{SoC}}{dt}\mathrm{dt}+{\int }_{t=t_1}^{t=t_2}\frac{\partial C_{f,\mathrm{lost}}}{\partial t}\mathrm{dt}={\int }_{t=t_1}^{t=t_2}\left[\frac{\partial C_{f,\mathrm{lost}}}{\partial \mathrm{SoC}}\frac{d\mathrm{SoC}}{dt}+\frac{\partial C_{f,\mathrm{lost}}}{\partial t}\right]\mathrm{dt}$

Once the battery capacity lost is determined over the future time domain by numerically integrating the above equation, the cost of operating the battery (e.g., a battery degradation cost) during that future time domain can be calculated as:

${\mathrm{BatteryCost}}_{t_1\dots t_2}={\mathrm{BatteryCost}}_{\mathrm{total}}*\frac{C_{f,\mathrm{lost}}}{}\left(1-C_{f,\mathrm{EOL}}\right),$

where BatteryCost_total is a total battery cost (for example an initial or net present cost) and C_f,EoL is the fractional battery capacity remaining at end of life. Stated otherwise, determining the battery degradation cost for a time period comprises multiplying the total battery degradation for the time period by a cost factor. The cost factor may be the total cost of the battery divided by the total fractional capacity loss during the battery's lifetime, BatteryCost_total/(1−C_f,EoL). In other words, the cost factor may be determined as a lifetime cost of the battery divided by the amount of battery degradation resulting in end of life of the battery.

Using the combined wear and aging model described above, coefficients can be found that result in a fit to a battery manufacturer's cycle life.

FIG. 20 is a pair of graphs 2010, 2020 that illustrate a battery's lifetime. Graph 2010 shows the manufacturer's data 2012 for battery cycle life (number of cycles) versus depth of discharge under continuous cycling conditions. Graph 2020 shows the manufacturer's data 2022 for battery lifetime (in years) versus depth of discharge assuming one cycle per day. The coefficients determined to match the data of this manufacturer's cycle life for the example battery may be:

• • A=4e-3, B=5.63, C=9e-4, D=27.4, E=3e-5, F=3e-5, G=1.95e-6 hr⁻¹, H=15.0, 1=0.2, J=2.5, and K=0.02.Graph 2010 includes a plot 2014 of the model with the above coefficients aligning with the manufacturer's data 2012 and providing a projected battery cycle life versus depth of discharge. Graph 2020 includes a plot 2024 of the model with the above coefficients aligning with the manufacturer's data 2022 and providing projected battery lifetime versus depth of discharge.

As can be appreciated, different coefficients and/or different battery degradation models may be used, depending on a type of battery deployed in an ESS, according to the present disclosure.

Other battery models may be used to estimate Coulombic and Ohmic efficiency or the maximum rates of charge and discharge. Similar to the degradation model described above, the efficiency and maximum charge and discharge rates may be parameterized with constants that achieve a substantial “fit” between the model and the expected battery performance. Once these battery performance models are defined and parameters are provided, they may be used in the cost function control simulation to better predict the outcome of application of various control parameter sets.

Apparatus Architectures

FIG. 21 is a diagram of an EO 2100 according to one embodiment of the present disclosure. The EO 2100 may determine a control plan for managing control of an electrical system 2118 during an upcoming time domain and provide the control plan as output. The determined control plan may include a plurality of sets of parameters each to be applied for a different time segment within an upcoming time domain. The EO 2100 may determine the control plan based on a set of configuration elements specifying one or more constraints of the electrical system 2118 and defining one or more cost elements associated with operation of the electrical system. The EO 2100 may also determine the control plan based on a set of process variables that provide one or more measurements of a state of the electrical system 2118. The EO 2100 may include one or more processors 2102, memory 2104, an input/output interface 2106, a network/COM interface 2108, and a system bus 2110.

The one or more processors 2102 may include one or more general purpose devices, such as an Intel®, AMD®, or other standard microprocessor. The one or more processors 2102 may include a special purpose processing device, such as ASIC, SoC, SiP, FPGA, PAL, PLA, FPLA, PLD, or other customized or programmable device. The one or more processors 2102 perform distributed (e.g., parallel) processing to execute or otherwise implement functionalities of the present embodiments. The one or more processors 2102 may run a standard operating system and perform standard operating system functions. It is recognized that any standard operating systems may be used, such as, for example, Microsoft® Windows®, Apple® MacOS®, Disk Operating System (DOS), UNIX, IRJX, Solaris, SunOS, FreeBSD, Linux®, ffiM® OS/2® operating systems, and so forth.

The memory 2104 may include static RAM, dynamic RAM, flash memory, one or more flip-flops, ROM, CD-ROM, DVD, disk, tape, or magnetic, optical, or other computer storage medium. The memory 2104 may include a plurality of program modules 2120 and a data 2140.

The program modules 2120 may include all or portions of other elements of the EO 2100. The program modules 2120 may run multiple operations concurrently or in parallel by or on the one or more processors 2102. In some embodiments, portions of the disclosed modules, components, and/or facilities are embodied as executable instructions embodied in hardware or in firmware, or stored on a non-transitory, machine-readable storage medium. The instructions may comprise computer program code that, when executed by a processor and/or computing device, cause a computing system to implement certain processing steps, procedures, and/or operations, as disclosed herein. The modules, components, and/or facilities disclosed herein may be implemented and/or embodied as a driver, a library, an interface, an API, FPGA configuration data, firmware (e.g., stored on an EEPROM), and/or the like. In some embodiments, portions of the modules, components, and/or facilities disclosed herein are embodied as machine components, such as general and/or application-specific devices, including, but not limited to: circuits, integrated circuits, processing components, interface components, hardware controller(s), storage controller(s), programmable hardware, FPGAs, ASICs, and/or the like. Accordingly, the modules disclosed herein may be referred to as controllers, layers, services, engines, facilities, drivers, circuits, subsystems and/or the like.

The system memory 2104 may also include the data 2140. Data generated by the EO 2100, such as by the program modules 2120 or other modules, may be stored on the system memory 2104, for example, as stored program data 2140. The data 2140 may be organized as one or more databases.

The input/output interface 2106 may facilitate interfacing with one or more input devices and/or one or more output devices. The input device(s) may include a keyboard, mouse, touch screen, light pen, tablet, microphone, sensor, or other hardware with accompanying firmware and/or software. The output device(s) may include a monitor or other display, printer, speech or text synthesizer, switch, signal line, or other hardware with accompanying firmware and/or software.

The network/COM interface 2108 may facilitate communication or other interaction with other computing devices (e.g., a dynamic manager 2114) and/or networks 2112, such as the Internet and/or other computing and/or communications networks. The network/COM interface 2108 may be equipped with conventional network connectivity, such as, for example, Ethernet (IEEE 802.3), Token Ring (IEEE 802.5), Fiber Distributed Datalink Interface (FDDI), or Asynchronous Transfer Mode (ATM). Further, the network/COM interface 2108 may be configured to support a variety of network protocols such as, for example, Internet Protocol (IP), Transfer Control Protocol (TCP), Network File System over UDP/TCP, Server Message Block (SMB), Microsoft® Common Internet File System (CIFS), Hypertext Transfer Protocols (HTTP), Direct Access File System (DAFS), File Transfer Protocol (FTP), Real-Time Publish Subscribe (RTPS), Open Systems Interconnection (OSI) protocols, Simple Mail Transfer Protocol (SMTP), Secure Shell (SSH), Secure Socket Layer (SSL), and so forth. The network/COM interface 2108 may be any appropriate communication interface for communicating with other systems and/or devices.

The system bus 2110 may facilitate communication and/or interaction between the other components of the system, including the one or more processors 2102, the memory 2104, the input/output interface 2106, and the network/COM interface 2108.

The modules 2120 may include a historic load shape learner 2122, a load predictor 2124, a control parameter definer 2126, a cost function preparer/initializer 2128, a cost function evaluator 2130, and an optimizer 2132.

The historic load shape learner 2122 may compile or otherwise gather historic trends to determine a historic profile or an average load shape that may be used for load prediction. The historic load shape learner 2122 may determine and update and an avg_load_shape array and an avg_load_shape_time_of_day array by recording load observations and using an approach to determine a suitable average of the historic load observations after multiple periods of time. The historic load shape learner 2122 may utilize a process or an approach to determining the historic average profile such as described above with reference to FIG. 8.

The load predictor 2124 may predict a load on the electrical system 2118 during an upcoming time domain. The load predictor 2124 may utilize a historic profile or historic load observations provided by the historic load shape learner 2122. The load predictor 2124 may utilize a load prediction method such as described above with reference to FIGS. 7 and 8.

The control parameter definer 2126 may generate, create, or otherwise define a control parameter set X, in accordance with a control law. The created control parameters 2150 may include a definition 2152 and a value 2154 and may be stored as data 2140.

The cost function preparer/initializer 2128 prepares or otherwise obtains a cost function to operate on the control parameter set X. The cost function may include the one or more constraints and the one or more cost elements associated with operation of the electrical system 2118. The cost function preparer/initializer 2128 pre-calculates certain values that may be used during iterative evaluation of the cost function involved with optimization.

The cost function evaluator 2130 evaluates the cost function based on the control parameter set X. Evaluating the cost function simulates operation of the electrical system for a given time period under a given set of circumstances set forth in the control parameter set X and returns a cost of operating the electrical system during the given time period.

The optimizer 2128 may execute a minimization of the cost function by utilizing an optimization algorithm to find the set of values for the set of control variables. Optimization (e.g., minimization) of the cost function may include iteratively utilizing the cost function evaluator 2130 to evaluate the cost function with different sets of values for a control parameter set X until a minimum cost is determined. In other words, the algorithm may iteratively change values for the control parameter set X to identify an optimal set of values in accordance with one or more constraints and one or more cost elements associated with operation of the electrical system.

The data 2140 may include configuration data 2142, external data 2144, process variables 2146, state data 2147, historic observations 2148, and control parameters 2150 (including definitions 2152 and values 2154).

The configuration data 2142 may be provided to, and received by, the EO 2100 to communicate constraints and characteristics of the electrical system 2118.

The external data 2144 may be received as external input (e.g., weather reports, changing tariffs, fuel costs, event data), which may inform the determination of the optimal set of values.

The process variables 2146 may be received as feedback from the electrical system 2118. The process variables 2146 are typically measurements of the electrical system 2118 state and are used to, among other things, determine how well objectives of controlling the electrical system 2118 are being met.

The state data 2147 would be any EO state information that may be helpful to be retained between one EO iteration and the next. An example is avg_load_shape.

The historic observations 2148 are the record of process variables that have been received. A good example is the set of historic load observations that may be useful in a load predictor algorithm.

As noted earlier, the control parameter definer may create control parameters 2150, which may include a definition 2152 and a value 2154 and may be stored as data 2140. The cost function evaluator 2130 and/or the optimizer 2132 can determine values 2154 for the control parameters 2150.

The EO 2100 may provide one or more control parameters 2150 as a control parameter set X to the dynamic manager 2114 via the network/COM interface 2108 and/or via the network 2112. The dynamic manager 2114 may then utilize the control parameter set X to determine values for a set of control variables to deliver to the electrical system 2118 to effectuate a change to the electrical system 2118 toward meeting one or more objectives (e.g., economic optimization) for controlling the electrical system 2118.

In other embodiments, the EO 2100 may communicate the control parameter set X directly to the electrical system 2118 via the network/COM interface 2108 and/or via the network 2112. In such embodiments, the electrical system 2118 may process the control parameter set X directly to determine control commands, and the dynamic manager 2114 may not be included.

In still other embodiments, the EO 2100 may determine values for a set of control variables (rather than for a control parameter set X) and may communicate the set of values for the control variables directly to the electrical system 2118 via the network/COM interface 2108 and/or via the network 2112.

One or more client computing devices 2116 may be coupled via the network 2112 and may be used to configure, provide inputs, or the like to the EO 2100, the dynamic manager 2114, and/or the electrical system 2118.

FIG. 22 is a diagram of a dynamic manager 2200, according to one embodiment of the present disclosure. The dynamic manager 2200, according to one embodiment of the present disclosure, is a second computing device that is separate from an EO 2215, which may be similar to the EO 2100 of FIG. 21. The dynamic manager 2200 may operate based on input (e.g., a control parameter set X) received from the EO 2215. The dynamic manager 2200 may determine a set of control values for a set of control variables for a given time segment of the upcoming time domain and provide the set of control values to an electrical system 2218 to effectuate a change to the electrical system 2218 toward meeting an objective (e.g., economical optimization) of the electrical system 2218 during an upcoming time domain. The dynamic manager 2200 determines the set of control values based on a control law and a set of values for a given control parameter set X. The dynamic manager 2200 may include one or more processors 2202, memory 2204, an input/output interface 2206, a network/COM interface 2208, and a system bus 2210.

