AbstractâIn the Payment Cost Minimization (PCM) mechanism [4] payment costs are minimized directly, thus the payment costs that results from selected offers ...
Payment Cost Minimization Using Lagrangian Relaxation and Modified Surrogate Optimization Approach Mikhail A. Bragin, Student Member, IEEE, Xu Han, Student Member, IEEE, Peter B. Luh, Fellow, IEEE, Joseph H. Yan, Senior Member, IEEE Abstract—In the Payment Cost Minimization (PCM) mechanism [4] payment costs are minimized directly, thus the payment costs that results from selected offers can be significantly reduced compared to the costs obtained by minimizing total bid costs. The PCM can be solved efficiently using standard LP software packages (e.g., CPLEX) only for a limited number of offers. Lagrangian relaxation (LR) has been a powerful technique to solve discrete and mixed-integer optimization problems. For complex problems, such as the PCM, the surrogate subgradient method is frequently used within Lagrangian relaxation approach to update multipliers (e.g., [6], [4]). In the surrogate subgradient approach a proper direction is obtained without fully minimizing the relaxed problem. This paper presents a modified Lagrangian relaxation and the surrogate optimization approach for obtaining a good feasible solution within a reasonable CPU time. The difficulty of the standard surrogate optimization method primarily arises due to the lack of prior knowledge about the optimal dual value, which is used in the definition of a step size. In order to overcome this difficulty, a new method is proposed. The main purpose of the modified surrogate subgradient approach is to obtain a “good” direction quickly and independently of the optimal dual value at each iteration. In this paper it is achieved by introducing a formula for updating the multipliers such that the exact minimization of the Lagrangian leads to a convergent result. Then an approximate formula for updating the multipliers is developed so that the exact optimization of the Lagrangian leads to a convergent result under certain optimality conditions. Lastly, the notion of the surrogate subgradient is used for ensuring the convergence within the reasonable CPU time. An analogue of the surrogate subgradient condition guarantees the convergence on the surrogate subgradient method. Numerical examples are provided to demonstrate the method’s effectiveness.
I. INTRODUCTION
C
URRENTLY, most ISOs in the United States adopt the bid cost minimization (BCM) settlement mechanism for minimizing total offer costs. In this setup, payment costs, which are determined by a mechanism that assigns uniform market clearing prices (MCPs), are different from the minimized offer costs. Owing to the fact that the direct minimization of payment costs results in lower payment costs than those obtained by the minimization of offer costs, the PCM mechanism is potentially attractive for potential use by power markets. However, in spite of its apparent advantages, the PCM problem presents new challenges both
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conceptual and computational. In the PCM setup the presence of the cross product between MCP and generation levels makes the objective function non-separable. Furthermore, system and MCP constraints couple individual offers. As a result, the number of branching operations sharply increases as number of offers increases. The Lagrangian relaxation has been a powerful technique to solve separable and non-separable discrete and mixed-integer problems. By relaxing coupling constraints, the original problem is transformed into its dual, which can be solved easier in the dual space. After the problem is transformed into its dual, the multipliers are updated and the surrogate subgradient direction is obtained. The convergence of the standard Lagrangian relaxation and surrogate optimization approach has been proved under certain conditions, namely, the upper bound on the step size and the surrogate optimization condition. The condition on the step size depends on the optimal dual value, which is usually unknown or requires additional computational effort to compute. The novelty of the presented approach is to prove the convergence of the Lagrangian relaxation and surrogate subgradient approach when the optimal dual value is unknown. In order to accomplish this task, we first consider a specific version of the subgradient approach, and prove its convergence under the subgradient framework. However, due to the difficulty of the problem, it is hard to obtain the exact subgradient direction, so the surrogate subgradient approach is used to obtain an approximate solution. In order to ensure the convergence of the surrogate subgradient method, an analogue of the “surrogate optimization” condition should be derived. 1 The rest of the paper is organized as follows. Section II presents the literature review. The mathematical formulation of the PCM problem is presented in Section III. The total payment cost, which is to be optimized with respect to offer levels and MCP, is subject to system demand, start-up cost, MCP and generation level This work was supported in part by grants from Southern California Edison and by the National Science Foundation under grants ECS-0621936 and ECCS-1028870. Mikhail Bragin, Xu Han and Peter B. Luh are with the Department of Electrical Engineering, University of Connecticut, Storrs, CT 06269-2157, USA Joseph H. Yan is the Manager of Bidding Strategy & Asset Optimization, Southern California Edison, Rosemead, CA 91770 USA
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constraints. Section IV presents the solution methodology. The novel surrogate subgradient approach is presented; the key idea is to update the multipliers without invoking the optimal dual value. The Lagrangian is formed by relaxing the system demand constraint. Further, the relaxed problem is be solved approximately to obtain an a fortiriori “good” surrogate subgradient direction. Section V presents numerical results that demonstrate the performance and effectiveness of the method. Section VI concludes.
