Patent Application
20200310369
The present invention relates generally to System and method for scheduling electric generators, in particular, generating System and method for scheduling electric generators using decision diagrams.
The UCP involves determining schedules for generators so as to meet a target power demand. As used herein, a schedule specifies for each time instant in a horizon of T time steps—Boolean variables that indicate whether generators should be OFF or ON for a particular time step and continuous variables indicating the power to be produced and the maximum power that can be produced at that time. Generators typically include nuclear, thermal, and renewable power sources. Generators are subject to constraints such as stable operating level, rate of ramping up or down, and the amount of time the generator is ON or OFF, which makes the UCP a difficult combinatorial optimization task, which arises when the operation of N individual generators is scheduled over T time steps, such that the total cost of producing electrical energy that meets the target power demand is minimized, while simultaneously observing the operational constraints of individual generators.
Conventionally, the UCP is typically formulated as a deterministic optimization problem where the outputs of the generators are assumed to be fully dispatchable, e.g., fossil-burned, nuclear, and the future power demand is assumed to be completely known or predictable. Various combinatorial optimization methods are known for solving deterministic UCP, including methods based on dynamic programming, Lagrange relaxation, and mixed integer programming. Despite the plethora of solution approaches there still exists a need to develop effective solution approaches for the UCP.
It is known that the UCP can be formulated as mixed integer program (MIP). It is also well known that the UCP can be formulated and solved using a state-space representation of the generator's operations and solved using dynamic programming. This invention provides a new formulation of the UCP that leverages the state space representation to formulate the UCP as a MIP.
The present invention discloses a novel representation of the space of the feasible operations of the generator using decision diagrams. Leveraging this decision diagram description, the present invention presents a novel network flow formulation for the UCP. Theoretically, the formulation leads to a stronger relaxation and leads to improved computational performance.
The embodiments of the invention provide a method for formulating a MIP by embedding a decision diagram representing the feasible operations of the generator units in the MIP formulation. The key insight from the embedding is that it allows to tighten the feasible set of operations for the generators by knowing the state of the generator, number of hours on or off, at each time period. This allows to readily incorporate the minimum up time and down time requirements on the generators.
Some embodiments of the present invention are based on realization of a power generation planning system for controlling on/off sequence of generators according to operational parameters. The power generation planning system includes an interface to receive the operational parameters including a power demand, state-data of the generators and operational histories of the generators from a power control system; a memory to store an objective function, a mixed-integer programming solver, generator parameters of each the generators and planning modules including a state-space representation module, a variable assignment module, a network flow module and a tight constraint module; a processor to perform the planning modules based on the operational parameters received by the interface. In this case, the processor is configured to construct decision diagrams for each of the generators according to the generator parameters by using the state-space representation module; generate arc-variables representing state-transformations of the generators by assigning binary variables to arcs of the decision diagrams by using the variable assignment module; generate network flow constraints to represent feasible operations of each of the generators; generate tight constraints of each of the generators by using the tight constraint module and formulate a mixed-integer problem; solve the mixed-integer problem, by using the mixed-integer programming solver, based on the states-data of the generators, the network flow constraints and the tight constraints; and transmit on/off sequence data of each of the generators obtained from the solved mixed-integer problem to a power control system via the interface.
According to embodiments of the present invention, the bounds that are used in the present invention can be substantially tightened based on the knowledge of the number of hours that the generator has been on or off at each time period.
According to another embodiment of the present invention, a power generation planning system for controlling on/off sequence of generators according to operational parameters, includes an interface to receive the operational parameters including as a set of scenarios wherein each scenario includes a demand and reserve pattern, state-data of the generators and operational histories of the generators from a power control system; a memory to store an objective function, a mixed-integer programming solver, generator parameters of each the generators and planning modules including a state-space representation module, a variable assignment module, a network flow module and a tight constraint module; a processor to perform the planning modules based on the operational parameters received by the interface. In this case, the processor is configured to construct decision diagrams for each of the generators according to the generator parameters by using the state-space representation module; generate path variables representing the state transformations of the generators using the variable assignment module and continuous variables for each scenario representing the power production of generators; generate restricted master problem constraints to represent feasible operations of each of the generators including the tight constraint module for each scenario and formulate a mixed-integer problem; solve the mixed-integer problem, by using Branch-and-Price algorithm, based on the states-data of the generators, the restricted master problem constraints including the tight constraints; and transmit on/off sequence data of each of the generators obtained from the solved mixed-integer problem to a power control system via the interface.
The embodiments of the invention allow for a tight representation of the feasible operations of the generators. Further, as some embodiments of the present invention can reduce enormous computing time, the present invention can substantially reduce the computational load and be improvements in computer functionality and improvements in an existing technology regarding computer implementation.
The embodiments of the invention allow for determining generator schedules in the presence of uncertainty in the power demand due to the inclusion of renewable energy sources such as solar and wind. The invention presents a scenario-based two-stage mixed integer stochastic programming formulation that allows for a tight representation of the feasible operations of the generators while including uncertainty in the power demand.
The embodiments of the invention present a Branch-and-Price algorithm for the scheduling of generators that also incorporates the tight representation of the feasible operations of the generators. The algorithm is applicable to UCP with known power demands and uncertain power demands.
The presently disclosed embodiments will be further explained with reference to the attached drawings. The drawings shown are not necessarily to scale, with emphasis instead generally being placed upon illustrating the principles of the presently disclosed embodiments.
While the above-identified drawings set forth presently disclosed embodiments, other embodiments are also contemplated, as noted in the discussion. This disclosure presents illustrative embodiments by way of representation and not limitation. Numerous other modifications and embodiments can be devised by those skilled in the art which fall within the scope and spirit of the principles of the presently disclosed embodiments.
Various embodiments of the present invention are described hereafter with reference to the figures. It would be noted that the figures are not drawn to scale elements of similar structures or functions are represented by like reference numerals throughout the figures. It should be also noted that the figures are only intended to facilitate the description of specific embodiments of the invention. They are not intended as an exhaustive description of the invention or as a limitation on the scope of the invention. In addition, an aspect described in conjunction with a particular embodiment of the invention is not necessarily limited to that embodiment and can be practiced in any other embodiments of the invention.
The above-described embodiments of the present invention can be implemented in any of numerous ways. For example, the embodiments may be implemented using hardware, software or a combination thereof. When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers. Such processors may be implemented as integrated circuits, with one or more processors in an integrated circuit component. Though, a processor may be implemented using circuitry in any suitable format.