The one or more processors 2202 may include one or more general purpose devices, such as an Intel®, AMD®, or other standard microprocessor. The one or more processors 2202 may include a special purpose processing device, such as ASIC, SoC, SiP, FPGA, PAL, PLA, FPLA, PLD, or other customized or programmable device. The one or more processors 2202 perform distributed (e.g., parallel) processing to execute or otherwise implement functionalities of the present embodiments. The one or more processors 2202 may run a standard operating system and perform standard operating system functions. It is recognized that any standard operating systems may be used, such as, for example, Microsoft® Windows®, Apple® MacOS®, Disk Operating System (DOS), UNIX, IRJX, Solaris, SunOS, FreeBSD, Linux®, ffiM® OS/2® operating systems, and so forth.

The memory 2204 may include static RAM, dynamic RAM, flash memory, one or more flip-flops, ROM, CD-ROM, DVD, disk, tape, or magnetic, optical, or other computer storage medium. The memory 2204 may include a plurality of program modules 2220 and a program data 2240.

The program modules 2220 may include all or portions of other elements of the dynamic manager 2200. The program modules 2220 may run multiple operations concurrently or in parallel by or on the one or more processors 2202. In some embodiments, portions of the disclosed modules, components, and/or facilities are embodied as executable instructions embodied in hardware or in firmware, or stored on a non-transitory, machine-readable storage medium. The instructions may comprise computer program code that, when executed by a processor and/or computing device, cause a computing system to implement certain processing steps, procedures, and/or operations, as disclosed herein. The modules, components, and/or facilities disclosed herein may be implemented and/or embodied as a driver, a library, an interface, an API, FPGA configuration data, firmware (e.g., stored on an EEPROM), and/or the like. In some embodiments, portions of the modules, components, and/or facilities disclosed herein are embodied as machine components, such as general and/or application-specific devices, including, but not limited to: circuits, integrated circuits, processing components, interface components, hardware controller(s), storage controller(s), programmable hardware, FPGAs, ASICs, and/or the like. Accordingly, the modules disclosed herein may be referred to as controllers, layers, services, engines, facilities, drivers, circuits, and/or the like.

The system memory 2204 may also include data 2240. Data generated by the dynamic manager 2200, such as by the program modules 2220 or other modules, may be stored on the system memory 2204, for example, as stored program data 2240. The stored program data 2240 may be organized as one or more databases.

The input/output interface 2206 may facilitate interfacing with one or more input devices and/or one or more output devices. The input device(s) may include a keyboard, mouse, touch screen, light pen, tablet, microphone, sensor, or other hardware with accompanying firmware and/or software. The output device(s) may include a monitor or other display, printer, speech or text synthesizer, switch, signal line, or other hardware with accompanying firmware and/or software.

The network/COM interface 2208 may facilitate communication with other computing devices and/or networks 2212, such as the Internet and/or other computing and/or communications networks. The network/COM interface 2208 may couple (e.g., electrically couple) to a communication path (e.g., direct or via the network) to the electrical system 2218. The network/COM interface 2208 may be equipped with conventional network connectivity, such as, for example, Ethernet (IEEE 802.3), Token Ring (IEEE 802.5), Fiber Distributed Datalink Interface (FDDI), or Asynchronous Transfer Mode (ATM). Further, the network/COM interface 2208 may be configured to support a variety of network protocols such as, for example, Internet Protocol (IP), Transfer Control Protocol (TCP), Network File System over UDP/TCP, Server Message Block (SMB), Microsoft® Common Internet File System (CIFS), Hypertext Transfer Protocols (HTTP), Direct Access File System (DAFS), File Transfer Protocol (FTP), Real-Time Publish Subscribe (RTPS), Open Systems Interconnection (OSI) protocols, Simple Mail Transfer Protocol (SMTP), Secure Shell (SSH), Secure Socket Layer (SSL), and so forth.

The system bus 2210 may facilitate communication and/or interaction between the other components of the system, including the one or more processors 2202, the memory 2204, the input/output interface 2206, and the network/COM interface 2208.

The modules 2220 may include a parameter selector 2222 and a control law applicator 2224.

The parameter selector may pick which set of parameters to be used from the control parameter set X, according to a given time segment.

The control law applicator 2224 may process the selected set of parameters from the control parameter set X and convert or translate the individual set of parameters into control variables (or values thereof). The control law applicator 2224 may apply logic and/or a translation process to determine a set of values for a set of control variables based on a given set of parameters (from a control parameter set X) for a corresponding time segment. For example, the control law applicator 2224 may apply a method and/or logic as shown in FIG. 16.

The data 2240 may include configuration data 2242, process variables 2246, control parameters 2250 (including definitions 2252 and values 2254), and/or control variables 2260 (including definitions 2262 and values 2264).

The configuration data 2242 may be provided to, and received by, the dynamic manager 2200 to communicate constraints and characteristics of the electrical system 2118.

The process variables 2246 may be received as feedback from the electrical system 2218. The process variables 2246 are typically measurements of the electrical system 2218 state and are used to, among other things, determine how well objectives of controlling the electrical system 2218 are being met. Historic process variables 2246 may be utilized by the HSC for example to calculate demand which may be calculated as average building power over the previous 15 or 30 minutes. The dynamic manager 2200 can determine the set of control values for the set of control variables based on the process variables 2246.

The control parameters 2250 may comprise a control parameter set X that includes one or more sets of parameters each for a corresponding time segment of an upcoming time domain. The control parameters 2250 may additionally, or alternately, provide a control plan for the upcoming time domain. The control parameters 2250 may be received from an EO 2215 as an optimal control parameter set X_opt.

The control variables 2260 may be generated by the parameter interpreter 2222 based on an optimal control parameter set X_opt.

The dynamic manager 2200 may receive the optimal control parameter set X_opt from the EO 2215 via the network/COM interface 2208 and/or via the network 2212. The dynamic manager 2200 may also receive the process variables from the electrical system 2218 via the network/COM interface 2208 and/or via the network 2212.

The dynamic manager 2200 may provide the values for the set of control variables to the electrical system 2218 via the network/COM interface 2208 and/or via the network 2212.

One or more client computing devices 2216 may be coupled via the network 2212 and may be used to configure, provide inputs, or the like to the EO 2215, the dynamic manager 2200, and/or the electrical system 2218.

Example Cases of Energy Costs

FIG. 23 is a graph 2300 illustrating how Time-of-Use (ToU) supply charges impact energy costs of a customer. ToU supply charges are time-specific charges customers pay for electrical energy consumed. The graph 2300 includes a plot 2302 of the load and a plot 2304 of a photovoltaic contribution. The graph 2300 includes a plot 2306 of ToU supply (or energy) rate. As can be seen in the illustrated example, ToU supply charges can vary by time of day, day of week, and season (summer vs. winter). ToU supply charges are calculated based on the NET energy consumed during specific meter read intervals (often 15 or 30 minutes). In the illustrated example, the supply rates are as follows:

Peak M-F 2 pm-8 pm $0.39/kWh,

Off-Peak 10p-7 am $0.08/kWh,

Shoulder $0.15/kWh.

In the example, based on the load, the photovoltaic generation, and the supply rates, the Supply Charge on July 5 is approximately: 12.2*0.39+1.6*0.08+4.1*0.15=$5.50.

FIG. 24 is a graph 2400 illustrating how demand charges impact energy costs of a customer. Demand charges are electrical distribution charges that customers pay based on their maximum demand (kW) during a specified window of time. The graph 2400 includes a plot 2402 of the load and a plot 2404 of the demand. The graph 2400 also includes a plot 2406 of the demand rate. Demand charges are typically calculated monthly but can also be daily. The maximum is often only taken for certain hours of the day. In the illustrated embodiment, a daily demand rate from 8:00 am to 10:00 pm on weekdays is $0.84/kW (daily). The peak demand on May 21 is 416 kW. Accordingly the Demand Charge=416*0.84=$349.

FIG. 25 is a graph 2500 illustrating the challenge of maximizing a customers' economic returns for a wide range of system configurations, building load profiles, and changing utility tariffs. The graph 2500 reflects consideration of a number of factors, including:

ToU Supply Charges (seasonal, hourly, for any number of time windows)

Demand Charges (daily, monthly, for any number of time windows)

Utilization of Renewable Generation (e.g., PV, CHP)

Contracted or Incentive Maneuvers (e.g., DMP and Demand Response)

Minimum Import Constraints

Battery Performance, Degradation Rate, and Cost.

The graph 2500 includes a plot 2501 for building load, a plot 2502 for PV full output, a plot 2503 for PV curtailed output, a plot 2504 for battery, a plot 2505 for net building demand, a plot 2506 for DMP battery power target, a plot 2507 for an energy supply rate (×1000), a plot 2508 for demand rate (×100), and a plot 2509 for the battery SoC (×100).

An EO according to one embodiment of the present disclosure optimizes overall energy economics by blending these factors (and any other factors) simultaneously in real time.

Updating Model Data and Making Forecasts

As discussed above with reference to FIGS. 5-7, load and generation predictions or forecasts may be used when the cost function is evaluated. More detail will here be discussed regarding load and generation predictions or forecasts.

FIG. 26 is a simplified block diagram illustrating an example of an electrical power system 2600, similar to the electrical power system 500 of FIG. 5. For example, the electrical power system 500A includes one or more loads 522A operably coupled to one or more sensors 528A, and a controller 510A operably coupled to the one or more sensors 528A. In some embodiments, the electrical power system 500A may include one or more generators 524A operably coupled to the one or more sensors 528A. The one or more loads 522A, the one or more generators 524A, the one or more sensors 528A, and the controller 510A may be similar to the loads 522, the generators 524, the sensors 528, and the controller 510 discussed above with reference to FIG. 5. For convenience, the one or more loads 522A, the one or more generators 524A, and the one or more sensors 528A may sometimes be referred to herein as “loads” 522A, “generators” 524A, and “sensors” 528A, respectively.

The controller 510A includes one or more processors 2612 operably coupled to one or more data storage devices 2614. The one or more processors 2612 and the one or more data storage devices 2614 may sometimes be referred to herein as “processors” 2612 and “storage” 2614, respectively. The storage 2614 is configured to store model data 2616 indicating, for time points of a time period of operation (e.g., one day) of the electrical power system 500A, a model load power for the loads 522A, a model generator power for the generators 524A, or a combination of the model load power and the model generator power. As used herein, the terms “model data,” “model load power,” and “model generator power” refer to models of typical data (e.g., data indicating typical load power consumed by the loads 522A, typical generator power provided by the generators 524A). This typical data may be produced (e.g., by the processors 2612) using actual measured data measured by the sensors 528A (e.g., average daily load/generator data), may be theoretical data provided by a user of the electrical power system 500A, theoretically estimated data based on estimated power consumption/generation, or any combination(s) thereof. A combination of the model load power and the model generator power may comprise a “net” (e.g., summation) of the model load power and the model generator power, or may comprise a dual model with a representation or model for the model load power and a representation or model for the model generator power. By way of non-limiting example, the processors 2612 may produce or update the model data 2616 using the method 2700 of FIG. 27.

In some embodiments, the model data 2616 includes an aggregation of a plurality of sets of previous data, each of the plurality of sets of previous data including data indicating, for time points of a different previous time period of operation of the electrical power system 500A, a previous load power consumed by the loads 522A, a previous generator power provided by the generators, or a combination of the previous load power and the previous generator power. In some embodiments, the aggregation of the plurality of sets of previous data includes an average of the plurality of sets of previous data. In some embodiments, the average of the plurality of sets of previous data includes a weighted average of the plurality of sets of previous data. In some embodiments, the model data 2616 includes a user defined estimate of a typical load power for the loads 522A, a typical generator power for the generators 524A, or a combination of the typical load power and the typical generator power for the time points of the time period. Many other methods are contemplated for generating the model data 2616. By way of non-limiting example, the model data 2616 may be generated using an output error model, AutoRegressive Moving Average (ARMA) models, other model generation techniques known in the art, or combinations thereof.

The processors 2612 are configured to determine, based on information received from the sensors, current data 2618, which can include a current load power consumed by the loads 522A, a current generator power provided by the generators 524A, or a combination of the current load power and the current generator power. The current data 2618 may be determined for time points of a current time period of operation of the electrical power system 500A. The current time period corresponds to the time period of the model data 2616.