II. LITERATURE REVIEW There are several approaches that have been developed and used for solving integer and mixed integer programming problems. They are Lagrangian relaxation [1], Lagrangian relaxation and surrogate subgradient optimization [6] and branch-and-cut. Even though the literature on Lagrangian relaxation and branch-and-cut method is vast, the surrogate subgradient framework, which is used within the Lagrangian relaxation method, is somewhat incomplete and is still not well understood. The Lagrangian relaxation and surrogate subgradient optimization approach was specifically treated in [3] and [2]. The former paper develops the surrogate subgradient method and proves its convergence. Compared to subgradient and gradient methods the surrogate subgradient approach finds better directions within less CPU time. The latter paper extends the methodology to solving coupled problems. Since the optimal dual value or the optimal multipliers remain unknown, there the need for development of the surrogate subgradient method, the convergence of which does not depend on the optimal dual value.
III. PROBLEM FORMULATION This section presents the Payment Cost Minimization auction mechanism. We use standard simplifying assumptions that are usually made in the existing literature (e.g. in [4]): system demand is deterministic at each hour t and is known, transmission constraints are omitted for simplicity and are not considered in this paper, startup costs are fully compensated and participants submit a single-block offer with startup cost, offer price, and maximum and minimum generation levels for each hour t. Consider a market with I participants enumerated by 1,2,…,I. For each offer i, pi(t) denotes the offer generation level at time t; Si(t) denotes the startup cost at time t, which is incurred if and only if offer i is turned “on” at time t, after being “off” at time t-1; The PCM problem can be formulated in the following way [4]: min
{MCP ( t ) , pi
T
I
∑∑ ( MCP ( t ) p ( t ) + S ( t ) u ( t ) ) ( )} t
t =1 i =1
i
i
i
(1)
Subject to: 1)
I
Pd ( t ) = ∑ pi ( t ) ∀ ( t = 1, 2,..., T )
(System
demand
i =1
constraint); The system demand Pd(t) has to be satisfied exactly by the selected offer at each time t. 2) ui ( t ) ≥ ( xi ( t ) − xi ( t − 1) ) , ∀ ( i ∈ I , t = 1, 2,..., T ) (Startup-cost constraint); The startup cost binary decision variable equals 1 at time t if and only if an offer i is turned “on” from an “off” state, and 0 otherwise. 3) MCP(t ) ≥ oi ( t ) xi ( t ) , ∀ ( i ∈ I , t = 1, 2,..., T ) (MCP constraint); MCP(t), which denotes a uniform market clearing price at time t, depends on the selected offers at time t and is greater than or equal to the most expensive selected offer at time t. If offer i is “off” at time t, the offer price oi ( t ) has no effect on the MCP(t). 4) xi ( t ) pi min ( t ) ≤ pi ( t ) ≤ xi ( t ) pi max ( t ) , ∀ ( i ∈ I , t = 1, 2,..., T ) , (Generation level constraint) where pi,min(t)/pi,max(t) are minimum/maximum generation levels at time t; Generation levels of selected offers should neither exceed the maximum generation level nor be below the minimum generation level. If an offer i is “off” at time t, then the generation level is zero. The feasible region of power levels is discontiguous since pi(t) exhibits a jump from 0 to pi,min(t). IV. SOLUTION METHODOLOGY A. Lagrangian Relaxation and surrogate optimization This section presents the theoretical basis of the Lagrangian relaxation and surrogate optimization approach. To present the key result of the paper, we introduce a particular subgradient approach under an assumption on the step size with convergence guaranteed. Since obtaining the exact values for the decision variables can be time consuming, the decision variables are obtained approximately by applying only a finite number of successive approximations at each iteration. Even when the decision variables are obtained approximately, the subgradient approach can still be computationally expensive. The analogue of the surrogate optimization condition is derived to remove the assumption that was made about the step size and to make the transition from subgradient to the surrogate subgradient approach possible. For brevity of exposition, all decision variables and relaxed constraints are collectively denoted by x and g(x) respectively. The objective function is denoted by f(x). In the standard surrogate subgradient method a proper subgradient direction toward the optimal λ* can be obtained without solving optimally all the subproblems, hence, with much less computational effort. The method’s efficiency enables to solve large problems [6].