Also, the embodiments of the invention may be embodied as a method, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.
Further, the generator operation schedules data obtained from the MIP solver are transmitted, via the interface 150 and the network 190, to the power control system 195, and the received generator operation schedules data are used to control the generators according to the generator operation schedules.
The generator parameters 116 associated with the operations of a generator g are:
Pming—the minimum power produced by generator when operational
Pmaxg—the maximum power produced by generator when operational
The parameters CHg, CCg, C0g and C1g are necessary for the cost of operations and are collectively referred as the cost information of a generator.
The decision diagram, similar to
Binary variables are associated with the arcs to indicate the state transition that is chosen at each time period. The UCP is formulated using these arc variables that represent state transformation in the decision diagram. The arc variables in the formulation associated with the decision diagram representation in
The arc transitioning between state Dnt−1 and Upt+UTg−1 can also be used to determine how long the generator has been in operation since turning on. For example, the arc 224 connects the states Dnt−1 and Upt+UTg−1. If this particular arc is chosen then it is clear that the generator g: (i) has been on for 0 time periods at time t; (ii) has been on for 1 time period at time (t+1); and so on so that the generator g is on for (UTg−1) time periods at time (t+UTg−1). Since the representation allows for obtaining additional information such as the number of hours the generator g has been in operation, this can be used to obtain a tight representation of the feasible space of operations for the generator. This is a key realization of the invention.
The continuous variables that are associated with the formulation are:
The constraints in the formulation for each generator and each time t are provided below. The variables with indices that are less than 1 are assumed to represent the generator's operational history 123 and are not variables in the optimization.
The operations of the generator that satisfy the minimum up and down time requirements are directly modeled using the constraints for g ∈ G, t ∈ T as
x
g,t−1
+s
g,t−UTg
=x
g,t
+z
g,t Eq (1)
w
g,t−1
+z
g,t−DTg
=w
g,t
+s
g,t Eq (2)
Eq (1)-(2) are the network flow constraints modeling the flow balance around the states—Upt, Dnt. The key realization leading to the strength of the relaxation is that the above constraints Eq (1)-(2) are the convex hull of the feasible on/off operations satisfying the minimum up/down time constraints for the generators.
The power productions limits on the generator are modeled for g ∈ G, t ∈ T as
p
g,t
+Pming(sg,t−Utg+1+ . . . +sg,tx+g,t)≤Pmg,t Eq (3)
Pm
g,t≤(PMg(UTg−1)sg,t−UTg+1+ . . . +PMg(0)sg,t+Pmaxgxg,t) +(SDg−Pmaxg)zg,t+1 Eq (4)
P
g,t≤(Pmaxg−Pming)(sg,t−UTg+1+ . . . +sg,t)+(Pmaxg−Pming)xg,t+Σj=1DTg−1min(0, SDg+(j−1)RDg−Pmaxg)zg,t+j) Eq (4a)
where the PMg(j) for j=UTg−1), . . . , 0 is defined for generators with UTg>1 as
PM
g(j)=Pmaxg for j=(UTg−1)
PM
g(j)=min(Pmaxg, SUg+j RUg) for j=(UTg−2), . . . , 0. Eq. (4b)
For generators with UTg=1 the parameter is set as
A key realization in Eq (4) is that the introduced start-up variables allow to tighten the bounds on the maximum power production that improves upon the previous formulations. A key realization in Eq (4a) is that the introduced start-up variables and shutdown variables allows to reduce the bounds on the pg,t based on the possible time of shutdown of the generator. For example, if the xg,t is 1 then the generator has been turned on for more than UTg time periods and suppose that the generator is turned off at time t+1 is zg,t+1. If zg,t1=1 then the right hand side of evaluates to (SDg−Pming). This is consistent with the shutdown ramp on generators. If zg,t+j=1 and so the generator is turned off at time t+j then the right hand side evaluates to (Pmaxg−Pming)+min(0,SDg+(j−1) RDg−Pmaxg). If the argument in the min is negative then right hand side evaluates to (SDg+(j−1) RDg−Pming).
The ramp-up limit constraints are modeled for g ∈ G, t ∈ T as
Pm
g,t
−p
g,t−1≤RUMg(UTg−1)sg,t−UTg+1+ . . . +RUMg(0)sg,t+(RUg+Pming)xg,t+(SDg−Pmaxg)zg,t+1 Eq. (5)
where the RUMg(j) for j=UTg−1), . . . , 0 is defined for generators with UTg>1 as
RUMg(j)=(RUg+Pming) for j=(UTg−1), . . . , 1. Eq (5a)
For generators with UTg=1 the parameter is set as
The ramp-down limit constraints are modeled for g ∈ G, t ∈ T as
P
g,t−1
−P
g,t≤RDMg(UTg−1)sg,t−UTg+1+ . . . +RDMg(0)sg,t+RDgxg,t+(SDg−Pming)zg,t Eq. (6)
where the RDMg(j) for j=(UTg−1), . . . , 0 is defined for generators with UTg>1 as
RDMg(j)=min(RDg, SUg+(j−1)RUg) for j=(UTg−1), . . . , 1. Eq (6a)
For generators with UTg=1 the parameter is set as
A key realization in Eq (6) is that the introduced start-up variables allow to tighten the bounds on the ramp-down limit that improves upon the previous formulations.
The cold start-up costs associated with the generator are modeled as
Cs
g,t≥(CHg−CCg)(sg,t−zg,t−DT− . . . −zg,t−DT−TC+1) Eq (7)
The demand and reserve satisfaction constraints are imposed for t ∈ T as
Σg ∈ G)Pg,t+Pming(sg,t−UTg+1+ . . . +sg,t+xg,t))≥Pdemt Eq (8)
Σg ∈ GPmg,t≥Prest Eq (9)
The start-up variables are associated with the costs of operation ηg,t as follows
ηg,t=CHg+min(T−t1,+UTg)(C0g+C1gPming)
Once the generator is turned on since it is required to be on for at least UTg periods the fixed costs of operation with the generator are C0g+UTg(C1g Pming). The minimum is applied to ensure that the operational costs associated with the generator are only for the time periods within the horizon T. Since the hot start cost is smaller than the cold start cost, switching on the generator requires that at least a cost of CHg is incurred.
