The present invention relates generally to large scale resource scheduling optimization, and more particularly, to general structure-based methods that are applicable to large scale resource scheduling optimization.
Resource scheduling is a special class of optimization problem, in which a mix of available resources are utilized to satisfy demand side requirements over a given time horizon at prescribed temporal or spatial resolutions. The decisions to be made involve the determination of optimal operation schedules for all resources involved. The operation schedule for a resource can be described by the startup and shutdown, and utilization at each time step over the scheduling horizon. The operation schedules for all the resources are determined such that various local and global constraints are satisfied and some generalized cost function is minimized. A well-known classic example of resource scheduling problem is the so called unit commitment problem in the electric power industry, where the resources are the thermal, hydro power plants in the system; the demand side requirements are the total customer hourly load over 24 or 168 hours. The startup, shutdown, and utilization of each plant is referred to as commitment, de-commitment. and dispatch. The operation schedule for each plant must satisfy the local constraints such as minimum up time, minimum down time, ramping constraint, and available capacity limitations. Collectively, all the plants committed must also satisfy global constraints such as various reserve requirements and energy balance constraints.
The resource scheduling problem, by its combinatorial nature, is very hard. The computational burden increases exponentially with the number of resources and the time steps in the scheduling horizon. To overcome the “curse of dimensionality”, decomposition techniques based on Lagrangian relaxation theory are generally used. In these decomposition methods, the original optimization problem is relaxed by removing the so called “complicating constraints”, also known as “coupling constraints”, to obtain a separable optimization problem, which can be divided into many smaller independent optimization problems, usually referred to as sub-problems. All the sub-problems are solved, e.g., one sub-problem for each unit in the unit commitment problem, and the Lagrangian multipliers are updated at a high level. The solutions of the sub-problems are coordinated by a set of price signals for the complicating constraints.
The performance of a resource scheduling method is measured by the following criteria: (1) optimality, (2) feasibility and (3) speed. Optimality measures how close the solution is to the theoretical best. A feasibility check ensures that all constraints have been satisfied. Speed is how fast the method is at finding the solution. The optimality of the solution is usually measured by the solution gap. The solution gap is a conservative estimate of the closeness of a solution to a theoretical optimum solution in terms of an objective value. Solution gap is defined by (OBJ−LB)/OBJ expressed as a percent, where OBJ is the best integer solution found and LB is the tightest lower bound known for the problem.
An optimization method based on Lagrangian relaxation typically includes two main steps: (1) Lagrangian dual optimization to get an optimal dual solution; and (2) construction of a primal feasible solution. The problem encountered in the first step is an optimization of the non-differentiable dual function, which is commonly solved by variations of subgradient (SG) and cutting plane (CP) methods. This is an iterative process plagued by slow convergence, requiring hundreds, sometimes thousands of iterations. Quickly getting to the optimal dual solution is an attractive and challenging objective in the design of Lagrangian relaxation-based algorithms.
The primal solution corresponding to the optimal dual solution in general is not ensured to be feasible for all the coupling constraints. The second step attempts to make the primal solution feasible by applying various problem dependent heuristics to adjust the primal solution until it becomes feasible. The heuristics rules are generally constructed from experience and insight into the specific problems at hand and lack generality. The method disclosed herein offers a huge performance advantage over the traditional Lagrangian relaxation-based methods by providing fast dual optimization and a problem independent process for feasibility enforcement.
The present invention provides a complete solution process for large-scale resource scheduling optimization and has the following unique advantages:
In an exemplary embodiment, the invention is directed to a method for optimization of large scale resource scheduling. An optimal solution is first found to the dual of the resource scheduling problem by sequentially finding a solution to a plurality of sub-problems and updating a set of multiplier values used in the dual problem formulation after each sub-problem solution is obtained. Lagrangian relaxation techniques can be used to obtain the dual solution and the set of values updated after each sub-problem is solved are Lagrangian multipliers. The solution to the dual optimization disclosed herein does not use any penalty factors. Coupling constraint violations are systematically reduced and the set of multiplier values are updated until a feasible solution to the primal resource scheduling problem is obtained. The method optionally includes estimating the lower bound to the primal resource scheduling problem which, in turn, is used as an estimate for the upper bound of the dual resource scheduling problem. An initial set of multiplier values is further determined by solving a relaxed version of the primal problem where most of the local constraints except the variable bounds are relaxed.
The structure-based algorithm of the invention is particularly well-suited for large scale resource scheduling such as the unit commitment problem in the electric power industry wherein hydro and thermal generating units are scheduled for operation to meet demand over a period of time while minimizing total operating costs. In the simplest case, the scheduling problem is decomposed into a single sub-problem for each thermal plant and for each hydroelectric plant.
The invention is better understood by reading the following detailed description of the invention in conjunction with the accompanying drawings.
The following description of the invention is provided as an enabling teaching of the invention in its best, currently known embodiment. Those skilled in the relevant art will recognize that many changes can be made to the embodiments described, while still obtaining the beneficial results of the present invention. It will also be apparent that some of the desired benefits of the present invention can be obtained by selecting some of the features of the present invention without utilizing other features. Accordingly, those who work in the art will recognize that many modifications and adaptations to the present invention are possible and may even be desirable in certain circumstances and are a part of the present invention. Thus, the following description is provided as illustrative of the principles of the present invention and not in limitation thereof, since the scope of the present invention is defined by the claims.