2
In a nutshell, in the surrogate subgradient method, the Lagrange multipliers are updated by: (2) λ k +1 = λ k + s k g% ( xˆ k ) ; where g% k = g% (λ k ) is the surrogate subgradient of L(λ) at λk and the step size sk satisfies 0 < sk
f ( x* ) + ( λ * ) g ( x* ) : T
2
2
λ * − λ k +1 ≤ λ * − λ k
(
T
−
) g ( x ) − f ( x ) − ( λ ) g ( x ) ) − c g ( xˆ )
* T
k +1 T
*
2
2
k
2
; (24)
2
λ * − λ k +1 ≤ λ * − λ k −
(
)
2c f ( x* ) + ( λ * ) g ( x* ) − f ( x* ) − ( λ k +1 ) g ( x* ) − c 2 g ( xˆ k ) ; (25) T
T
2
Due to the complementary slackness condition: λ * − λ k +1 ≤ λ * − λ k + 2c ( λ k +1 ) g ( x* ) − c 2 g ( xˆ k ) ; 2
Since ( λ
2
2
T
(26)
) g(x ) ≤ 0 ,
k +1 T
*
2
2
λ * − λ k +1 ≤ λ * − λ k . □
(27)
Proof of Corollary 1: The surrogate optimization condition can be rewritten as
VI. CONCLUSION This paper aims to reduce the gap in the knowledge concerning the convergence, applicability and the efficiency of the surrogate subgradient approach. A novel surrogate subgradient approach is presented, and its convergence is proved. The Lagrangian Relaxation and the surrogate subgradient optimization method provides reasonably good bounds on the optimal solution, which can be computed within reasonable amount of CPU time and is especially efficient for solving large-scale optimization problems. VII. APPENDIX.
L% ( λ k +1 , xˆ k ) ≤ L% ( λ k +1 , xˆ k −1 )
By the definition of xˆ
k
(
T
(28)
)
(
)
min f ( x) + λ k + cg ( xˆ k ) g ( x ) = f ( xˆ k ) + λ k + cg ( xˆ k ) g ( xˆ k ) , (29)
which implies
(
)
(
T
)
f ( x ) + λ k + cg ( xˆ k ) g ( x ) ≥ f ( xˆ k ) + λ k + cg ( xˆ k ) g ( xˆ k ) , ∀( x ) (30) T
T
Since the latter inequality holds for any x, it also holds for xˆ k −1 . In other words,
(
)
(
)
f ( xˆ k −1 ) + λ k + cg ( xˆ k ) g% ( xk −1 ) ≥ f ( xˆ k ) + λ k + cg ( xˆ k ) g ( xˆ k ) . (31) T
Proof of Corollary 2:
(
)
T
By the definition of xˆ k
(
)
f ( xˆ k ) + λ k + cg ( xˆ k ) g ( xˆ k ) ≤ f ( x) + λ k + cg ( xˆ k ) g ( x) , ∀( x) (32)
Proof of Proposition 1: Using (5),
T
T
(18)
Since the latter inequality holds for any x, it also holds for xˆ k +1 .