The optimization problem for the UCP can be formulated as
sg,t, xg,t, wg,t·zg,t ∈ {0,1}pPg,t, Pmg,t, CSg,t≥0. Eq (UCP-1)
The objective function in Eq (UCP-1) models the cost of operating the generators. In the above formulation in Eq (UCP-1) the constraints Eq (1)-Eq (9) are the network flow constraints and tight constraints and are a key realization of the invention.
The decision diagram, similar to
Binary variables are associated with the arcs to indicate the state transition that is chosen at each time period. The UCP is formulated using these arc variables that represent state transformation in the decision diagram.
The arc variables in the formulation associated with the state-space representation in
The arc transitioning between state Dnt−t,k and Upt+UTg−1 can also be used to determine how long the generator has been in operation since turning on. For example, the arc 364 connects the states Dnt−1,1 and Upt+UTg−1. The arc 372 connects the states Dnt−1,0 and Upt+UTg−1. If this particular arc is chosen then it is clear that the generator g: (i) has been on for 0 time periods at time t; (ii) has been on for 1 time period at time (t+1); and so on so that the generator g is on for (UTg−1) time periods at time (t+UTg−1). Since the representation allows for obtaining additional information such as the number of hours the generator g has been in operation, this can be used to obtain a tight representation of the feasible space of operations for the generator.
In addition, the arc transitioning between state Dnt−DTg+1,k and Upt+1 can be associated with different start-up costs. For instance, the arcs (Dnt−DTg+1,k,Upt+1) with 0≤k≤(TCg−2) are associated with hot start-up costs. On the other hand, the arc (Dnt−DTg+1,TCg−t, Upt+1)is associated with the higher cold start-up costs. A key realization is that this representation is the convex hull of the feasible schedules of the generator satisfying the minimum up and down requirements including the start-up costs.
The continuous variables that are associated with the formulation are:
The constraints in the formulation for each generator and each time t are provided below. The variables with indices that are less than 1 are assumed to represent the generator's operational history and are not variables in the optimization.
The operations of the generator that satisfy the minimum up and down time requirements are directly modeled using the constraints for g ∈ G, t ∈ T as
x
g,t−1 +Σk=0TCg−1Sg,t−UTg,k =Zg,tZg,t Eq (10)
z
g,t−DTg
=W
g,t,0
+S
g,0,t Eq (11a)
w
g,t−1,k−1
=w
g,t,k
+S
g,t,k for k=1, . . . , (TCg−2) Eq (11b)
w
g,t−1,TCg−1
+w
g,t−1,TCg−2
=w
g,t,TCg−1
+s
g,t,TCg−1 Eq (11c)
Eq (10)-(11) are the network flow constraints modeling the flow balance around the states—Upt, Dnt,0, . . . , Dnt,TC−1. The key realization leading to the strength of the relaxation is that the above constraints Eq (10)-(11) are the convex hull of the feasible on/off schedules for the generators including the cold start-up costs for the generator.
The power productions limits on the generator are modeled for g ∈ G, t ∈ T as
p
g,t
+Pming(Σk=0TCg−1(sg,t=UTg+1,k+ , . . . +sg,t,k)+xg,t)≤Pmg,t Eq (12)
Pm
g,t≤(Σk=0TCg−1 (PMg(UTg−1)sg,t−UTg+1,k+ . . . +PMg(0)sg,t,k)+Pmaxgxg,t) +(SDg−Pmaxg)zg,t+1 Eq. (13)
p
g,t≤(Pmaxg−Pming)Σk=1TCg−1(sg,t−UTg+1,k+ . . . sg,t,k)+Pmaxg−Pming)xg,t+Σj=1DTg−1min (0,SDg+(j−1)RDg−Pmaxg)zg−Pmaxg)zg,t+j Eq (13a)
where the PMg(j) for j=(UTg−1), . . . , 0 is defined for generators as in Eq. (4b).
A key realization in Eq (13) is that the introduced start-up variables allow to tighten the bounds on the maximum power production that improves upon the previous formulations. A key realization in Eq (13a) is that the introduced start-up variables and shutdown variables allows to reduce the bounds on the pg,t based on the possible time of shutdown of the generator. For example, if the xg,t is 1 then the generator has been turned on for more than UTg time periods and suppose that the generator is turned off at time t+1 is zg,t+1. If zg,t+1=1 then the right hand side of evaluates to (SDg−Pming). This is consistent with the shutdown ramp on generators. If zg,t+j=1 and so the generator is turned off at time t+j then the right hand side evaluates to (Pmaxg−Pming)+min(0,SDg+(j−1) RDg−Pmaxg). If the argument in the min is negative then right hand side evaluates to (SDg+(j−1) RDg−Pming).
The ramp-up limit constraints are modeled for g ∈ G, t ∈ T as
Pm
g,t
−p
g,t−1≤ΣTk=0TCg−1(RUMg(UTg−1)sg,t−UTg+1+ . . . +RUMg(0)sg,t,k)+(RUg+Pming)xg,t+(SDg−Pmaxg)zg,t+1 Eq (14)
where the RUMg(j) for j=(UTg−1, . . . 0 is defined for generators as in Eq (5a).
The ramp-down limit constraints are modeled for g ∈ G, t ∈ T as
P
g,t−1
−p
g,t≤Σk+0TCg−1(RDMg(UTg−1)sg,t−UTg+1,k+ . . . + RDMg(0)sg,t,k)+RDgxg,t+(SDg−Pming)zg,t Eq (15)
where the RDMg(j) for j=(UTg−1), . . . , 0 is defined for generators as in Eq (6a).
A key realization in Eq (15) is that the introduced start-up variables allow to tighten the bounds on the ramp-down limit that improves upon the previous formulations.
The demand and reserve satisfaction constraints are imposed for t ∈ T as
Σg ∈ G (pg,t+Pming(Σk=0TCg−1(sg,t−UTg+1,k+ . . . , +sg,t,k)+xg,t))≥Pdemt Eq (16)
Σg ∈ G PMg,t≥Prest Eq (17)
The start-up variables are associated with the costs of operation ηg,t,k as follows
ηg,t,k=CHg+min(T−t+1,UTg)(C0g+C1gPming) for k=0, . . . , (TCg−2)
ηg,t,k=CCg+min(T−t+1,UTg)(C0g+C1gPming) for k=(TCg−1)
Once the generator is turned on since it is required to be on for at least UTg periods the fixed costs of operation with the generator are C0g+UTg (C1g Pming). The minimum is applied to ensure that the operational costs associated with the generator are only for the time periods within the horizon T.