The first step (block 100) is the estimation of the lower bound of the problem to be solved. If the estimation is available from other sources, this step is optional. When the resource scheduling is formulated as a MIP problem, the lower bound can be quickly obtained by removal of sub-problem level constraints and integer relaxation. The resulting lower bound will be used as an estimate of the upper bound of the dual problem. If this step is executed, the multipliers for the complicating constraints can be estimated from the resulting linear programming (LP) problem.
In the second step (block 102), all sub-problems are solved once for the initial set of multipliers to get the initial primal solution. If the first step is skipped, an initial multiplier set of zero can be used with very little effect on the convergence of the solution process.
In the third step (block 104), the dual problem is optimized by sequentially solving sub-problems and updating the multipliers after each sub-problem solution. The Lagrangian dual function is evaluated after each sub-problem solution, and an update for the multipliers is calculated. This differs from the traditional approach where multipliers are updated after all sub-problems are solved for a given set of multipliers. This design also differs in fundamental ways from a prior art method that uses a set of penalty factors to reduce violation of the coupling constraints and requires some follow up process that uses problem dependent heuristics to make the primal solution feasible. The presence of penalty factors causes significant bias in the multiplier update process and reduces the optimality of the dual solution. The solution disclosed herein does not use any penalty factors in the dual optimization phase and does not concern itself with primal feasibility until the next step.
The fourth step (block 106) searches for primal feasibility by systematically reducing violations of coupling constraints. It updates the multipliers based on the modified subgradients and solve the corresponding sub-problems until feasibility is achieved.
The method successively solves resource scheduling problems with hundreds of thousands of variables and several hundred thousand constraints, producing solutions that are 100% feasible with very good solution gap, thereby reducing solution time by orders of magnitude as compared with optimization by a standard solver directly.
For simplicity of notation, the inventive algorithm is described with respect to a reference model for resource scheduling optimization. The reference model is expressed as a mixed integer programming (MIP) problem, which is done solely to describe the principles of the invention.
The Reference Model and Notations
Minimize
cTx
Ax=b
Acx>=bc
Subject to
Bl<=x<=Bu
x>=0
The above problem will be referred to as problem P. Ac is the constraint matrix for the coupling rows. A is the constraint matrix for the non-coupling rows. The inclusion of only greater than constraints for the coupling rows will not affect the generality of the method. By adding slack variables to the coupling rows, we get the following problem PS:
Minimize
cTx+csTSc
Ax=b
Acx+Sc>=bc
Subject to
Bl<=x<=Bu
x>=0
The Lagrangian is defined as:
L(λ,x)=cTx+λT(bc−Acx)
The Lagrangian dual function is defined as:
Ax=b
Subject to
Bl<=x<=Bu
x>=0
The Lagrangian dual function is separable into sub-problems. The k-th sub-problem, referred to as P(k), is defined by the following:
[A]kxk=bk
Subject to
Bl<=xk<=Bu
xk>=0
The processing logic for the structure-based algorithm for optimization of large scale separable resource scheduling illustrated in
Pseudocode for the lower bound and initial multiplier estimate step of the algorithm is as follows:
Lower Bound and Initial Multiplier Estimate Step (block 100)
ok=ckT−λT[Ac]k
where ok are the objective coefficients of the sub-problem k, and λ is the given multiplier.
Pseudocode for the initialization sub-problem solution for initial multiplier step of the algorithm is as follows:
Initial Sub-Problem Solution for Initial Multipliers Step (Block 102)
g=bc−Acx.
The step size θ for multiplier update (logic block 510) is calculated as:
The multiplier is updated (logic block 512) by using equation:
λnew=λold+θ·g,
The objective coefficients of the current sub-problem are updated (logic block 514) in the same way as in the “initial sub-problem solution” step of the algorithm.
Pseudocode for the dual optimization by sub-problem level dual function evaluation and multiplier update step of the algorithm is as follows:
Dual Optimization by Sub-Problem Level Dual Function Evaluation and Multiplier Update Step (Block 104)
g=[g]+.
The multiplier is updated as:
λnew=λold+β·g,
Based on the new multiplier, objective coefficients of each of the sub-problems are updated similarly to step “initial sub-problem solution”, and the sub-problems are solved to obtain a new dual solution. Then pass the values of the integer variables of the dual solution to problem P1. This procedure is repeated until a primal feasible solution is achieved.
Pseudocode for the primal feasible solution generation step of the algorithm is as follows:
Primal Feasible Solution Generation Step (Block 106)
The method for optimization of large scale separable resource scheduling of the present invention has been described as a computer implemented process using application programs resident on a computer system that can process large scale optimization problems. It is important to note, however, that those skilled in the art will appreciate that the mechanisms of the present invention are capable of being distributed as a program product in a variety of forms, and that the present invention applies regardless of the particular type of signal bearing media utilized to carry out the distribution. Examples of signal bearing media include, without limitation, recordable-type media such as diskettes or CD ROMs, and transmission type media such as analog or digital communications links. Furthermore, those skilled in the art will appreciate that the broad concept of large scale resource scheduling optimization by sequential Lagrangian dual optimization and meta-heuristics to enforce a feasible solution are applicable to a wide range of computationally complex optimization problems and that the reference model described herein in the form of a mixed integer programming problem is used solely to describe an exemplary embodiment.
Those skilled in the art will appreciate that many modifications to the preferred embodiment of the present invention are possible without departing from the spirit and scope of the present invention. In addition, it is possible to use some of the features of the present invention without the corresponding use of other features. Accordingly, the foregoing description of the preferred embodiment is provided for the purpose of illustrating the principles of the present invention and not in limitation thereof, since the scope of the present invention is defined solely by the appended claims.
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Number | Date | Country | |
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20060089864 A1 | Apr 2006 | US |