λ * − λ k +1 = λ * − λ k − 2c ( λ * − λ k ) g ( xˆ k ) + c 2 g ( xˆ k ) ; (19)
f ( xˆ k ) + λ k + cg ( xˆ k ) g ( xˆ k ) ≤ f ( xˆ k +1 ) + λ k + cg ( xˆ k ) g ( xˆ k +1 ) , (33)
λ * − λ k +1 = ( λ * − λ k ) − cg ( xˆ k ) ; 2
2
2
2
T
2
Using (5) one more time gives: 2
(
)
(
T
)
T
adding the positive term c g ( xˆ k +1 ) to the right hand side of 2
)
λ* − λk +1 = λ* − λk − 2c λ* − λk+1 + cg ( xˆk ) g ( xˆk ) + c2 g ( xˆk ) ; (20) 2
(
T
2
the inequality will make it bigger, so
5
(
)
(
)
f ( xˆk ) + λk + cg ( xˆk ) g ( xˆk ) ≤ f ( xˆk +1 ) + λk + cg ( xˆk ) g( xˆk +1 ) + c g ( xˆk +1 ) ,(34) T
T
2
The latter inequality is equivalent to
(
f ( xˆ ) + λ + cg ( xˆ k
k
k
)) g ( xˆ ) ≤ f ( xˆ ) + ( λ T
k +1
k
k +1
+ cg ( xˆ
k +1
k +1
(35)
(
)
(
)
f ( xˆ k ) + λ k + cg ( xˆ k ) g ( xˆ k ) ≤ f ( x) + λ k + cg ( xˆ k ) g ( x) , ∀( x) (36) T
Since the latter inequality holds for any x, it also holds for xˆ * . f ( xˆ
k
) + (λ
k
+ cg ( xˆ
k
) ) g ( xˆ ) ≤ f ( x ) + ( λ T
*
k
+ cg ( xˆ
k
)) g ( x ) T
k
*
(37)
Since ( λ k +1 ) g ( x* ) ≤ 0 and ( λ * ) g ( x* ) = 0 , T
T
(
therefore,
(
f ( xˆ k ) + λ k + cg ( xˆ k )
) g ( xˆ ) ≤ f ( x ) + ( λ ) g ( x ).
(39)
T
* T
*
*
* T
*
k
T
0 ≥ f ( x ) + ( λ ) g( x ) − ( λ ) g( xˆ ) − f ( x ) − ( λ ) g( x ) + ( λ * T
* T
*
k 1
* T
*
Given (14), f ( xˆ k +1 ) + ( λ k +1 ) g% ( xˆ k +1 ) + c g% ( xˆ k +1 ) ≤ L* ,
in view of non-negativity of c g% ( xˆ k +1 ) ,
[2] [3]
k 1
0 ≥ − ( λ* ) g ( xˆ1k ) + ( λ k +1 ) g ( xˆ1k ) T
(42)
[4]
The latter inequality implies ( λ * − λ k +1 ) g ( xˆ1k ) ≥ 0 .□ T
[5]
Proof of Proposition 3: Similar to Proposition 1,
(
)
λ* − λ k +1 ≤ λ* − λ k − 2c λ* − λ k − cg ( xˆ0k ) g ( xˆ1k ) − c2 g ( xˆ1k ) (43) 2
2
T
2
By proposition 2,
(λ
*
− λ k − cg ( xˆ0k )
therefore,
(λ
*
− λ − cg ( xˆ k
k 1
) g ( xˆ ) ≥ 0 , T
k 1
IX. BIOGRAPHIES
k 1
k 1
k 0
k 1
(45)
so provided the following condition
(
)
g ( x1k ) g ( x0k ) − g ( x1k ) ≥ 0
(46)
holds and the multipliers are updates as λ then the inequality (43) implies λ −λ *
k +1 2
≤ λ −λ *
k 2
(
k +1
= λ + cg ( xˆ1k ) , k
)
− 2c λ − λ − cg ( xˆ1k ) g ( xˆ1k ) − c2 g ( xˆ1k ) , (47) *
k
T
2
so 2
[6]
D. P. Bertsekas, Nonlinear Programming, Massachusetts: Athena Scientific, 1999. T. S. Chang, “Comments on Surrogate Gradient Algorithm for Lagrangian Relaxation,” Journal of Optimization Theory and Applications, 2007. X. Han, P. B. Luh, J. H. Yan and G. A. Stern, “Payment cost minimization with transmission capacity constraints and losses using the objective switching method,” Proceedings of the IEEE Power Engineering Society 2010 General Meeting, Minneapolis, Minnesota, July 2010. P. B. Luh, W. E. Blankson, Y. Chen, J. H. Yan, G. A. Stern, S. C. Chang and F. Zhao, "Payment Cost Minimization Auction for the Deregulated Electricity Markets Using Surrogate Optimization," IEEE Transactions on Power Systems, Vol. 21, No. 2, 2006, pp. 568578. F. Zhao, P. B. Luh, J. H. Yan, G. A. Stern and S. C. Chang, “Payment Minimization Auction for Deregulated Electricity Markets with Transmission Capacity Constraints,” IEEE Transactions on Power Systems, Vol. 23, No. 2, May 2008, pp. 532-544. X. Zhao, P. B. Luh, and J. Wang, “Surrogate Gradient Algorithm for Lagrangian Relaxation,” Journal of Optimization Theory and Applications, Vol. 100, No. 3, March 1999, pp. 699-712.