The optimization problem for the UCP to determine the optimal schedules for the generators can be formulated as
Sg,t,k, Xg,t, wg,t,k, zg,t ∈ {0,1}
pg,t, PMg,t≥0. Eq (UCP-2)
The objective function in Eq (UCP-2) models the cost of operating the generators. The constraints in Eq (10)-(17) in the above formulation in Eq(UCP-2) are the network flow constraints and tight constraints.
In another embodiment of the invention, a so-called Exponential Formulation (EF) may be used to find the optimal scheduling of the generators. The decision diagram representation of the generator's operations consists of paths that start from the initial state of the generators and follows the sequence of arcs in the decision diagram leading up to the end of the time horizon of operations.
Denote by g the set of paths in the decision diagram that is associated with generator g ∈ G. In
Denote by Ong={ON0, . . . , ONUTg−1}, Offg={OFFDTg+TCg−1, OFFDTg+TCg−2, . . . ,OFFDTg−1, . . . , OFF0} and OnOffg=Ong∪Offτ for τ≤(DTg+TCg−2) represents that the generator has been turned off for (DTg+TCg−2) time periods; OFFτ for τ=(DTg+TCg−1) represents that the generator has been turned off for (DTg+TCg−1) or more time periods; ONτfor τ≤(UTg−2) represents that the generator has been turned on for (UTg−2) time periods; and ONτfor τ=(UTg−1) represents that the generator has been turned on for (UTg −1) or more time periods.
Given a path p ∈ g, defining a sequence of Up/Dn states for the generator g on that path p a sequence of states s(p)={s0,s1, . . . ,sT} for each time instant in the horizon starting with a given initial state s0 and st∈OnOffg. For example, the path (Dnt−1,0,Dnt,1,Upt+2) in the decision diagram, of UCP-DDwCS formulation, of
In addition, for each such sequence of states s(p) associate at each time instant a number indicating the number of time periods prior to turning the generator off. This can be obtained by proceeding as follows from the last time period. Define Time2OffT(p)=∞. For all other times t=(T−1), . . . , 1 define Time2Offt(p) as
Further, with each such sequence of states s(p)={s0,s1, . . . , sT} for st∈OnOffg
Based on the definition of the parameter ηt(p) a cost coefficient η(p) is associated with each path as is defined as η(p)=Σt=1T η(p).
In the EF formulation, the path variables are:
The continuous variables that are associated with the formulation are:
The power productions limits on the generator are modeled for g ∈ G, t ∈ T as
p
g,t
+Pming (αt(p) zp)≤Pmg,t Eq (18)
Pmg,t≤(βt(p)zp) Eq (19)
Pg,t≤(δt(p)zp). Eq (20)
The ramp-up limit constraints are modeled for g ∈ G, t ∈ T as
Pm
g,t
−p
g,t−1≤(θt(p)zp). Eq (21)
The ramp-down limit constraints are modeled for g ∈ G, t ∈ T as
pg,t−1−Pg,t≤ (ϕt(p)zp). Eq (22)
The demand and reserve satisfaction constraints are imposed for t ∈ T as
Σg ∈G(pg,t+Pming (αt(p)zp))≥Pdemt Eq (23)
Σg ∈G PMg,t≥Prest Eq (24)
The optimization problem for the (EF) is
z
p=1∀g ∈ G
zp∈{0,1}
pg,t, Pmg,t≥0. Eq (UCP-EF)
The constraints Eq (18)-(24) in the formulation in Eq (UCP-EF) are the path-based operational constraints of the generators.
The solution of the optimization problem in (UCP-EF) is challenging since the number of variables in the formulation is potentially exponential.
In another embodiment of the invention, a Branch-and-Price (BP) algorithm is proposed wherein the number of variables in the optimization problem are progressively increased. Further, branching is incorporated in order obtain a complete method that obtains a solution.
The BP algorithm proceeds by defining an initial search-tree node with no branching decisions and choose for g ∈ G, a subset of paths g⊂
g are chosen representing a subset of the generator's on/off operations satisfying the minimum up/down time constraints. Denote by
:=∪g ∈ G
g. With the subset of paths an optimization problem called as the restricted master problem (RMP) is defined. The variables in the RMP are the path variables:
zp ∈ {0,1}∀p
∈ g indicating the choice of a path from the decision diagram from generator g
The constraints in the RMP formulation are:
=1 ∀g ∈ G (RMP.2)
where the constraint enforces exactly one path is chosen from each decision diagram for each generator.
The constraints in Eq (18)-(24) where the summation is replaced over the paths is replaced by the set g instead of the set
g. (RMP.3)
zp ∈ {0,1} ∀g ∈ G, ∀p ∈ g (RMP.4)
where the constraint enforces that the variables are binary valued.
The RMP formulation can be posed as
The constraints in Eq (RMP.2)-(RMP.4) are the restricted master problem constraints.
The linear programming relaxation of the RMP, denoted as LPRMP, is obtained by replacing the binary requirement in (RMP.4) with zp≥0 ∀g ∈ G, ∀p ∈g.
The LPRMP is solved using column generation where the paths p ∈ g\
g are added if the associated variable in (EF) has a reduced cost that is negative at the solution corresponding to LPRMP for the chosen paths in
. This is accomplished using a pricing problem that is described below.
Denote by μg∀g ∈ G ∈ the Lagrange multiplier associated with (RMP.2) at the optimal the solution of the LPRMP. Denote by λg,α,t,λg,βt,λg,δ,t, λg,φ,t,λg,θ,t, λd,t, λr,t∀t ∈ the Lagrange multiplier for Eq, (18)-(24) respectively in (RMP.3) at the optimal solution of the LPRMP.
The pricing problem (PP) to identify paths that have negative reduced cost is:
For each g ∈ G, define arc-costs θ(α) for all arcs in the decision diagram representation, UDP-DD formulation, where the cold start-up costs are not included as:
In the case of decision diagram representation with cold start-up costs, UCP-DDwCS formulation, the cost ηg,t is replaced by ηg,t,k according to the off state (in Offg) from which the generator is started.
For each g ∈ G, find the minimum cost from the initial states of the generators to reach a state at final time pg path is determined using the arc costs θ(α). Such calculations can be performed using the well known Djikstra's algorithm.
For each g ∈ G, the path p9 is added to g the reduced cost defined as, Σα∈pd θ(α)−μg<0.