In some embodiments, it may be useful to track how accurately the current data 2618 tracks the model data 2616, or how uncertain the current data 2618 or model data 2616 is (e.g., even in the case of data for which no prediction or forecast has been created). In such embodiments, the storage 510A may include uncertainty data 2619. The uncertainty data 2619 includes data indicating how much the current data 2618 historically deviates from the model data 2616, or some other uncertainty metrics. By way of non-limiting example, the uncertainty data 2619 may include error (e.g., the model data 2616 minus the current data 2618), a mean square of the error, a variance of the error, a standard deviation of the error, other error metrics, or combinations thereof. By way of non-limiting example, the uncertainty data 2619 may include a historic mean square error of measured data (e.g., the current data 2618 over multiple different measured time periods) for a plurality of points along the time period. Also by way of non-limiting example, the uncertainty data 2619 may include a historic variance and/or a historic standard deviation of the error of the measured data for a plurality of points along the time period. In some embodiments, the uncertainty data 2619 may include an average historic error, an average historic mean square error, an average historic variance, an average historic standard deviation, or combinations thereof for the entire period of time.

In some embodiments, the uncertainty data 2619 may include an uncertainty metric other than an error metric. By way of non-limiting example, the uncertainty data 2619 may include a sample variance and/or a sample standard deviation of historic measured data at a plurality of points of time during the time period. An example of an expression to determine a sample variance, σ², is as follows:

${\sigma }^2=\frac{1}{N}\sum _{i=1}^N{\left(x_i-\stackrel{\_}{x}\right)}^2$

where N is the number of samples corresponding to a particular point in time, x_i is the value of a sample at the point in time, and x is the mean of all samples corresponding to the point in time. An example of an expression to determine a sample standard deviation, σ, is as follows:

$\sigma =\sqrt{\frac{1}{N-1}\sum _{i=1}^N{\left(x_i-\stackrel{\_}{x}\right)}^2}$

In some embodiments, the uncertainty data 2619 may include averages of uncertainty metrics for the entire period of time. By way of non-limiting example, the sample variance and or sample standard deviation may be averaged over the entire time period.

The uncertainty data 2619 may be used to analyze uncertainty in various variables that relate to operation of the electrical power system 2600. By way of non-limiting example, the uncertainty data 2619 may be used to generate random variables that relate to operation of the electrical power system 2600, as will be discussed in more detail below.

In some embodiments, the uncertainty data 2619 may include data indicating a probability distribution that has been fitted to an uncertainty metric and/or an error metric. A probability distribution may eliminate the need to store a large amount of data as an error metric or an uncertainty metric could be modeled or estimated using only a few values. By way of non-limiting example, if it is determined that the uncertainty data 2619 may be modeled using a normal distribution, a mean value and a standard deviation or variance value would be sufficient to provide a probability distribution for the uncertainty data 2619.

The processors 2612 are also configured to modify (e.g., update) the model data 2616 by aggregating the model data 2616 with the current data 2618. In some embodiments, the processors 2612 are configured to aggregate the model data 2616 with the current data 2618 by determining a weighted average between the model data 2616 and the current data 2618. In some embodiments, the model data 2616 is weighted more heavily than the current data 2618 in the weighted average. By way of non-limiting example, an IIR filter may be used to update the model data 2616. An expression for the IIR filter may be given as follow:

Data_Model(n+1,i)=a₁Data_Model(n,i)+a₂Data_current(i)

where Data_Model(n+1,i) is the updated model data 2616, Data_Model(n, i) is the model data 2616 prior to being updated, Data_current(i) is the current data 2618, and a₁ and a₂ are coefficients of the IIR filter. As a specific non-limiting example, the model data 2616 may be weighed more heavily than the current data 2618 by selecting a₁>a₂ (e.g., a₁=0.95 and a₂=0.05).

With the model data 2616 weighed more heavily than the current data 2618 in updating the model data 2616, the model data 2616 may be resilient to non-recurring artifacts of any given current data 2618. If, however, the behavior of the loads 522A and/or the generators 524A manifests recurrent changes, the model data 2616 will gradually incorporate that behavior over several updates of the model data 2616. For example, if a₁=0.95 and a₂=0.05, it would take approximately twenty consecutive recurrences (e.g., days) of a changed behavior of the loads 522A and/or the generators 524A for that behavior to show up in 60% of its strength in the model data 2616.

In some embodiments, a number of the time points of the period of time corresponding to the model data 2616 may be different from a number of the time points of the current period of time corresponding to the current data 2628. In such embodiments, the processor 2612 may be configured to interpolate the current data 2618 to include the same number of time points as the model data 2616 before aggregating the model data 2616 with the current data 2618. Moreover, in some embodiments, the time points corresponding to the model data 2616, the time points corresponding to the current data 2618, or a combination thereof may be spaced at non-uniform time intervals. In some embodiments, the time points of the model data 2616, the current data 2618, or a combination thereof may be uniformly spaced at intervals from anywhere between five (5) minutes to 120 minutes.

In some embodiments, the uncertainty data 2619 may also be updated with the current data 2618. Similarly as discussed above with respect to the model data 2616, the uncertainty data 2619 may be updated using weighted averages. For example, a long-term probability distribution may be stored along with only a few most recent current data 2618 profiles for a period of time. The error or uncertainty metrics of the long-term probability distribution may be updated with error or uncertainty metrics of the few most recent current data 2618 profiles by computing a weighted average between the error or uncertainty metrics of the long-term probability distribution and the error or uncertainty metrics of the few most recent current data 2618 profiles.

The processors 2612 may also, as discussed above, determine a set of control values for a set of control variables to effectuate a change to operation of the electrical power system 500A based, at least in part, on the model data 2616.

FIG. 27 is a simplified flowchart illustrating a method 2700 of operating an electrical power system (e.g., the electrical power system 500A of FIG. 26). Referring to FIGS. 26 and 27 together, the method includes storing 2710 model data 2616 indicating, for time points of a time period of operation of the electrical power system 500A, a model load power for the loads 522A of the electrical power system 500A, a model generator power for the generators 524A of the electrical power system 500A, or a combination of the model load power and the model generator power.

The method 2700 also includes determining 2720, based on information received from the sensors 528A of the electrical power system 500A, current data 2618, which can include a current load power consumed by the loads 522A of the electrical power system 500A, a current generator power provided by the generators 524A of the electrical power system 500A, or a combination of the current load power and the current generator power for time points of a current time period of operation of the electrical power system. The current time period can correspond to the time period of the model data 2616. By way of non-limiting example, the current time period may be a current day and the time period of the model data 2616 may also be a day.

The method 2700 includes updating 2730 uncertainty data 2619. In some embodiments, updating uncertainty data 2619 comprises computing an error (e.g., mean square error (MSE), variance and/or standard deviation of the error, sample variance, sample standard deviation, etc.) between the model data 2616 and the current data 2618. In some embodiments, updating uncertainty data 2619 includes determining variance and/or standard deviation of current data 2616 from a plurality of different time periods. By way of non-limiting example, a variance of the current data for a particular time of day for a plurality of different days may be determined.

The method 2700 further includes updating 2740 the model data 2616 by modifying (e.g., updating) the model data 2616 with the current data 2618. In some embodiments, updating 2740 the model data 2616 includes determining a weighted average between the model data 2616 and the current data 2618 (e.g., using an IIR filter). In some embodiments, determining a weighted average between the model data 2616 and the current data 2618 includes weighing the model data 2616 more heavily than the current data 2618 in the weighted average.

The method 2700 also includes determining 2750 a set of control values (e.g., for a set of control variables or decision variables) to effectuate a change to operation of the electrical power system 500A based, at least in part, on the model data 2616, as discussed above. In some embodiments, determining 2750 a set of control values includes analyzing an uncertainty of the model data 2616 based on the uncertainty data 2619.

Referring once again to FIG. 26, the model data 2616 may be used to predict or forecast behavior of the loads 522A, the generators 524A, or a combination thereof. An example of a method 700 for predicting or forecasting this behavior is discussed above with reference to FIG. 7. For example, before current data 2618 for the entire period of time corresponding to the model data 2616 has been recorded, future data predicted or forecasted for the current data may be generated using the method 700 of FIG. 7. For example, the processor 2612 may be configured to determine, based on information received from the sensors 528A, the current data 2618 for time points of the current time period corresponding to an early portion of the time period of the model data 2616. The processor 2612 may also be configured to fit the early portion of the model data 2616 to the current data to produce predicted or forecasted data. A future portion of the predicted or forecasted data corresponds to time points occurring in the future with reference to the current data 2618. The processor 2612 may further be configured to determine a set of control values (e.g., for a set of control variables) to effectuate a change to operation of the electrical power system based, at least in part, on the future portion of the predicted or forecasted data.

In practice, the prediction or forecast of the future portion of the predicted or forecasted data may share some similarities with applying a Kalman filter. For example, the prediction takes into account both model data 2616 and innovations in the form of the current data 2618. The prediction may also involve weighting of the model data 2616 and the current data 2618 in updating the model data 2616, and weighted regressions that favor more recent samples or older samples of the current data 2618. As a result, behavior of the predictions may be somewhat similar to what may be observed if the predictions instead were made using a Kalman filter. The processor 2612, however, may perform the predictions without using a linearized model in the form of x(k+1)=A*x(k)+B*u, as is used in Kalman filter implementations. Rather, the processor 2612 may instead use a table of historic average values that represents the model data 2618, which may be an evolution of load power and/or generator power that may not necessarily be expressed in linear equation form.

Stochastic Optimization

In some embodiments, uncertainty regarding one or more elements of a cost function or objective function may exist. By way of non-limiting examples, there may be a degree of uncertainty as to the accuracy of a predicted or forecasted load or a predicted or forecasted generator profile over a future time domain. FIG. 8 illustrates that an actual load shape 810 (in graphical form) observed after prediction or forecast is not identical to the predicted load 808. A stochastic approach to dealing with uncertain elements may be used in economic optimization of an electrical system.

Random variables are one approach for accounting for uncertainty. In some embodiments, one or more of the elements of the cost function or objective function may include one or more random variables. As used herein, the term “random variable” refers to a variable having a probability distribution function that indicates probabilities that the value of the random variable will take on certain values. In other words, a “random variable” may be a function defined on a sample space of a probability space. As a result, in some embodiments, the cost function or objective function may include a probability distribution function (e.g., a probability density function for continuous-time cost or objective functions, a probability mass function for discrete-time cost or objective functions). Optimization of the cost function or objective function may involve computing an integral or sum of the cost function or objective function to determine an optimal expected value (e.g., a minimum expected value) of the cost function or objective function. When taking the integral or sum over the random variables to yield the expected value, the integrand may be the multiplied by a probability distribution or density function representing the probability of occurrence of each value of the random variable.

In some embodiments, the one or more random variables of the cost function or objective function may be determined using historically observed input data resulting from operation of an electrical power system (e.g., the electrical power system 2600 of FIG. 26). By way of non-limiting example, the uncertainty data 2619 may be used to generate the one or more random variables. More detail about how the one or more random variables are derived will be discussed below with reference to FIGS. 29A-29G.

FIG. 28 is a simplified block diagram of an electrical system controller 2800, according to some embodiments. In some embodiments, the electrical system controller 2800 may be (or may replace) the controllers 110, 510 of the electrical systems 100, 500 of FIGS. 1 and 5, respectively. The electrical system controller 2800 may be similar to the controllers 110, 510 except that the controller 2800 employs random variables in constructing a cost or objective function, and determines an optimal expected value instead of an optimal value by employing an optimization algorithm.

The electrical system controller 2800 may be combined with any of the embodiments discussed herein. In such embodiments, it will be understood that the electrical system controller 2800 applies by merely modifying any reference to an optimized value (e.g., a minimized value) of the cost or objective function to instead refer to an optimized expected value of the cost or objective function. Correspondingly, any variable of the cost or objective function, according to any of the embodiments discussed herein, may be replaced with a random variable, as will be discussed in more detail below. Accordingly, it is within the scope of the present disclosure to modify any embodiment disclosed herein, and any of its features, to employ stochastic optimization.

The electrical system controller 2800 includes one or more data storage devices 2810 (sometimes referred to herein as “storage devices” 2810) operably coupled to one or more processors 2820 (sometimes referred to herein as “processors” 2820). The storage devices 2810 are configured to store data corresponding to one or more random variables (sometimes referred to herein as “random variables”) associated with operation of an electrical system (e.g., a building electrical system). The processors 2820 are configured to determine a set of control values for a set of control variables to effectuate a change to the electrical system toward meeting a controller objective for economical optimization of the electrical system. The set of control values are determined by the processors 2820 utilizing an optimization algorithm to determine the set of control values as a function of the one or more random variables. The processors 2820 are configured to control the electrical system based on the control values.

In some embodiments, the storage devices 2810 may be configured to store observed input data. The observed input data corresponds to observed values of one or more input variables during one or more previous periods of time of operation. Generation of the random variables may, in some instances, be based on this observed input data. By way of non-limiting example, the processors 2820 may be configured to generate one or more uncertainty metrics based on the observed input data. The one or more uncertainty metrics are configured to indicate an uncertainty of the one or more input variables. The random variables may be generated based on the one or more uncertainty metrics to forecast input data during a future period of time.