(44)
)) g ( xˆ ) ≥ g ( x ) ( g ( x ) − g ( x ) ) , T
(52)
Inequality (13) follows likewise.
) g( xˆ ) ( 41)
k +1 T
*
(51)
2
which leads to T
2
T
T
By the arguments similar to those of the Proposition 1, *
(50)
T
[1]
0 = f ( xˆ1k ) + ( λ* ) g( xˆ1k ) −( λ* ) g( xˆ1k ) − f ( xˆ1k ) −( λk+1) g( xˆ1k ) + ( λk+1) g( xˆ1k ) (40) T
f ( xˆ k ) + ( λ k +1 ) g% ( xˆ k ) ≤ L* ,
VIII. REFERENCES
*
Proof of Proposition 2: T
(49)
2
T
T
(38)
*
f ( xˆ k ) + ( λ k ) g% ( xˆ k ) + c g% ( xˆ k ) ≤ L* ,
f ( xˆ k +1 ) + ( λ k +1 ) g% ( xˆ k +1 ) ≤ L* .
) g ( x ) ≤ f ( x ) + (λ ) g ( x ) ,
f ( x* ) + λ k + cg ( xˆ k )
T
(48)
Taking into account λ k +1 = λ k + cg% ( xˆ k )
Proof of Corollary 3: By the definition of xˆ k T
2
T
For iteration k-1,
)) g ( xˆ ). T
f ( xˆ1 ) + ( λ 1 ) g% ( xˆ1 ) + c g% ( xˆ1 ) ≤ L*.
2
λ * − λ k +1 ≤ λ * − λ k .
Therefore, g ( xˆ2k ) is a proper subgradient direction. □ Proof of Corollary 4: This corollary is proved by induction. For iteration 0, by an appropriate choice of c, λ1, and xˆ1 the following inequality can be satisfied:
Mikhail Bragin received the B.S. degree in mathematics from the Voronezh State University, Russia, in 2004, the M.S. degree in physics and astronomy from the University of Nebraska-Lincoln, USA, in 2006. He is currently pursuing the Ph.D. degree in electrical and computer engineering at the University of Connecticut, Storrs. His research interests include optimization, operations, and economics of electricity markets. Xu Han received the B.S. degree in automation the M.S. degree in control theory and control engineering from the University of Science and Technology Beijing, China, in 2005 and 2008. He is currently pursuing the Ph.D. degree in electrical and computer engineering at the University of Connecticut, Storrs. His research interests include optimization, operations, and economics of electricity markets. Peter B. Luh (S’77–M’80–SM’91–F’95) received the B.S. degree in electrical engineering from National Taiwan University, Taipei, in 1973, the M.S. degree in aeronautics and astronautics engineering from Massachusetts Institute of Technology (M.I.T.), Cambridge, in 1977, and the Ph.D. degree in applied mathematics from Harvard University, Cambridge, in 1980. Since then, he has been with the University of Connecticut, and currently is the SNET Professor of Communications and
6
Information Technologies and the Head of the Department of Electrical and Computer Engineering. He is also a member of the Chair Professors Group at the Center for Intelligent and Networked Systems, the Department of Automation, Tsinghua University, Beijing, China. He is interested in planning, scheduling, and coordination of design, manufacturing, and supply chain activities; configuration and operation of elevators and HVAC systems for normal and emergency conditions; schedule, auction, portfolio optimization, and load/price forecasting for power systems; and decisionmaking under uncertain or distributed environments. Dr. Luh is Vice President of Publication Activities for the IEEE Robotics and Automation Society, an Associate Editor of IIE Transactions on Design and Manufacturing, an Associate Editor of Discrete Event Dynamic Systems, and was the founding Editor-in-Chief of the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING (2003–2007) and the Editor-in-Chief of IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION (1999–2003). Joseph H. Yan (M’02–SM’06) received the Ph.D. degree in electrical and systems engineering from the University of Connecticut, Storrs. He is currently the Manager in the Department of Bidding Strategy and Asset Optimization of Southern California Edison. He was involved in areas of the electricity system operations, wholesale energy market analysis for both regulated and nonregulated affiliates, and market monitoring and market design for ten years in California. His research interest includes operation research, optimization, unit commitment/scheduling and transaction evaluation, and optimal simultaneous auction in deregulated ISO/RTO markets.
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