Solving the (RMP) as an integer program results in a feasible solution to the scheduling of passengers. A branch-and-bound search is conducted to complete the BP algorithm. A queue of search-tree nodes Γ is defined, initialized as a singleton γ′. At any point in the execution of the algorithm, each search node γ∈ Γ is defined by a set of branch decisions out(γ), in(γ). The branch-and-bound search maintains the best known solution z* and its objective value ƒ*.
While Γ≠∅, a search node γ is selected to explore. The chosen node is the one with the worst LPRMP objective value of the search node from which it was created. The LPRMP for the search node γ is solved using column generation as described before. If the optimal objective value of the LPRMP(γ) is greater than ƒ* then the node is pruned, and the search continues by selecting another node in Γ. Otherwise, the integer program in (RMP) is solved and the solution z′ with objective value ƒ′ is obtained. If ƒ′ is lower than ƒ* then z*, ƒ* are replaced by z′, ƒ′ respectively. Let y*p denote the optimal value to the LPRMP. The path p=(γ|y*p−0.5|with with the fractional is selected to branch on. Two nodes γ0,yγ1 are created with in(γ0)=in(γ),out(γ0)=out(γ)∪ {p} and in(γ1)=in(γ)∪{p}) and out(γ1)=out(γ), and update the search tree as Γ=Γ∪ {y0,γ1}\{γ}.
Finding an initial feasible solution an initial feasible solution to RMP is obtained by defining a path pg,0 for each g ∈ G starting from the initial state perform the following steps: (a) choose the arc that allows to turn the generator on tat the earliest time if the generator is currently off or choose the arc that keeps he generator on if the generator is already on while satisfying the state transitions; (b) repeating the step in (a) from the resultant state in the next time period until the end of the time horizon. Using with just this singleton element in the sets g solve the (LPRMP). If this problem is feasible then obtain an upper bound on the cost of operation. Note that the above choice corresponds to choosing the most expensive solution in terms of the cost of the operations since all the generators are kept on for most periods. If this problem is infeasible then it is determined that the demand and reserve requirements cannot be satisfied using the given generators.
Suppose that y*p∀p ∈ is the optimal solution the LPRMP and the solution is not integral. Then solve the (RMP) to obtain a feasible solution. This does not need to be solved to optimality. Instead, the (RMP) can be solved to generate better feasible solutions as opposed to proving optimality.
In another embodiment of the invention is considered the scheduling of generators in the presence of uncertainty in the demand. This is particularly important in the face of incorporation of renewables such as wind and solar in the power generation sources. Due to this the demand variability is significant based on the weather pattern that is realized in the day. However, the commitment of the units is performed at the beginning of the day and hence, (i) the power from the renewable sources must be predicted using the predictions on the weather pattern for that day; and (ii) the possible demand patterns must be derived for the day based on the power from renewables. Hence, in the face of renewables the UCP problem has demand uncertainty. The demand uncertainty is represented as a set of scenarios Q={1, . . . , Q} where for each scenario q a different demand and reserve pattern: (Pdemt(q),Prest(q)) is assumed.
The binary variables in UCP-DD-DU are identical to those in UCP-DD. Binary variables are associated with the arcs to indicate the state transition that is chosen at each time period. The UCP is formulated using these arc variables that represent state transformation in the decision diagram. The arc variables in the formulation associated with the decision diagram representation in
The arc transitioning between state Dnt−1 and Upt+UTg−1 can also be used to determine how long the generator has been in operation since turning on. For example, the arc 224 connects the states Dnt−1 and Upt+UTg−1. If this particular arc is chosen then is is clear that the generator g: (i) has been on for 0 time periods at time t; (ii) has been on for 1 time period at time (t+1); and so on so that the generator g is on for (UTg−1) time periods at time (t+UTg−1). Since the representation allows for obtaining additional information such as the number of hours the generator g has been in operation, this can be used to obtain a tight
representation of the feasible space of operations for the generator. This is a key realization of the invention.
The continuous variables that are associated with the formulation are now dependent on the scenario q∈Q:
The constraints in the formulation for each generator and each time t are provided below. The variables with indices that are less than 1 are assumed to represent the generator's operational history 123 and are not variables in the optimization.
The operations of the generator that satisfy the minimum up and down time requirements are directly modeled using the constraints for g ∈ G, t ∈ T are identical to Eq (1)-(2).
The constraints are identical to the tight constraints in UCP-DD but are now applied for each scenario q∈ Q. The power productions limits on the generator are modeled for g ∈ G, t ∈ T, q ∈ Q as
p
g,t(q)+Pming (sg,t−UTg+1+ . . . +sg,t+xg,t)≤Pmg,t(q) Eq (3q)
Pm
g,t(q)≤(PMg(UTg−1)sg,t−UTg+1+ . . . +PMg(0)sg,t+Pmax gxg,t) +(SDg−Pmaxg)zg,t+1 Eq (4q)
g,t(q)≤(Pmaxg−Pming) (sg,t−UTg+1+ . . . +sg,t)+Σj=1DTg−1 min(0,SDg+(j−1)RDg−Pmax g)zg,t+k Eq (4aq)
where the PMg(j) for j=(UTg-−1), . . . , 0 is defined in Eq (4b).
The ramp-up limit constraints are modeled for g ∈ G, t ∈ T, q ∈ Q as
Pm
g,t(q)−pg,t−1(q)≤RUMg(UTg−1)sg,t−UTg+1+ . . . +RUMg(0)sg,t+(RUg+Pming)xg,t+(SDg−Pmaxg)zg,t+1 Eq (5q)
where the RUMg(j) for j=(UTg−1), . . . , 0 is defined in Eq (5a).
The ramp-down limit constraints are modeled for g ∈ G, t ∈ T, q ∈ Q as
p
g,t−1(q)−pg,t(q)≤RDMg(UTg−1)sg,t−uTg+1+ . . . +RDMg(0)sg,t+RDgxg,t+(SDg−Pming)zg,t Eq (6q)
where the RDMg(j) for j=(UTg−1), . . . , 0 is defined in Eq (6a).
The cold start-up costs associated with the generator are modeled as in Eq (7).