In some embodiments, the processors 2820 are configured to construct a cost function including an expected value of an economic cost of operating an electrical system. The cost function may be a function of the random variables. The cost function may include one or more control variables or decision variables of the electrical system. The control variables may correspond to controllable features of the electrical system.

In some embodiments, the processors 2820 are configured to utilize the optimization algorithm by generating a cost function including the one or more random variables and minimizing an expected value of a cost of operating the electrical system. The processors 2820 may be configured to determine control values for the control variables corresponding to a minimum value of the expected value of the economic cost of operating the electrical system. In some embodiments, the cost function comprises an integral of a probability density function multiplied by a cost. By way of non-limiting example, the cost function may be given by:

Cost=∫_−∞^∞f(l)c(l)dl,

where l is a random variable, f(l) is a probability density function representing the probability of occurrence of each value of the random variable l, and c(l) is the operating cost evaluated over some future period of time as a function of random variable l. In some embodiments the cost function includes a sum of such integrals. In some embodiments, the cost function comprises a sum of a probability mass function. In some embodiments, the cost function takes into consideration time-dependence of cost elements of operating the electrical system over a future period of time.

In embodiments where the electrical system controller 2800 is similar to the controller 510 of FIG. 5, the electrical system controller 2800 (e.g., the processors 2810 and the data storage devices 2810) may be distributed between an economic optimizer (EO) 530 and a dynamic manager or high speed controller (HSC) 540, as discussed above with reference to FIG. 5.

In some embodiments, the random variables of the cost or objective function include a predicted load random variable corresponding to a predicted load of the electrical system. In some embodiments, the random variables of the cost or objective function include a predicted generator profile random variable corresponding to a predicted generator power generation profile of an electrical power generator of the electrical system.

Other values besides a predicted load and a predicted generator power may also be accounted for using random variables. As a result, uncertainty data 2619 (FIG. 26) may be collected for these other values and stored in storage 2614 (FIG. 26). This error data may be used to generate the random variables.

In some embodiments, the random variables of the cost or objective function include a system constraint random variable corresponding to a system constraint of the electrical system. By way of non-limiting example, the system constraint random variable may correspond to one or more of an ESS configuration, an ESS efficiency, an ESS degradation, an electricity supply tariff, an electricity demand tariff, a minimum power input of the electrical system, an ESS SoC limit, an ESS power limit, weather data, building occupancy, other constraint, or combinations thereof.

In some embodiments, the one or more random variables of the cost or objective function include one or more cost element random variables corresponding to one or more costs of operating the electrical system. Examples of cost elements that may be included in the cost function or objective function are a net cost of electrical power from the electrical grid and a net cost of operating electrical equipment of the electrical system. The net cost of electrical power from the electrical grid may include a ToU supply charge, a demand charge, a local contracted or incentive maneuver benefit, or combinations thereof. One or more cost element random variables may be generated from historic data to account directly for any one or more of the ToU supply charge, the demand charge, or the local contracted or incentive maneuver benefit. The cost element random variables may account directly for the uncertainty of these cost elements themselves, in contrast to sub-elements of these cost elements. For example, a predicted load and a predicted power generation profile may both be factors that influence ToU supply charge, the demand charge, and the local contracted or incentive maneuver benefit. Uncertainty of the cost elements themselves may also be accounted for using random variables for the sub-elements; ultimately, the cost function will include random variables if a cost element is a function of those random variables. In some embodiments, the one or more cost element random variables include two or more of a ToU supply charge random variable, a demand charge random variable, and a local contracted or incentive maneuver benefit random variable.

The net cost of operating electrical equipment of the electrical power system may include an equipment degradation cost (e.g., wear and tear) and other equipment costs/benefits. By way of non-limiting example, equipment degradation random variables may include a random variable to account for a cost associated with degradation of a battery or other energy storage device. Also by way of non-limiting example, equipment degradation random variables may include a random variable to account for a cost of degradation of a generator. Degradation of some generators may be modeled as proportional to time of operation of the generator, subject to some uncertainty. As a result, costs of degradation for such generators may also be proportional to time of use of the generators (e.g., the product of the cost of the generator and the ratio of time of use to expected time of use during the lifetime of the generator), subject to some level of uncertainty. As a further non-limiting example, equipment degradation random variables may include a random variable to account for a cost of degradation of production equipment. Although production equipment may be operated to generate economic benefits (e.g., to produce products that can be sold for financial profit), production equipment also undergoes wear and tear when used. Uncertainty in costs associated with this equipment degradation can be accounted for using one or more equipment degradation random variables.

The other equipment costs/benefits may include a benefit arising from selling products produced by a piece of equipment that operates using electricity from the electrical power system. Although such equipment experiences wear and tear during operation, there is often a quantifiable benefit that arises from its use. One or more random variables may be used to account for uncertainty in such benefits. Sources of uncertainty may include uncertainty in a price that products produced by the equipment will command, uncertainty in a quantity of sellable products that the equipment can produce, uncertainty in cost of materials the equipment uses to produce the products, and other costs/benefits. Another example of other equipment costs/benefits includes other financial and/or non-financial costs or benefits. For example, benefit values may be assigned for keeping a building's ambient temperature within predetermined ranges. As a result, a benefit may be obtained for running an air conditioning system. Also, costs may be assigned to an unpleasantness experienced by occupants of the building resulting from the building's ambient temperature rising or falling outside the predetermined ranges. By way of non-limiting example, a random variable may be used to account for the uncertainty that the building will be occupied at any given time. Also by way of non-limiting example, a random variable may be used to account for the uncertainty that certain temperature ranges would be considered comfortable or uncomfortable to occupants.

Random variables may also be used to account for uncertainty in other financial consequences of operating or not operating equipment that consumes electrical power. For example, a random variable may be used to account for the uncertainty that certain products will go bad or spoil if the ambient temperature of a building falls outside of predetermined ranges. Also, an uncertainty in the cost of cooling a building back down after shutting down an air conditioning unit and allowing the building to heat to temporarily reduce the amount of power consumed may be accounted for using a random variable.

Other random variables may be associated with electricity market bids (e.g., bids to sell electrical power back to the grid). For example, a random variable may be a utility or ISO's offered electricity price at a point in time in the future. In another example, if an optimizer is configured such that one or more decision variables are market bids, a random variable may be developed that relates to the likelihood that a given bid is accepted or rejected (e.g., the probability that the grid will accept a bid to sell electrical power back to the grid at a future point in time).

The processors 2820 may be configured to generate the one or more random variables of the cost or objective function. There are a variety of ways that the processors 2820 may generate the random variables. For example, the processors 2820 may observe uncertainty in previous measurements and generate the random variables based on the observed uncertainty. Another approach is to program the processors 2820 with assumptions regarding the uncertainty, or to accept a user input regarding the uncertainty via a user interface. In this way, if information is known or estimated regarding the uncertainty, this information can be provided to the processors 2820.

FIGS. 29A-29G illustrate an example of generating random variables to account for uncertainty in a load, according to some embodiments.

FIG. 29A is a simplified flowchart illustrating an example method 2900 of generating random variables to account for uncertainty in a load, according to some embodiments. The method 2900 includes obtaining 2912 observed input data, fitting 2920 a probability distribution function (PDF) to the observed input data, and generating 2930 an uncertainty metric for the load.

In some embodiments, fitting 2920 a PDF to the observed input data includes fitting the PDF to the currently and/or historically observed current data 2618 (FIG. 26) measured from a load 522A (FIG. 26) from a current and/or multiple previous periods of time (e.g., days). In some embodiments, fitting 2920 a PDF to the observed input data includes fitting the PDF to error data generated by comparing currently observed data (e.g., the current data 2618) measured from the load 522A to predicted or forecasted data (e.g., forecasted data generated based on the model data 2616 of FIG. 26).

FIG. 29B is a simplified plot 2902 illustrating a predicted or forecasted load 2910 derived from the model data 2616 (FIG. 26) and an actual or observed load 2908 (e.g., currently observed current data 2618 of FIG. 26) for a period of time. The term “predicted load 2910” may be used interchangeably herein with the term “forecasted load 2910.” Also, the term “actual load 2908” may be used interchangeably herein with the term “observed load 2908.”

The period of time illustrated in FIG. 29B is one day (day 182 as seen from the horizontal axis). As previously discussed, in some embodiments fitting 2920 a PDF to the observed input data includes fitting the PDF to the currently and/or historically observed current data 2618 (FIG. 26) measured from a load 522A (FIG. 26) from a current and/or multiple previous periods of time (e.g., days). In such embodiments, the PDF may be fit 2920 to the observed load 2908 itself of one or more periods of time. In some embodiments, the PDF may be fit 2920 to observed load 2908 data from multiple periods of time.

As also previously discussed, in some embodiments fitting 2920 a PDF to the observed input data includes fitting the PDF to error data generated by comparing currently observed data measured from the load 522A to forecasted data. In such embodiments, the PDF may be fit to an error between the forecasted load 2910 and the observed load 2908. In some embodiments, the PDF may be fit to errors observed between forecasted loads 2910 and observed loads 2908 of multiple different time periods (e.g., multiple different previous time periods).

FIG. 29C is a simplified plot 2904 illustrating an error 2914 between the forecasted load 2910 of FIG. 29B and the observed load 2908 of FIG. 29B during a period of time. As discussed above, in some embodiments this error 2914 may be used as the observed input data that is obtained 2912, and that the PDF is fit 2920 to in the method 2900. In some embodiments, error from multiple time periods may be used as the observed input data.

Referring again to FIG. 29A, as previously discussed, the method 2900 includes fitting 2920 a probability distribution function (PDF) to the observed input data (e.g., to the actual observed load 2908 data itself or to error 2914 between the forecasted load 2910 and the observed load 2908). In some embodiments, fitting 2920 a PDF to the observed input data includes fitting a PDF to the observed input data for each of multiple points of time during the time period, based on variance of the observed input data over multiple different time periods.

FIG. 29D is a simplified plot 2922 illustrating a fit of a PDF 2924 to observed input data 2924. The observed input data 2924 in this example is a histogram of error between the forecasted load 2910 of FIG. 29B and the observed load 2908 of FIG. 29B at the end of the day for one thousand days. The observed input data 2924 in this case is very close to a normal distribution, so the PDF 2924 is selected to be a normal distribution. In other cases, the observed input data 2924 may be closer to some other type of distribution, such as a uniform distribution, a Laplace distribution, a chi-squared distribution, or some other distribution. Although FIG. 29D illustrates a fit of the PDF 2924 to the observed input data 2924, which in this case is error data, in some embodiments a PDF may be fit to the actual load 2908 data itself. In such embodiments, the PDF would be the similar to the PDF 2924 except that there would be a different mean value for the PDF (corresponding to a horizontal shift in the plot 2922).

Referring again to FIG. 29A, in some embodiments, the method 2900 further includes generating 2930 an uncertainty metric for the load. Examples of uncertainty metrics that may be used include a sample variation of the observed input data, a sample standard deviation of the observed input data, a mean square error (MSE) between the observed input data and forecasted data (see FIG. 29E), a variance of the error (see FIG. 29F), a standard deviation of the error (see FIG. 29G), or combinations thereof.

FIG. 29E is a simplified plot 2932 illustrating an MSE 2934 between the forecasted load 2910 of FIG. 29B and the observed load 2908 of FIG. 29B at the end of the day for ten days.

FIG. 29F is a simplified plot 2936 illustrating a variance 2938 of the error between the forecasted load 2910 of FIG. 29B and the observed load 2908 of FIG. 29B at the end of the day for ten days. The variance 2938 would be similar to the sample variance of the actual load 2908 at the end of the day for ten days.

FIG. 29G is a simplified plot 2937 illustrating a standard deviation 2939 of the error between the forecasted load 2910 of FIG. 29B and the observed load 2908 of FIG. 29B at the end of the day for ten days. The standard deviation 2939 would be similar to the sample standard deviation of the observed load 2908 at the end of the day for ten days.

In some embodiments, observed input data measured from multiple different periods of time (e.g., days) may be stored by the storage 2614, 2810 (FIGS. 26, 28) to enable the processors 2612, 2820 (FIGS. 26, 28) to determine the uncertainty of the observed input data at various points of time during the period of time (i.e., generate random variables for each of the various points of time). In some embodiments, observed input data from only a predetermined number of the most recent periods of time may be stored for use in determining the uncertainty. For example, the uncertainty may be determined based on the most recent 5, 10, 100, or 1,000 periods of time.

In some embodiments, rather than store the actual observed input data from the periods of time, an uncertainty metric may be stored. For example, the MSE at various points of time of the period of time may be stored, amounting to an MSE profile over the period of time. The MSE data may be updated based on newly measured error 2914 from a new period of time. By way of non-limiting example, a weighted average between a historic MSE profile for the period of time and a square of a newly measured error may be taken. The historic MSE profile may be weighed more heavily than the newly-measured error to prevent non-recurrent artifacts of the newly-measured error from having a large influence on the updated MSE profile. By way of non-limiting example, the PDF for each of the points in the period may be estimated using the updated MSE profile. In this way, uncertainty may be analyzed without storing a large amount of observed input data. Other uncertainty metric profiles such as error variance profiles, error standard deviation profiles, sample variance profiles, sample standard deviation profiles, or combinations thereof may be used.

It should be noted that although FIGS. 29A-29G discuss generation of random variables using uncertainty profiles for a load profile, a similar approach may be taken to generate random variables for other system metrics. By way of non-limiting example, a similar approach may be taken for generating random variables for one or more generators. Also by way of non-limiting example, a similar approach may be taken for generating any other random variables mentioned herein.

FIG. 30 is a graphical representation 3000 illustrating an accounting for uncertainty in a cost of operating an electrical system during an upcoming time domain, according to some embodiments. As previously discussed, the cost function is a function of one or more random variables. In the example of FIG. 30, a random variable is used to account for the uncertainty in a forecasted load 3002. Thus, the graphical representation 3000 includes a plot of forecasted load 3002 over the entire upcoming time domain. Possible values for control variables or decision variables for controlling an electrical system may be fluctuated to minimize an expected value of the cost function.

The uncertainty of certain elements of the cost function may be modeled at any given point in time using random variables. As a result, each point in time of the forecasted load 3002 may have a PDF 3010 associated therewith. The PDF 3010 may include an estimate of the probability of the value of the predicted load 3002 at the given time. For example, the PDF 3010 corresponding to point in time t₁ is shown in the graphical representation 3000.

In some embodiments, the PDF 3010 may include a normal distribution, a Laplace distribution, a uniform distribution, a chi-squared distribution, or some other distribution constructed based on observations of operation of the electrical system (e.g., the observed input data). In some embodiments, the PDF 3010 may include a single-variable distribution or a multi-variable distribution.

An expected value for the cost function at each of the points in time (e.g., t₁) during a future period of time may be determined based on the corresponding PDF or other uncertainty metric for the points (e.g., PDF 3010). By way of non-limiting example, a sum or integral of the PDF 3010 may be computed or estimated (e.g., using closed form or numeric integration techniques) to determine an expected value of the cost function at the points. The PDF 3010 may be determined, in some embodiments, as discussed above with reference to FIGS. 29A-29G.

In some embodiments, the expected value of the cost function at point t₁ may be determined by determining a probability weighted mean of the corresponding PDF 3010. This value is marked on the PDF 3010 by the symbol L^e. In some embodiments, however, the expected value of the cost function at point t₁ may be determined by computing or estimating a probability weighted mean on only a predetermined subset of the values of the PDF 3010 (e.g., the subset may include one or more sub-ranges, one or more discrete points, or combinations thereof). For example, a probability weighted average of the values of the PDF 3010 at points L⁻, L^e, and L⁺ may be determined instead of the probability weighted average of the entire PDF 3010.

In some embodiments, L⁻ and L⁺ may be located at the cumulative 10% and 90% locations, respectively, of the PDF 3010. In some embodiments, L⁻ and L⁺ may be located at the cumulative 30% and 70% locations, respectively, of the PDF 3010. In some embodiments, L⁻ and L⁺ may be located at integer multiples of a standard deviation away from L^e on the PDF 3010. By way of non-limiting example, a simple expected value calculation can involve running three scenarios weighted by each's relative probability: 1) mean −1 standard deviation, 2) mean, and 3) mean +1 standard deviation. In some embodiments, L⁻ and L⁺ may be located at non-equal distances from L^e on the PDF 3010 (e.g., L⁻ may be located a distance A % from L^e and L⁺ may be located at distance B % from L^e, where A % is different from B %). In embodiments where a relatively high degree of uncertainty is present, L⁻ and L⁺ may be located at relatively large distances from L^e. In embodiments where a relatively low degree of uncertainty is present, L⁻ and L⁺ may be located at relatively small distances from L^e.

FIG. 31 is a simplified flowchart illustrating a method 3100 of operating an electrical system, according to some embodiments. The method 3100 includes generating 3110 probability distribution functions (referenced as “PDFs” in FIG. 31) corresponding to probability density of a given random variable value occurring at different points in time of a future period of time. The PDFs may be the probability densities of given values of the random variables occurring. To get an expected value of a cost function, a PDF (e.g., f(l)) is multiplied by a cost function integrand, which is also a function of the random variable, and an integral is taken over a random variable range to get the expected value. Generating 3110 PDFs may include a process similar to that discussed above with respect to FIG. 29A. The probability distribution functions each take into consideration uncertainty of one or more random variables associated with the economic cost of operating the electrical system. In some embodiments, generating 3110 probability distribution functions corresponding to probabilities of occurrence of a given random variable value at different points of time of a future period of time includes accounting for a predicted load of the electrical system using a random variable. In some embodiments, generating 3110 probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time includes accounting for predicted power generated by a generator of the electrical system using a random variable. In some embodiments, generating 3110 probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time includes accounting for a predicted cost of operating a battery in the electrical system using a random variable.

In some embodiments, generating 3110 probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time includes accounting for predicted fluctuations in one or more external inputs using one or more random variables corresponding to the one or more external inputs. By way of non-limiting example, the one or more external inputs may include one or more of predicted weather data, predicted building occupation, predicted demand rates for energy supplied by an electrical grid, or combinations thereof.

In some embodiments, generating 3110 probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time includes accounting for predicted fluctuations in one or more process variables of the electrical system using one or more random variables corresponding to the one or more process variables. By way of non-limiting example, the one or more process variables may include one or more of an unadjusted net power, an unadjusted demand, an adjusted net power, an adjusted demand, a load, a generation, an energy storage system (ESS) charge, a generation rate for an ESS, an energy storage device state of charge (SoC), an energy storage device temperature, an electrical meter output, or combinations thereof.

In some embodiments, generating 3110 probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time includes accounting for predicted fluctuations in one or more configuration elements of the electrical system. By way of non-limiting example, the one or more configuration elements of the electrical system may include one or more of an energy storage system (ESS) configuration, an ESS efficiency, an ESS degradation, an electricity supply tariff, an electricity demand tariff, a minimum power input of the electrical system, an ESS state of charge (SoC) limit, an ESS power limit, weather data, or building occupancy.

The method 3100 also includes determining 3120 a set of control values for a set of control variables that correspond to a minimum expected value of the economic cost of operating the electrical system over the future period of time. It should be noted that in some embodiments the control variables themselves may be varied in a cost function to determine the minimum expected values. In some embodiments, however, other variables that effect the control variables may be varied. These other variables may be referred to as “decision variables.” Accordingly, decision values of the decision variables may be varied to determine the minimum expected value of the cost of operating the electrical system in order to determine an optimal set of control values for the control variables. The method 3100 further includes controlling 3130 the electrical system based on the determined set of control values.

In some embodiments, at least a portion of the electrical system controller 2800 (FIG. 28), operations discussed above with reference to the graphical representation 3000 of FIG. 30, the method 3100 of FIG. 31, or combinations thereof may be implemented as computer-readable instructions stored by one or more computer-readable storage media (e.g., non-transitory computer-readable storage media) (e.g., the storage device 2810 of FIG. 28). These computer-readable instructions are configured to instruct one or more processors (e.g., the processors 2820 of FIG. 28) to perform operations discussed above with reference to FIGS. 28-31.

Example Embodiments

The following are some example embodiments within the scope of the disclosure. In order to avoid complexity in providing the disclosure, not all of the examples listed below are separately and explicitly disclosed as having been contemplated herein as combinable with all of the others of the examples listed below and other embodiments disclosed hereinabove. Unless one of ordinary skill in the art would understand that these examples listed below (and the above disclosed embodiments) are not combinable, it is contemplated within the scope of the disclosure that such examples and embodiments are combinable.

Example 1

An electrical system controller, comprising: one or more data storage devices configured to store data corresponding to one or more random variables associated with operation of an electrical system; and one or more processors configured to: determine a set of control values for a set of control variables to effectuate a change to the electrical system toward meeting a controller objective for economical optimization of the electrical system, wherein the set of control values is determined as a function of the one or more random variables by the one or more processors utilizing an optimization algorithm; and control the electrical system based on the control values.

Example 2

The electrical system controller of Example 1, wherein the one or more random variables comprise a predicted load random variable corresponding to a predicted load of the electrical system.

Example 3

The electrical system controller according to any one of Examples 1 and 2, wherein the one or more random variables comprise a predicted generator profile random variable corresponding to a predicted generator power generation profile of an electrical power generator of the electrical system.

Example 4

The electrical system controller according to any one of Examples 1-3, wherein the one or more random variables comprise a system constraint random variable corresponding to a system constraint of the electrical system.

Example 5

The electrical system controller according to any one of Examples 1-4, wherein the one or more random variables comprise a cost element random variable corresponding to a cost of operating the electrical system.

Example 6

The electrical system controller according to any one of Examples 1-5, wherein the one or more processors are configured to utilize the optimization algorithm by generating a cost function including the one or more random variables and minimize an expected value of a cost of operating the electrical system.

Example 7

The electrical system controller of Example 6, wherein the cost function comprises an integral of a probability density function.

Example 8

The electrical system controller according to any one of Examples 6 and 7, wherein the cost function comprises a sum of a probability mass function.

Example 9

The electrical system controller according to any one of Examples 6-8, wherein the cost function takes into consideration time-dependence of cost elements of operating the electrical system over a future period of time.

Example 10

A method of operating an electrical system, the method comprising: generating probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time, the probability distribution functions each taking into consideration uncertainty of one or more random variables associated with the economic cost of operating the electrical system; determining a set of control values for a set of control variables that correspond to a minimum expected value of the economic cost of operating the electrical system over the future period of time; and controlling the electrical system based on the determined set of control values.

Example 11

The method of Example 10, wherein generating probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time comprises accounting for a predicted load of the electrical system using a random variable.

Example 12

The method according to any one of Examples 10 and 11, wherein generating probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time comprises accounting for predicted power generated by a generator of the electrical system using a random variable.

Example 13

The method according to an one of Examples 10-12, wherein generating probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time comprises accounting for a predicted cost of operating a battery in the electrical system using a random variable.

Example 14

The method according to any one of Examples 10-13, wherein generating probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time comprises accounting for predicted fluctuations in one or more external inputs using one or more random variables corresponding to the one or more external inputs.

Example 15

The method of Example 14, wherein the one or more external inputs comprise one or more of predicted weather data, predicted building occupation, predicted demand rates for energy supplied by an electrical grid, predicted time-of-use (ToU) supply charges, predicted local contracted or incentive maneuvers, or combinations thereof.

Example 16

The method according to any one of Examples 10-15, wherein generating probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time comprises accounting for predicted fluctuations in one or more process variables of the electrical system using one or more random variables corresponding to the one or more process variables.

Example 17

The method of Example 16, wherein the one or more process variables comprise one or more of an unadjusted net power, an unadjusted demand, an adjusted net power, a demand, a load, a generation, an energy storage system (ESS) charge, a generation rate for an ESS, an energy storage device state of charge (SoC), an energy storage device temperature, or an electrical meter output.

Example 18

The method according to any one of Examples 10-17, wherein generating probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time comprises accounting for predicted fluctuations in one or more configuration elements of the electrical system with one or more random variables.

Example 19

The method of Example 18, wherein the one or more configuration elements of the electrical system include one or more of an energy storage system (ESS) configuration, an ESS efficiency, an ESS degradation, an electricity supply tariff, an electricity demand tariff, a minimum power input of the electrical system, an ESS state of charge (SoC) limit, an ESS power limit, weather data, or building occupancy.

Example 20

One or more non-transitory computer-readable storage media including computer-readable instructions stored thereon, the computer-readable instructions configured to instruct one or more processors to: construct a cost function comprising an expected value of an economic cost of operating an electrical system, the cost function further including one or more control variables of the electrical system, the control variables corresponding to controllable features of the electrical system; determine control values for the control variables corresponding to a minimum value of the expected value of the economic cost of operating the electrical system; and issue one or more commands to the electrical system to operate according to the determined control values for the control variables.

Example 21

The one or more non-transitory computer-readable storage media of Example 20, wherein the minimum value of the expected value of the economic cost comprises a global minimum value.