The demand and reserve satisfaction constraints are imposed for t ∈ T, q ∈ Q as
Σg ∈ G(Pg,t(q)+Pming(sg,t−UTg+1+ . . . +sg,t+xg,t))≥Pde,t(q) Eq (8q)
Σg ∈ GPMg,t(q)≥Prest(q) Eq (9q)
The start-up variables are associated with the costs of operation ηg,t as follows
ηg,t=CHg+min(T−t+1, UTg)(C0g+C1gPming)
Once the generator is turned on since it is required to be on for at least UTg periods the fixed costs of operation with the generator are C0g+UTg(C1g Pming). The minimum is applied to ensure that the operational costs associated with the generator are only for the time periods within the horizon T. Since the hot start cost is smaller than the cold start cost, switching on the generator requires that at least a cost of CHg is incurred.
The optimization problem for the UCP can be formulated as
sg,t, xg,t, wg,t, zg,t ∈ {0,1}pg,t(q), Pmg,t(q), CSg,t≥0. Eq (UCP-1-DU)
The objective function in Eq (UCP-1-DU) models the expected cost of operating the generators. The constraints Eq (1)-Eq (2), are the network flow constraints and the Eq (7), Eq(3q), (4q) (4aq), (5q), (6q), (8q), (9q) for q ∈ Q are the tight constraints in the formulation in Eq (UCP-1-DU).
Note that when the number of scenarios is one (|Q|=1) then the formulation is Eq (UCP-1-DU) is identical to the formulation in Eq (UCP-1). In other words, the formulation with demand uncertainty reduces to the formulation with no uncertainty when the number of scenarios is just one.
The binary variables in the formulation are identical to the UCP-DDwCS formulation. Binary variables are associated with the arcs to indicate the state transition that is chosen at each time period.
The arc variables that represent state transformation in the decision diagram in the formulation associated with the state-space representation in
The arc transitioning between state Dnt−t,k and Upt+UTg−1 can also be used to determine how long the generator has been in operation since turning on. For example, the arc 364 connects the states Dnt−1,1 and Upt+UTg−1. The arc 372 connects the states Dnt−1,0 and Upt+UTg−1. If this particular arc is chosen then it is clear that the generator g: (i) has been on for 0 time periods at time t; (ii) has been on for 1 time period at time (t+1); and so on so that the generator g is on for (UTg−1) time periods at time (t+UTg−1). Since the representation allows for obtaining additional information such as the number of hours the generator g has been in operation, this can be used to obtain a tight representation of the feasible space of operations for the generator.
In addition, the arc transitioning between state Dnt−DTg+1,k and Upt+1 can be associated with different start-up costs. For instance, the arcs (Dnt−DTg+t,k,Upt+1) with 0≤k≤(TCg−2) are associated with hot start-up costs. On the other hand, the arc (Dnt−DTg+1,TCg−1,Upt+1) is associated with the higher cold start-up costs. A key realization is that this representation is the convex hull of the feasible schedules of the generator satisfying the minimum up and down requirements including the start-up costs.
The continuous variables that are associated with the formulation are now defined for each scenario q ∈ Q:
The constraints in the formulation for each generator and each time t are provided below. The variables with indices that are less than 1 are assumed to represent the generator's operational history and are not variables in the optimization.
p The operations of the generator that satisfy the minimum up and down time requirements are directly modeled using the constraints for g ∈ G, t ∈ T are identical to Eq (10)-Eq (11).
The power productions limits on the generator are modeled for g ∈ G, t ∈ T, q ∈ Q as
p
g,t(q)+Pming(Σk=0TCg−1 (sg,t−UTg+1,k+ . . . +sg,t,k)+xg,t)≤Pmg,t(q) Eq (12q)
Pm
g)q)≤(Σk=1TCg−1(PMg(UTg−1)sg,t−UTg+1,k+ . . . +PMg(0)sg,t,k)+Pmaxgg,t)+(SDg−Pmaxg)zg,(t+1 Eq (13q)
PM
g(q)≤(Pmaxg−Pming)Σk=0TCg−1(sg,t−UTg+1,k+ . . . +sg,t,k)+(Pmaxg−Pming)xg,t+Σj=1UTg−1 min(0, SDg+(RDg−Pmaxg)zg,1+j Eq (13aq)
where the PMg(j) for j=(UTg−1), . . . , 0 is defined for generators as in Eq. (4b).
The ramp-up limit constraints are modeled for g ∈ G, t ∈ T, q ∈ Q as
Pm
g,t(q)−Pg,t−1(q)≤Σk=0TCg−1 (RUMg(UTg−1)sg,t−UTd+1+ . . . ′RUMg(0)sg,t,k)+(RUg+Pming)xg,t+(ISDg−Pmaxg)zg,t+1 Eq (14q)
where the RUMg(j) for j=(UTg−1), . . . , 0 is defined for generators as in Eq (5a).
The ramp-down limit constraints are modeled for g ∈ G, t ∈ T, q ∈ Q as
P
g,t−1(q)−pg,t(q)≤Σk=0TCg−1 (RDMg(UTg−1)sg,−utG+1,K+ . . . +RDMg(0)sg,T,k)+RDgxg,t+(SDg−Pming)zg,t Eq (15q)
where the RDMg(j) for j=(UTg−1), . . . , 0 is defined for generators as in Eq (6a).
A key realization in Eq (15) is that the introduced start-up variables allow to tighten the bounds on the ramp-down limit that improves upon the previous formulations.
The demand and reserve satisfaction constraints are imposed for t ∈ T, q ∈ Q as
Σg ∈ G(pg,t(q)+Pming(Σk=0TCg−1(sg,t−UTg+1,k+ . . . +sg,t,k)+xg.t))≥Pdemt(q) (16q)
Σg ∈ GOmg,t(q)≥Prest (Eq17q)
The start-up variables are associated with the costs of operation ηg,t,k as follows
ηg,t,kCHg+min(T−t+1,UTg)(C0g+C1gPming) for k=0, . . . ,(TCg−2)
ηg,t,kCCg+min(T−t+1,UTg)(C0g+C1gPming) for k=0, . . . ,(TCg−1)
Once the generator is turned on since it is required to be on for at least UTg periods the fixed costs of operation with the generator are C0g+UTg(C1gPming). The minimum is applied to ensure that the operational costs associated with the generator are only for the time periods within the horizon T.
The optimization problem for the UCP to determine the optimal schedules for the generators can be formulated as
sg,t,k, xg,t, wg,t,k·zg,t ∈ {0,1} Pg,t(q), Pmg,t(q)≥0. Eq (UCP-2-DU)
The objective function in Eq (UCP-2-DU) models the expected cost of operating the generators. The constraints Eq (10) Eq (11) are the network flow constraints, and Eq (12q), (13q), (13aq), (14q) (17q) for q ∈ Q are the tight constraints in the formulation in Eq (UCP-2-DU).