Example 22

The one or more non-transitory computer-readable storage media of Example 20, wherein the minimum value of the expected value of the economic cost comprises a local minimum value.

Example 23

A method of operating an electrical system controller, the method comprising: storing, on one or more data storage devices, data corresponding to one or more random variables associated with operation of an electrical system; determining a set of control values for a set of control variables to effectuate a change to the electrical system toward meeting a controller objective for economical optimization of the electrical system, wherein the set of control values are determined as a function of one or more random variables by the one or more processors utilizing an optimization algorithm; and controlling the electrical system based on the control values.

Example 24

The method of Example 23, wherein the one or more random variables comprise a predicted load random variable corresponding to a predicted load of the electrical system.

Example 25

The method according to any one of Examples 23 and 24, wherein the one or more random variables comprise a predicted generator profile random variable corresponding to a predicted generator power generation profile of an electrical power generator of the electrical system.

Example 26

The method according to any one of Examples 23-25, wherein the one or more random variables comprise a system constraint random variable corresponding to a system constraint of the electrical system.

Example 27

The method according to any one of Examples 23-26, wherein the one or more random variables comprise a cost element random variable corresponding to a cost of operating the electrical system.

Example 28

The method according to any one of Examples 23-27, further comprising utilizing the optimization algorithm by generating a cost function including the one or more random variables and minimizing an expected value of the cost of operating the electrical system.

Example 29

The method of Example 28, wherein the cost function comprises an integral of a probability density function.

Example 30

The method according to any one of Examples 28 and 29, wherein the cost function comprises a sum of a probability mass function.

Example 31

The method according to any one of Examples 28-30, wherein the cost function takes into consideration time-dependence of cost elements of operating the electrical system over a future period of time.

Example 32

An electrical system controller configured to: generate probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time, the probability distribution functions each taking into consideration uncertainty of one or more random variables associated with the economic cost of operating the electrical system; determine a set of control values for a set of control variables that correspond to a minimum expected value of the economic cost of operating the electrical system over the future period of time; and control the electrical system based on the determined set of control values.

Example 33

The electrical system controller of Example 32, wherein the electrical system controller is configured to generate probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time by accounting for a predicted load of the electrical system using a random variable.

Example 34

The electrical system controller according to any one of Examples 32 and 33, wherein the electrical system controller is configured to generate probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time by accounting for predicted power generated by a generator of the electrical system using a random variable.

Example 35

The electrical system controller according to any one of Examples 32-34, wherein the electrical system controller is configured to generate probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time by accounting for a predicted cost of operating a battery in the electrical system using a random variable.

Example 36

The electrical system controller according to any one of Examples 32-35, wherein the electrical system controller is configured to generate probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time by accounting for predicted fluctuations in one or more external inputs using one or more random variables corresponding to the one or more external inputs.

Example 37

The electrical system controller of Example 36, wherein the one or more external inputs comprise one or more of predicted weather data, predicted building occupation, or predicted demand rates for energy supplied by an electrical grid.

Example 38

The electrical system controller according to any one of Example 32-37, wherein the electrical system controller is configured to generate probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time by accounting for predicted fluctuations in one or more process variables of the electrical system using one or more random variables corresponding to the one or more process variables.

Example 39

The electrical system controller of Example 38, wherein the one or more process variables comprise one or more of an unadjusted net power, an unadjusted demand, an adjusted net power, a demand, a load, a generation, an energy storage system (ESS) charge, a generation rate for an ESS, an energy storage device state of charge (SoC), an energy storage device temperature, or an electrical meter output.

Example 40

The electrical system controller according to any one of Examples 32-39, wherein the electrical system controller is configured to generate probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time by accounting for predicted fluctuations in one or more configuration elements of the electrical system.

Example 41

The electrical system controller of Example 40, wherein the one or more configuration elements of the electrical system include one or more of an energy storage system (ESS) configuration, an ESS efficiency, an ESS degradation, an electricity supply tariff, an electricity demand tariff, a minimum power input of the electrical system, an ESS state of charge (SoC) limit, an ESS power limit, weather data, or building occupancy.

Example 42

A method of operating an electrical system, the method comprising: constructing a cost function comprising an expected value of an economic cost of operating an electrical system, the cost function further including one or more control variables of the electrical system, the control variables corresponding to controllable features of the electrical system; determining control values for the control variables corresponding to a minimum value of the expected value of the economic cost of operating the electrical system; and issuing one or more commands to the electrical system to operate according to the determined control values for the control variables.

Example 43

The method of Example 42, wherein the minimum value of the expected value of the economic cost comprises a global minimum value.

Example 44

The method of Example 42, wherein the minimum value of the expected value of the economic cost comprises a local minimum value.

Example 45

A controller to optimize overall economics of operation of an electrical system, the controller comprising a communication interface and one or more processors. The communication interface is to provide a communication path with an electrical system. The one or more processors are to determine a set of control values for a set of control variables to effectuate a change to the electrical system toward meeting a controller objective for economical optimization of the electrical system during an upcoming time domain, wherein the set of control values are determined by the one or more processors utilizing an optimization algorithm (e.g., a continuous optimization algorithm) to identify the set of control values in accordance with one or more constraints and one or more cost elements associated with operation of the electrical system. The optimization algorithm takes into consideration one or more random variables. The one or more processors are also to provide the values for the set of control variables to the electrical system via the communication interface.

Example 46

A computer-implemented method of a controller to optimize overall economics of operation of an electrical system, the method comprising: receiving at a computing device a set of configuration elements specifying one or more constraints of the electrical system and defining one or more cost elements associated with operation of the electrical system; receiving at a computing device a set of process variables that provide one or more measurements of a state of the electrical system, the set of process variables to be used by the controller to determine progress toward meeting a controller objective for economical optimization of the electrical system; determining on a computing device a set of control values for a set of control variables to provide to the electrical system to effectuate a change to the electrical system toward meeting the controller objective for economical optimization of the electrical system (e.g., during an upcoming time domain), the determining the set of control values including utilization of an optimization algorithm to identify the set of control values in accordance with the one or more constraints and the one or more cost elements; providing the set of control values for the set of control variables to the electrical system; wherein determining a set of control values for a set of control variables to provide to the electrical system to effectuate a change to the electrical system toward meeting the controller objective for economical optimization of the electrical system comprises taking into consideration one or more random variables in determining the set of control values.

Example 47

A controller to improve operation of an electrical system, the controller comprising: a communication interface to couple to a communication path to an electrical system; and one or more processors to: determine a set of control values for a set of control variables to effectuate a change to the electrical system toward meeting a controller objective for optimization of the electrical system, wherein the set of control values are determined by the one or more processors utilizing an optimization algorithm to identify the set of control values in accordance with one or more constraints associated with operation of the electrical system, the optimization algorithm including one or more probability distribution functions; and provide the values for the set of control variables to the electrical system via the communication interface.

Example 48

A controller to improve operation of an electrical system, the controller comprising: an economic optimizer to: prepare an objective function comprising one or more random variables, the objective function to operate on a current set of values for a control parameter set X comprising a plurality of sets of control variables each to be applied during different time segments of an upcoming time domain, the objective function including one or more constraints associated with operation of the electrical system; and execute an optimization of the objective function based on the current set of values for a control parameter set X to determine an optimal set of values for control parameter set X; and a dynamic manager to: receive the values for the control parameter set X from the economic optimizer; determine a set of control values for a set of control variables to effectuate a change to the electrical system toward meeting a controller objective for operation of the electrical system, the set of control values determined based on a control law and the values for the control parameter set X; and provide the values for the control variable to the electrical system via the a communication interface coupling to a communication path to the electrical system.

Example 49

A computer-implemented method of a controller of an electrical system including a battery, the method comprising: receiving from a sensor of an electrical system a state of charge of a battery of the electrical system; determining on a computing device a battery degradation cost of a battery of the electrical system based on the state of charge of the battery; determining values for a set of control variables to provide to the electrical system to effectuate a change to the electrical system toward meeting a controller objective for economical optimization of the electrical system, wherein the values are determined based on the battery degradation cost, one or more cost elements, and/or one or more constraints on the electrical system, wherein the values are also determined based on one or more random variables; providing the values for the set of control variables to the electrical system.

Example 50

An electrical system controller to optimize overall economics of operation of an electrical system, the controller comprising: a first computing device (e.g., an economic optimizer) to determine a control plan for managing control of the electrical system during an upcoming time domain and provide the control plan as output, the control plan including a plurality of sets of parameters each to be applied for a different time segment within the upcoming time domain, each of the plurality of sets of parameters determined based on one or more random variables; and a second computing device (e.g., a dynamic manager, or high speed controller) to determine a set of control values for a set of control variables for a given time segment of the upcoming time domain and provide the set of control values to the electrical system to effectuate a change to the electrical system toward meeting the controller objective for economical optimization of the electrical system during the upcoming time domain, wherein the second computing device determines the set of control values based on a control law and a set of values for a given set of parameters of the plurality of sets of parameters of the control plan, wherein the given set of parameters corresponds to an upcoming time segment.

Example 51

A computer-implemented method to optimize overall economics of operation of an electrical system, the method comprising: determining on a first computing device a control plan that includes values of a plurality of sets of control parameters, each set of the plurality of sets to be applied for a different time segment within an upcoming time domain, each set of the plurality of sets determined based at least in part on one or more random variables; providing the control plan to a second computing device for managing control of the electrical system during the upcoming time domain; and providing, by the second computing device, a set of values to the electrical system to effectuate a change to the electrical system toward meeting an objective for economical optimization of the electrical system during the upcoming time domain, the set of values provided according to the control plan.

Example 52

An electrical system controller to optimize overall economics of operation of an electrical system, the controller comprising: a first computing device (e.g., an economic optimizer) to determine an optimal set of values for a control parameter set X based at least in part on one or more random variables and provide the optimal set of values as output for managing control of the electrical system during an upcoming time domain, the control parameter set X including a plurality of sets of parameters each to be applied for a different time segment within the upcoming time domain, wherein the first computing device determines the optimal set of values for the control parameter set X based on a set of configuration elements and a set of process variables, the set of configuration elements specifying one or more constraints of the electrical system and defining one or more cost elements associated with operation of the electrical system, the set of process variables providing one or more measurements of a state of the electrical system; a second computing device (e.g., dynamic manager, or high speed controller) to determine a set of control values for a set of control variables for a given time segment of the upcoming time domain and to provide the set of control values to the electrical system, wherein the second computing device determines the set of control values based on a control law and a set of values for a given set of parameters of the plurality of sets of parameters of the control parameter set X, wherein the given set of parameters corresponds to an upcoming time segment.

Example 53

A computer-implemented method to optimize overall economics of operation of an electrical system, the method comprising: determining on a first computing device, based at least in part on one or more random variables, an optimal set of values for a control parameter set X that includes a plurality of sets of control parameters each to be applied for a different time segment within an upcoming time domain; providing the optimal set of values for the control parameter set X to a second computing device for managing control of the electrical system during the upcoming time domain; determining on a second computing device a set of control values for a set of control variables corresponding to a given time segment of the upcoming time domain, the determining based on a control law and a set of values for a given set of control parameters of the plurality of sets of control parameters of the control parameter set X, wherein the given set of control parameters corresponds to the given time segment; providing, by the second computing device, the set of control values for the set of control variables to the electrical system to effectuate a change to the electrical system toward meeting a controller objective for economical optimization of the electrical system during the upcoming time domain.

Example 54

A computer-implemented method to optimize overall economics of operation of an electrical system, the method comprising: receiving at a first computing device a set of configuration elements specifying one or more constraints of the electrical system and defining one or more cost elements associated with operation of the electrical system; receiving at the first computing device a set of process variables that provide one or more measurements of a state of the electrical system, the set of process variables to be used by the controller to determine progress toward meeting a controller objective for economical optimization of the electrical system; determining on the first computing device based on one or more random variables an optimal set of values for a control parameter set X that includes a plurality of sets of parameters each to be applied for a different time segment within an upcoming time domain; providing the optimal set of values for the control parameter set X to a second computing device for managing control of the electrical system during the upcoming time domain; receiving at the second computing device the set of values for the control parameter set X; receiving at the second computing device the set of process variables; determine on the second computing device a set of control values for a set of control variables for a given time segment of the upcoming time domain a control law and a set of values for a given set of parameters of the plurality of sets of parameters of the control parameter set X, wherein the given set of parameters corresponds to an upcoming time segment; providing, by the second computing device, the set of control values for the set of control variables to the electrical system to effectuate a change to the electrical system toward meeting the controller objective for economical optimization of the electrical system during the upcoming time domain.

Example 55

A controller to control an electrical system, the controller comprising: a communication interface to provide a communication path with an electrical system; and one or more processors to: receive, via the communication interface, one or more measurements of a state of the electrical system, each measurement designated as a value for a process variable in a set of process variables; obtain a set of control parameters to be applied for an extended time segment, each parameter of the set of control parameters specifying a behavior of a control law of the controller, each parameter of the set of control parameters determined based on one or more random variables; apply the control law by comparing the set of process variables to the set of control parameters to determine a value for each control variable of a set of control variables, each control variable controlling a component of the electrical system, the value of each control variable determined by the control law; and provide the values for the set of control variables to the electrical system, via the communication interface, to effectuate a change to one or more components of the electrical system.

Example 56

A computer-implemented method of a controller of an electrical system, the method comprising: obtaining a set of control parameters to be applied for an extended time segment of an upcoming time domain, each parameter of the set of control parameters specifying a behavior (or property) of a control law of the controller under various conditions, each parameter of the set of control parameters determined based on one or more random variables; receiving a set of process variables that provide one or more measurements of a state of the electrical system; applying the control law, including comparing the set of process variables to the set of control parameters according to the control law, to determine a value for each control variable of a set of control variables, each control variable controlling a component of the electrical system, the value of each control variable determined by the control law (e.g., to promote and maintain compliance of the electrical system with one or more conditions of the control law specified by the set of control parameters); and providing the values for the set of control variables to the electrical system to effectuate a change to one or more components of the electrical system (e.g., to promote and maintain compliance of the electrical system with one or more conditions of the control law specified by the set of control parameters).

Example 57

A computer-readable storage medium including computer-readable instructions stored thereon, the computer-readable instructions configured to instruct one or more processors to perform at least a portion of the method according to any one of Examples 10-19, 23-31, and 42-44.

Example 58

A means for performing at least a portion of the method according to any one of Examples 10-19, 23-31, and 42-44.

Example 59

The electrical system controller of Example 1, wherein the one or more random variables comprise one or more cost element random variables corresponding to one or more costs of operating the electrical system.

Example 60

The electrical system controller of Example 59, wherein the one or more costs of operating the electrical system comprise a net electrical cost of electrical power from an electrical grid and a net equipment operation cost of operating equipment of the electrical system.

Example 61

The electrical system controller of Example 60, wherein the net equipment operation cost of operating the equipment of the electrical system comprises an equipment degradation costs of one or more energy storage systems, one or more generators, or combinations thereof.

Example 62

The electrical system controller of Example 59, wherein the one or more costs of operating the electrical system comprise a net electrical cost of electrical power from an electrical grid, the net electrical cost of electrical power from the electrical grid comprising two or more different cost elements.

Example 63

The electrical system controller of Example 62, wherein the two or more different cost elements comprise two or more of a time-of-use (ToU) supply charge, a demand charge, or a local contracted or incentive maneuver.

Example 64

The electrical system controller of Example 1, wherein the one or more processors are configured to utilize the optimization algorithm by generating a cost function including the one or more random variables and minimize an expected value of the cost function.

Example 65

The electrical system controller of Example 1, wherein the one or more random variables are determined by analyzing past error in predicting one or more system behaviors.

Example 66

The method of Example 10, wherein generating a probability distribution function corresponding to probability density of a given random variable value occurring at different points in time of a future period of time comprises generating one or more random variables associated with operating the electrical system, wherein generating one or more random variables comprises determining uncertainties of the one or more random variables by comparing past predicted values associated with operating the electrical system to observed values associated with operating the electrical system.

Example 67

The method of Example 66, wherein comparing past predicted values to observed values comprises: measuring error between the past predicted values and the observed values; and correlating the measured error to a probability distribution function.

Example 68

One or more non-transitory computer-readable storage media including computer-readable instructions stored thereon, the computer-readable instructions configured to instruct one or more processors to: construct a cost function comprising an expected value of an economic cost of operating an electrical system over a future period of time, the cost function further including one or more decision variables, the one or more decision variables related to controllable features of the electrical system, the cost function including cost elements corresponding to net electricity payments to an electrical grid plus net equipment operating costs; determine optimal values for the decision variables corresponding to a minimum value of the expected value of the economic cost of operating the electrical system over the future period of time; and issue one or more commands to the electrical system to operate according to the determined optimal values for the decision variables.

Example 69

The one or more non-transitory computer-readable storage media of Example 68, wherein the net electricity payments to the electrical grid comprise two or more of a time-of-use (ToU) supply charge, a demand charge, or a local contracted or incentive maneuver.

Example 70

The one or more non-transitory computer-readable storage media of Example 68, wherein the net equipment operating costs include a cost of degradation of electrical equipment of the electrical system.

Example 71

An electrical power system controller, comprising: one or more data storage devices; one or more observed input data interfaces configured to receive observed input data, the observed input data corresponding to observed values of one or more input variables during one or more previous periods of time; and one or more processors configured to: store the observed input data on the one or more data storage devices; generate one or more uncertainty metrics based on the observed input data, the one or more uncertainty metrics configured to indicate an uncertainty of the one or more input variables; generate one or more random variables based on the one or more uncertainty metrics to forecast input data during a future period of time; generate a cost function corresponding to a cost of operating the electrical power system during the future period of time, wherein the cost function is a function of the one or more random variables; and control operation of the electrical power system during the future period of time based on an optimization of the cost function.

Example 72

The electrical power system controller of Example 71, wherein the observed input data comprises observed electrical load data of one or more loads of the electrical power system.

Example 73

The electrical power system controller according to any one of Examples 71 and 72, wherein the observed input data comprises observed electrical power generation data of one or more electrical power generators of the electrical power system.

Example 74

The electrical power system controller according to any one of Examples 71-73, wherein the observed in put data comprises observed market data.

Example 75

The electrical power system controller according to any one of Examples 71-74, wherein the one or more uncertainty metrics comprise functions of sample variances of the observed values as functions of time of day.

Example 76

The electrical power system controller according to any one of Examples 71-76, wherein one or more uncertainty metrics comprise functions of error between the observed values and forecasted values.

Example 77

The electrical power system controller according to any one of Examples 71-76, wherein: the one or more data storage devices are configured to store historic forecasted input data, the historic forecasted input data including one or more previously forecasted values of the one or more input variables as previously forecasted for the one or more previous periods of time; and the one or more uncertainty metrics are generated based on a comparison between the observed input data and the historic forecasted input data.

Example 78

The electrical power system controller of Example 77, wherein the one or more uncertainty metrics are based on an error between the observed input data and the historic forecasted input data.

Example 79

The electrical power system controller of Example 78, wherein the one or more uncertainty metrics comprise a mean square of the error between the observed input data and the historic forecasted input data.

Example 80

The electrical power system controller of Example 78, wherein the one or more uncertainty metrics comprise a variance or standard deviation of the error between the observed input data and the historic forecasted input data.

Example 81

The electrical power system controller of Example 71, wherein the one or more uncertainty metrics comprise a sample variance or sample standard deviation of the observed input data.

Example 82

The electrical power system controller according to any one of Examples 71, wherein the one or more uncertainty metrics comprise an uncertainty metric profile including varying values of an uncertainty metric at different points of time during a period of time, the period of time correlated to the future period of time and the one or more previous periods of time.

Example 83

The electrical power system controller of Example 82, wherein the uncertainty metric profile comprises an uncertainty of a forecasted load profile, the forecasted load profile corresponding to a forecasted load of the electrical power system at the different points of time during the future period of time.

Example 84

The electrical power system controller of Example 82, wherein the uncertainty metric profile comprises a predicted electrical power generation profile, the predicted electrical power generation profile corresponding to a predicted electrical power generation of one or more electrical power generators of the electrical power system at the different points of time during the future period of time.

Example 85

The electrical power system controller according to any one of Examples 71-84, wherein the observed input data includes external input data, process variable data including measurements fed back from an electrical power system, or a combination of the external input data and the process variable data.

Example 86

A method of controlling an electrical power system, the method comprising: receiving observed input data comprising external input data, the observed input data corresponding to observed values of one or more input variables during one or more previous periods of time; storing the observed input data on the one or more data storage devices; generating one or more uncertainty metrics based on the observed input data, the one or more uncertainty metrics configured to indicate an uncertainty of the one or more input variables; generating one or more random variables based on the one or more uncertainty metrics to forecast the input data during a future period of time; generating a cost function corresponding to a cost of operating the electrical power system during the future period of time, wherein the cost function is a function of the one or more random variables; and controlling operation of the electrical power system during the future period of time based on an optimization of the cost function.

Example 87

The method of Example 86, wherein generating one or more random variables comprises generating a random variable corresponding to an uncertainty of one or more system constraints.

Example 88

The method according to any one of Examples 86 and 87, wherein generating one or more random variables comprises generating a random variable corresponding to an uncertainty of one or more cost elements.

Example 89

The method of Example 88, wherein the one or more cost elements comprise a net electrical cost of electrical power from an electrical grid and a net equipment operation cost of operating equipment of the electrical system.

Example 90

The method of Example 89, wherein the net equipment operation cost of operating the equipment of the electrical system comprises an equipment degradation cost of one or more energy storage systems, one or more electrical power generators, or combinations thereof.

Example 91

The method of Example 88, wherein the one or more cost elements comprise a net electrical cost of electrical power from an electrical grid, the net electrical cost of electrical power from the electrical grid including two or more of a time-of-use (ToU) supply charge, a demand charge, or a local contracted maneuver or incentive maneuver.

Example 92

The method of Example 88, wherein the one or more cost elements comprise a net electrical cost of electrical power from an electrical grid, the net electrical cost of electrical power from the electrical grid including a time-of-use (ToU) supply charge and a demand charge.

Example 93

The method according to any one of Examples 86-92, wherein generating one or more random variables comprises generating a random variable corresponding to an uncertainty of a cost of operating a battery.

Example 94

The method according to any one of Examples 86-93, wherein generating one or more random variables comprises generating a random variable corresponding to an uncertainty of fluctuations in one or more external inputs.

Example 95

The method of Example 94, wherein the one or more external inputs comprise one or more of predicted weather data, predicted building occupation, predicted demand rates for energy supplied by an electrical grid, predicted time of use (ToU) supply charges, predicted local contracted or incentive maneuvers, or combinations thereof.

Example 96

The method according to any one of Examples 86-95, wherein generating one or more random variables comprises generating a random variable corresponding to an uncertainty of fluctuations in one or more process variables at the different points of time during the future period of time.

Example 97

The method of Example 96, wherein the one or more process variables comprise one or more of an unadjusted net power, an unadjusted demand, an adjusted net power, a demand, a load, a generation, an energy storage system (ESS) charge, a generation rate for an ESS, an energy storage system (ESS) state of charge (SoC), an ESS temperature, or an electrical meter output.

Example 98

The method according to any one of Examples 86-97, wherein the optimization of the cost function comprises minimization of an expected value of an economic cost of operating the electrical power system during the future time period.

Example 99

The method according to any one of Examples 86-98, wherein the observed input data comprises observed electrical load data of one or more loads of the electrical power system.

Example 100

The method according to any one of Examples 86-99, wherein the observed input data comprises observed electrical power generation data of one or more electrical power generators of the electrical power system.

Example 101

The method according to any one of Examples 86-100, wherein the observed in put data comprises observed market data.

Example 102

The method according to any one of Examples 86-101, wherein the one or more uncertainty metrics comprise functions of sample variances of the observed values as functions of time of day.

Example 103

The method according to any one of Examples 86-102, wherein one or more uncertainty metrics comprise functions of error between the observed values and forecasted values.

Example 104

The method according to any one of Examples 86-103, wherein: the one or more data storage devices are configured to store historic forecasted input data, the historic forecasted input data including one or more previously forecasted values of the one or more input variables as previously forecasted for the one or more previous periods of time; and the one or more uncertainty metrics are generated based on a comparison between the observed input data and the historic forecasted input data.

Example 105

The method of Example 104, wherein the one or more uncertainty metrics are based on an error between the observed input data and the historic forecasted input data.

Example 106

The method of Example 105, wherein the one or more uncertainty metrics comprise a mean square of the error between the observed input data and the historic forecasted input data.

Example 107

The method controller of Example 105, wherein the one or more uncertainty metrics comprise a variance or standard deviation of the error between the observed input data and the historic forecasted input data.

Example 108

The method according to any one of Examples 86-107, wherein the one or more uncertainty metrics comprise a sample variance or sample standard deviation of the observed input data.

Example 109

The method according to any one of Examples 86-108, wherein the one or more uncertainty metrics comprise an uncertainty metric profile including varying values of an uncertainty metric at different points of time during a period of time, the period of time correlated to the future period of time and the one or more previous periods of time.