Note that when the number of scenarios is one (|Q|=1) then the formulation is Eq (UCP-2-DU) is identical to the formulation in Eq (UCP-2). In other words, the formulation with demand uncertainty reduces to the formulation with no uncertainty when the number of scenarios is just one.
In another embodiment of the invention, the Branch-and-Price formulation is extended to the scheduling of generators with demand uncertainty. The definition of paths in the decision diagrams and the parameters Time2OffT(p), αt(p), βt(p), δt(p), φt(p), θt(p), ηt(p) are as defined in the BP formulation.
Based on the definition of the parameter ηt(p) a cost coefficient η(p) is associated with each path as is defined as η(p)=Σt=1T ηt(p).
In the EF formulation with demand uncertainty, the path variables are: zp ∈({0,1} ∀ p ∈ g indicating the choice of a path p from the decision diagram.
The continuous variables that are associated with the formulation in each scenario q ∈ Q are:
pg,t(q)—continuous variable modeling the excess over the minimum that is produced by generator at time t
Pmg,(q)—continuous variable modeling the maximum power that the generator can potentially produce if required at time t
The power productions limits on the generator are modeled for g ∈ G, t ∈ T, q∈ Q as
p
g,t(q)+Pming (αt(p)zp)≤Pmg,t(q) Eq (18q)
Pm
g,(q)≤ (βt(p)zp) Eq (19q)
P
g,t(q)≤ δt(p)zp). Eq (20q)
The ramp-up limit constraints are modeled for g ∈ G, t ∈ T, q∈ Q as
Pm
g,t(q)−pg,t−1)q) (74t(p)zp) Eq. (21)
The ramp-down limit constraints are modeled for g ∈ G, t ∈ T, q ∈ Q as
p
g,t−1(q)−pg,t(q)≤ (ϕt(p)zp). Eq (22)
The demand and reserve satisfaction constraints are imposed for t ∈ T, q∈ Q as
Σg ∈ G(pg,t(q)+Pming (αt(p)zp)≤Pdemt(q) Eq (23q)
Σg ∈ G Pmg,t(q)≥Prest(q) Eq (24q)
The optimization problem for the (EF-DU) is
zp1∀ g ∈ G zp ∈ {0,1}zp,t, Pmg,t≥0. Eq (UCP-EF-DU)
The solution of the optimization problem in (UCP-EF-DU) is challenging since the number of variables in the formulation is potentially exponential.
Note that when the number of scenarios is one (|Q|=1) then the formulation is Eq (UCP-EF-DU) is identical to the formulation in Eq (UCP-EF). In other words, the formulation with demand uncertainty reduces to the formulation with no uncertainty when the number of scenarios is just one.
In another embodiment of the invention, a Branch-and-Price (BP-DU) algorithm is proposed wherein the number of variables in the optimization problem are progressively increased. Further, branching is incorporated in order obtain a complete method that obtains a solution.
The BP-DU algorithm proceeds by defining an initial search-tree node with no branching decisions and choose for g ∈ G, a subset of paths g⊂
g are chosen. Denote by
:=∪g ∈ G
g. With the subset of paths an optimization problem called as the restricted master problem (RMP) is defined. The variables in the RMP are the path variables:
zp ∈ {0,1} ∀p. g indicating the choice of a path from the decision diagram from generator g The objective function in the RMP formulation is
The continuous variables that are associated with the formulation for each scenario ∈ Q are:
The constraints in the RMP formulation are:
zp=1∀g ∈ G (RM)-DU.2)
where the constraint enforces exactly one path is chosen from each decision diagram for each generator.
The constraints in Eq (18q)-(24q) where the summation is replaced over the paths is replaced by the set instead of the set
. (RMP-DU. 3)
The RMP formulation can be posed as
The constraints Eqs. (RMP-DU.2)-(RMP-DU.4) are the restricted master problem constraints under demand uncertainty.
Note that when the number of scenarios is one (|Q|Q=1) then the formulation is Eq (RMP-DU) is identical to the formulation in Eq (RMP). In other words, the formulation with demand uncertainty reduces to the formulation with no uncertainty when the number of scenarios is just one.
The linear programming relaxation of the RMP-DU, denoted as LPRMP-DU, is obtained by replacing the binary requirement in (RMP-DU.4) with zp≥0 ∀g∈ G, ∀p ∈ g.
The LPRMP-DU is solved using column generation where the paths p ∈ g\
g are added if the associated variable in (EF-DU) has a reduced cost that is negative at the solution corresponding to LPRMP-DU for the chosen paths in
. This is accomplished using a pricing problem that is described below.
Denote by μg ∀g ∈ G E the Lagrange multiplier associated with (RMP.2) at the optimal the solution of the LPRMP-DU. Denote by λg,α,t(q), λg,β,y(q), λg,δ,t(q), λgφ,t(q), λg,θ,t(q), λd,t(q), λr,t(q)∀t ∈ the Lagrange multiplier for Eq, (18q)-(24q) respectively in (RMP.3∈) for each q∈ Q at the optimal solution of the LPRMP-DU.
The pricing problem (PP) to identify paths that have negative reduced cost is:
For each g ∈ G, define arc-costs θ(α) for all arcs in the decision diagram representation, UCP-DD formulation, where the cold start-up costs are not included as:
θ(α)=ηg,t+PmingΣq∈QΣj=0UTg−1 λg,α, t+j(q)−Σq∈QΣj=0UTg−1 PMg(j)λg,β,t+j(q)−Σq∈QΣj=0UTg−1 RUMg(j)λg,φt+j(q)−Σq∈QΣj=0UTg−1 RDMg(j)λg,θ,t+k(q)−PmingΣq∈QΣj−0UT−1 λd,t(q)-−(Pmaxg−Pming)Σq∈QΣj+0UTg−1 λg,δ,t+j(q) if the arc α turns the generator on at time t i.e. satisfies the transition (ST1). θ(α)=(C0g+C1gPimg)+Pming Σq∈Qλg,α,t(q)−Pmaxg Σq∈Qλg,β,t(t)(q)−(Pmaxg−Pming) Σq∈Qλg,δ,t(q)−(RUg+Pming) Σq∈Qλg,φ,t(q)−RDgΣq∈Qλg,θ,t(q)−Pming Σq∈Qλd,t(q) if the arc α keeps the generator on at time t i.e. satisfies the transition (ST2).
θ(α)=−(SDg−Pmaxg) Σq∈Q λg,β,t−1(q)−(−(RUg−Pming) Σq∈QΣj=0DTg−1 min (0, SDg+(j−1)RDg−Pmaxg)λg,δ,t−j(q)−Σq∈Q(SDg−RUg−Pming) λg,φ,t−1)q)−(SDg−Pmaxg)Σq∈Qλg,θ,t(q) if the arc α turns the generator off at time t i.e. satisfies the transition (ST3).
θ(α)=0 for all other arcs.
In the case of decision diagram representation with cold start-up costs, UCP-DDwCS formulation, the cost ηg,t is replaced by ηg,t,k according to the off state (in Offg) from which the generator is started.
For each g ∈ G, find the minimum cost from the initial states of the generators to reach a state at final time p9 path is determined using the arc costs θ(α). Such calculations can be performed using the well known Djikstra's algorithm.
For each g ∈ G, the path p9 is added to g if the reduced cost defined as, Σβ∈pd θ)α)−μg<0.
Solving the (RMP-DU) as an integer program results in a feasible solution to the scheduling of passengers. A branch-and-bound search is conducted to complete the BP algorithm. A queue of search-tree nodes Γ is defined, initialized as a singleton γ′. At any point in the execution of the algorithm, each search node γ ∈ Γ is defined by a set of branch decisions out(γ), in(γ). The branch-and-bound search maintains the best known solution z* and its objective value ″*.
While Γ≠∅, a search node γ is selected to explore. The chosen node is the one with the worst LPRMP-DU objective value of the search node from which it was created. The LPRMP-DU for the search node γ is solved using column generation as described before. If the optimal objective value of the LPRMP-DU(γ) is greater than ƒ* then the node is pruned, and the search continues by selecting another node in Γ. Otherwise, the integer program in (RMP-DU) is solved and the solution z′ with objective value ƒ′ is obtained. If ƒ′ is lower than ƒ* then z*, ƒ* are replaced by z′, f′ respectively. Let y; denote the optimal value to the LPRMP-DU. The path p=(γ)|γ*p−0.5 with with the fractional is selected to branch on. Two nodes γ0, γ1 are created with)in(γ0)=in(γ)), out(γ0)=out(γ)∪[p] and in(γ1)=in(γ)∉ [p] and out(γ1)=outγy), and update the search tree as Γ=Γ ∪{γ0,γ1}\{γ}.
An initial feasible solution to RMP-DU is obtained by defining a path pg,0 for each g ∈ G starting from the initial state perform the following steps: (a) choose the arc that allows to turn the generator on at the earliest time if the generator is currently off or choose the arc that keeps the generator on if the generator is already on while satisfying the state transitions; (b) repeating the step in (a) from the resultant state in the next time period until the end of the time horizon. Using with just this singleton element in the sets g solve the (LPRMP-DU). If this problem is feasible then obtain an upper bound on the cost of operation. Note that the above choice corresponds to choosing the most expensive solution in terms of the cost of the operations since all the generators are kept on for most periods. If this problem is infeasible then it is determined that the demand and reserve requirements cannot be satisfied using the given generators.
Suppose that γ*p∀p ∈ is the optimal solution the LPRMP and the solution is not integral. Then solve the (RMP) to obtain a feasible solution. This does not need to be solved to optimality. Instead, the (RMP) can be solved to generate better feasible solutions as opposed to proving optimality.
The branch and bound algorithm for solving the BP-DU is identical to the algorithm outlines in
Further, the power generation planning system 100 can be applied to a different set of feasible on/off operations of the generators in the presence of start-up patterns for generating a decision diagram using the state-space representation module 124.
The operation states of the generator are indicated by a shut-down state SS, a turn-on state including a start-up pattern AP, a continuous operation (driving) state CDS, and a shut-down pattern SP. A continuous shut-down state CSS can be a variable that indicates a period of time in which the generator has been shut-down since the latest end of the SP. A power output during the CDS can be controlled or adjusted between a predetermined maximum-minimum power range.
The AP, CDS, and SP states may simply be referred to as an operation state (DS), and the CSS may be referred to as SS.
In general, each power generator has its own AP and SP as generator parameters, which can be included in the generator parameters 116. The CSS and CDS are variables, and the CDS can be controlled/adjusted in response to a power demand profile.
In some cases, the AP of a generator can be changed according to a period of time of the CSS. For instance, when the period of time of the CSS is longer than a threshold, then the AP can be a cold-start pattern as temperature of the generator is cool. In this case the generator is scheduled to more gradually be driven up to a CDS. When the period of time of the CSS is shorter than a threshold, then the AP can be a warm-start pattern. When the period of time of the CSS is much shorter than a threshold, the AP can be a hot-start pattern.
In accordance with embodiments of the present invention, the generator parameters (AP, SP) of power generators can be represented by decision diagrams constructed using the state-space representation module124. Further, the system 100 generates arc-variables representing state-transformations of the generators by assigning binary variables to arcs of the decision diagrams by using the variable assignment module 122, and generates network flow constraints to represent feasible operations of each of the generators, and then the system 100 obtains feasible operations (operation schedules) of the generators, in which part of the generator parameters can be used to represent constraints, network flow constraints and tight constraints for obtaining the feasible operations (operation schedules) of the generators.
In other words, the demanded-power-profile is achieved by integrating the start-up patterns, continuous operation states, shut-down patterns and continuous shut-down states of the generators.
Further, according to embodiments of the present invention, linear objective functions can be optimized over in linear time (with respect to the size of the Decision Diagram) by computing a shortest path, which can lead to orders-of-magnitude improvement gains over other techniques.
Although the invention has been described by way of examples of preferred embodiments, it is to be understood that various other adaptations and modifications can be made within the spirit and scope of the invention.
Therefore, it is the object of the appended claims to cover all such variations and modifications as come within the true spirit and scope of the invention.
Use of ordinal terms such as “first,” “second,” in the claims to modify a claim element does not by itself connote any priority, precedence, or order of one claim element over another or the temporal order in which acts of a method are performed, but are used merely as labels to distinguish one claim element having a certain name from another element having a same name (but for use of the ordinal term) to distinguish the claim elements.