Example 110

The method of Example 109, wherein the uncertainty metric profile comprises an uncertainty of a forecasted load profile, the forecasted load profile corresponding to a forecasted load of the electrical power system at the different points of time during the future period of time.

Example 111

The method of Example 109, wherein the uncertainty metric profile comprises a predicted electrical power generation profile, the predicted electrical power generation profile corresponding to a predicted electrical power generation of one or more electrical power generators of the electrical power system at the different points of time during the future period of time.

Example 112

The method according to any one of Examples 86-111, wherein the observed input data includes external input data, process variable data including measurements fed back from an electrical power system, or a combination of the external input data and the process variable data.

Example 113

One or more non-transitory computer-readable storage media comprising computer-readable instructions stored thereon, the computer readable instructions configured to instruct one or more processors to perform at least a portion of the method of any one of Examples 86-112.

Example 114

A means for performing at least a portion of the method according to any one of Examples 86-112.

The described features, operations, or characteristics may be arranged and designed in a wide variety of different configurations and/or combined in any suitable manner in one or more embodiments. Thus, the detailed description of the embodiments of the systems and methods is not intended to limit the scope of the disclosure, as claimed, but is merely representative of possible embodiments of the disclosure. In addition, it will also be readily understood that the order of the steps or actions of the methods described in connection with the embodiments disclosed may be changed as would be apparent to those skilled in the art. Thus, any order in the drawings or Detailed Description is for illustrative purposes only and is not meant to imply a required order, unless specified to require an order.

Embodiments may include various acts, which may be embodied in machine-executable instructions to be executed by a general-purpose or special-purpose computer (or other electronic device). Alternatively, the acts may be performed by hardware components that include specific logic for performing the acts, or by a combination of hardware, software, and/or firmware.

Embodiments may also be provided as a computer program product including a computer-readable storage medium having stored instructions thereon that may be used to program a computer (or other electronic device) to perform processes described herein. The computer-readable storage medium may include, but is not limited to: hard drives, floppy diskettes, optical disks, CD-ROMs, DVD-ROMs, ROMs, RAMs, EPROMs, EEPROMs, magnetic or optical cards, solid-state memory devices, or other types of medium/machine-readable medium suitable for storing electronic instructions.

As used herein, a software module or component may include any type of computer instruction or computer executable code located within a memory device and/or computer-readable storage medium. A software module may, for instance, comprise one or more physical or logical blocks of computer instructions, which may be organized as a routine, program, object, component, data structure, etc., that performs one or more tasks or implements particular abstract data types.

In certain embodiments, a particular software module may comprise disparate instructions stored in different locations of a memory device, which together implement the described functionality of the module. Indeed, a module may comprise a single instruction or many instructions, and may be distributed over several different code segments, among different programs, and across several memory devices. Some embodiments may be practiced in a distributed computing environment where tasks are performed by a remote processing device linked through a communications network. In a distributed computing environment, software modules may be located in local and/or remote memory storage devices. In addition, data being tied or rendered together in a database record may be resident in the same memory device, or across several memory devices, and may be linked together in fields of a record in a database across a network.

The foregoing specification has been described with reference to various embodiments, including the best mode. However, those skilled in the art appreciate that various modifications and changes can be made without departing from the scope of the present disclosure and the underlying principles of the invention. Accordingly, this disclosure is to be regarded in an illustrative rather than a restrictive sense, and all such modifications are intended to be included within the scope thereof. Likewise, benefits, other advantages, and solutions to problems have been described above with regard to various embodiments. However, benefits, advantages, solutions to problems, and any element(s) that may cause any benefit, advantage, or solution to occur or become more pronounced are not to be construed as a critical, required, or essential feature or element.

As used herein, the terms “comprises,” “comprising,” or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Also, as used herein, the terms “coupled,” “coupling,” or any other variation thereof, are intended to cover a physical connection, an electrical connection, a magnetic connection, an optical connection, a communicative connection, a functional connection, and/or any other connection.

Principles of the present disclosure may be reflected in a computer program product on a tangible computer-readable storage medium having computer-readable program code means embodied in the storage medium. Any suitable computer-readable storage medium may be utilized, including magnetic storage devices (hard disks, floppy disks, and the like), optical storage devices (CD-ROMs, DVDs, Blu-Ray discs, and the like), flash memory, and/or the like. These computer program instructions may be loaded onto a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions that execute on the computer or other programmable data processing apparatus create means for implementing the functions specified. These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified. The computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer-implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified.

Principles of the present disclosure may be reflected in a computer program implemented as one or more software modules or components. As used herein, a software module or component (e.g., engine, system, subsystem) may include any type of computer instruction or computer-executable code located within a memory device and/or computer-readable storage medium. A software module may, for instance, comprise one or more physical or logical blocks of computer instructions, which may be organized as a routine, a program, an object, a component, a data structure, etc., that perform one or more tasks or implement particular data types.

In certain embodiments, a particular software module may comprise disparate instructions stored in different locations of a memory device, which together implement the described functionality of the module. Indeed, a module may comprise a single instruction or many instructions, and may be distributed over several different code segments, among different programs, and across several memory devices. Some embodiments may be practiced in a distributed computing environment where tasks are performed by a remote processing device linked through a communications network. In a distributed computing environment, software modules may be located in local and/or remote memory storage devices. In addition, data being tied or rendered together in a database record may be resident in the same memory device, or across several memory devices, and may be linked together in fields of a record in a database across a network.

Suitable software to assist in implementing the invention is readily provided by those of skill in the pertinent art(s) using the teachings presented here and programming languages and tools, such as Java, Pascal, C++, C, database languages, APIs, SDKs, assembly, firmware, microcode, and/or other languages and tools.

Embodiments as disclosed herein may be computer-implemented in whole or in part on a digital computer. The digital computer includes a processor performing the required computations. The computer further includes a memory in electronic communication with the processor to store a computer operating system. The computer operating systems may include, but are not limited to, MS-DOS, Windows, Linux, Unix, AIX, CLIX, QNX, OS/2, and Apple. Alternatively, it is expected that future embodiments will be adapted to execute on other future operating systems.

In some cases, well-known features, structures or operations are not shown or described in detail. Furthermore, the described features, structures, or operations may be combined in any suitable manner in one or more embodiments. It will also be readily understood that the components of the embodiments as generally described and illustrated in the figures herein could be arranged and designed in a wide variety of different configurations.

Various operational steps, as well as components for carrying out operational steps, may be implemented in alternate ways depending upon the particular application or in consideration of any number of cost functions associated with the operation of the system, e.g., one or more of the steps may be deleted, modified, or combined with other steps.

While the principles of this disclosure have been shown in various embodiments, many modifications of structure, arrangements, proportions, the elements, materials and components, used in practice, which are particularly adapted for a specific environment and operating requirements, may be used without departing from the principles and scope of this disclosure. These and other changes or modifications are intended to be included within the scope of the present disclosure.

The scope of the present invention should, therefore, be determined only by the following claims.

Claims

1. An electrical system controller, comprising: one or more data storage devices configured to store data corresponding to one or more random variables associated with operation of an electrical system; andone or more processors to: determine a set of control values for a set of control variables to effectuate a change to the electrical system toward meeting a controller objective for economical optimization of the electrical system, wherein the set of control values is determined as a function of the one or more random variables by the one or more processors utilizing an optimization algorithm; andcontrol the electrical system based on the control values.

2. The electrical system controller of claim 1, wherein the one or more random variables comprise a predicted load random variable that represents an uncertainty in a predicted load of the electrical system.

3. The electrical system controller of claim 1, wherein the one or more random variables comprise a predicted generator profile random variable corresponding to a predicted generator power generation profile of an electrical power generator of the electrical system.

4. The electrical system controller of claim 1, wherein the one or more random variables represent an uncertainty in predicted power generation of the electrical system.

5. The electrical system controller of claim 1, wherein the one or more random variables comprise a system constraint random variable corresponding to a system constraint of the electrical system.

6. The electrical system controller of claim 1, wherein the one or more random variables comprise one or more cost element random variables corresponding to one or more costs of operating the electrical system.

7. The electrical system controller of claim 6, wherein the one or more costs of operating the electrical system comprise a net electrical cost of electrical power from an electrical grid and a net equipment operation cost of operating equipment of the electrical system.

8. The electrical system controller of claim 7, wherein the net equipment operation cost of operating the equipment of the electrical system comprises an equipment degradation costs of one or more energy storage systems, one or more generators, or combinations thereof.

9. The electrical system controller of claim 6, wherein the one or more costs of operating the electrical system comprise a net electrical cost of electrical power from an electrical grid, the net electrical cost of electrical power from the electrical grid comprising two or more different cost elements.

10. The electrical system controller of claim 9, wherein the two or more different cost elements comprise two or more of a time-of-use (ToU) supply charge, a demand charge, or a local contracted or incentive maneuver.

11. The electrical system controller of claim 1, wherein the one or more processors are configured to utilize the optimization algorithm by generating a cost function including the one or more random variables and minimize an expected value of the cost function.

12. The electrical system controller of claim 1, wherein the one or more random variables are determined by analyzing past error in predicting one or more system behaviors.

13. A method of operating an electrical system, the method comprising: generating probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time, the probability distribution functions each taking into consideration uncertainty of one or more random variables associated with the economic cost of operating the electrical system;constructing a cost function based on the one or more random variables;determining a set of control values for a set of control variables that correspond to a minimum expected value of the economic cost of operating the electrical system over the future period of time based on the cost function; andcontrolling the electrical system based on the determined set of control values.

14. The method of claim 13, wherein generating probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time comprises accounting for a predicted load of the electrical system using a random variable.

15. The method of claim 13, wherein generating probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time comprises accounting for predicted power generated by a generator of the electrical system using a random variable.

16. The method of claim 13, wherein generating probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time comprises accounting for a predicted cost of operating a battery in the electrical system using a random variable.

17. The method of claim 13, wherein generating probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time comprises accounting for predicted fluctuations in one or more external inputs using one or more random variables corresponding to the one or more external inputs.

18. The method of claim 17, wherein the one or more external inputs comprise one or more of predicted weather data, predicted building occupation, predicted demand rates for energy supplied by an electrical grid, predicted time-of-use (ToU) supply charges, predicted local contracted or incentive maneuvers, or combinations thereof.

19. The method of claim 13, wherein generating probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time comprises accounting for predicted fluctuations in one or more process variables of the electrical system using one or more random variables corresponding to the one or more process variables.

20. The method of claim 19, wherein the one or more process variables comprise one or more of an unadjusted net power, an unadjusted demand, an adjusted net power, a demand, a load, a generation, an energy storage system (ESS) charge, a generation rate for an ESS, an energy storage device state of charge (SoC), an energy storage device temperature, or an electrical meter output.

21. The method of claim 13, wherein generating probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time comprises accounting for predicted fluctuations in one or more configuration elements of the electrical system with one or more random variables.

22. The method of claim 21, wherein the one or more configuration elements of the electrical system include one or more of an energy storage system (ESS) configuration, an ESS efficiency, an ESS degradation, an electricity supply tariff, an electricity demand tariff, a minimum power input of the electrical system, an ESS state of charge (SoC) limit, or an ESS power limit.

23. The method of claim 13, wherein generating a probability distribution functions corresponding to probability density of a given random variable value occurring at different points in time of a future period of time comprises generating one or more random variables associated with operating the electrical system, wherein generating one or more random variables comprises determining uncertainties of the one or more random variables by comparing past predicted values associated with operating the electrical system to observed values associated with operating the electrical system.

24. The method of claim 23, wherein comparing past predicted values to observed values comprises: measuring error between the past predicted values and the observed values; andcorrelating the measured error to a probability distribution function.

25. The method of claim 13, wherein the one or more random variables are associated with a probability that an electrical grid supplying power to the electrical system will accept a bid to sell electrical power back to the grid at a future time.

26. One or more non-transitory computer-readable storage media including computer-readable instructions stored thereon, the computer-readable instructions configured to instruct one or more processors to: construct a cost function comprising an expected value of an economic cost of operating an electrical system over a future period of time, the cost function further including one or more decision variables, the one or more decision variables related to controllable features of the electrical system, the cost function including cost elements corresponding to net electricity payments to an electrical grid plus net equipment operating costs;determine optimal values for the decision variables corresponding to a minimum value of the expected value of the economic cost of operating the electrical system over the future period of time; andissue one or more commands to the electrical system to operate according to the determined optimal values for the decision variables.

27. The one or more non-transitory computer-readable storage media of claim 26, wherein the net electricity payments to the electrical grid comprise two or more of a time-of-use (ToU) supply charge, a demand charge, or a local contracted or incentive maneuver.

28. The one or more non-transitory computer-readable storage media of claim 26, wherein the net equipment operating costs include a cost of degradation of electrical equipment of the electrical system.