Several Notes on Lagrangian Relaxation for Unit Commitment

Proceedings of the 29th Chinese Control Conference July 29-31, 2010, Beijing, China

Several Notes on Lagrangian Relaxation for Unit Commitment ZHAI Qiaozhu, WU Hongyu . Systems Engineering Institute, SKLMS Laboratory, Xian Jiaotong University, Shaanxi 710049, P. R. China E-mail: [email protected], [email protected] Abstract: Lagrangian Relaxation (LR) based algorithms are among the most successful methods for unit

commitment (UC) and short term hydro-thermal scheduling. In this paper, some important problems related to LR are studied and summarized. Such problems include the possibility of positive duality gap, the difference between relative and absolute duality gap, and economical interpretation of the optimal multipliers, and so on. The results obtained can be viewed as relevant addition to LR for UC and are useful to lead to a better understanding of the nature of LR.

Key Words: Duality Gap, Hydro-Thermal Scheduling, Lagrangian Relaxation, Unit Commitment

1

INTRODUCTION

Unit commitment (UC) is to determine the commitment states and generation levels of all generating units (thermal and hydro units) over a period of time to minimize the total operating cost ([1-5]). With the optimal or near optimal unit commitment, it is possible to significantly reduce the total operating cost and pollution. The potential cost savings of a power system with capacity of 1000MW to 3000MW by optimizing the unit commitment may amount to millions of dollars one year if only the operating cost is reduced by 1%. In the newly deregulated electric power market, unit commitment is integrated with the generation supply bidding and still plays a key role in determining the market bidding strategies. In the electricity market with simplified biding rule, a power supply company needs to aggregate its power generation awards as the system demand after the auction, and performs unit commitment to meet its total obligation at the minimum cost while satisfying operation constraints ([1]). Developing an efficient algorithm for Unit Commitment is therefore very important. Lagrangian Relaxation (LR) based methods and Mixed Integer Programming (MIP) based methods are among the most successful approaches for unit commitment ([2, 4-8]). The basic idea of LR is to use Lagrange multipliers to relax system-wide constraints such as demand and reserve requirements. The problem is then converted into a two-level structure. Given a set of multipliers, all the subproblems are solved at the low level, one for each unit, and the multipliers are updated at the high level. The procedure will be stopped when the multipliers are near the optimal multipliers. A feasible unit commitment is generally obtained based on the dual solution. One of the

most obvious advantages of LR is its quantitative measure of the solution quality since the cost of the dual function is a lower bound on the cost of the primal problem. That is, the duality gap can be used to evaluate the quality of the feasible solution obtained. Besides, according to the theory of mathematical programming, the (near) optimal multipliers are believed very important since they give the information of shadow prices ([8-10]). An upper bound to the relative duality gap for UC problems is given in [11] and it is also pointed out that the upper bound tends to zero with the increasing of problem scale. The result is very important theoretically. However, the possibility of positive duality gap for UC problem has not been discussed in literature to the best of our knowledge. Hence, it should be clarified that whether there are some UC problems for which the duality gap is greater than zero. Meanwhile, the optimal multipliers can be interpreted as shadow prices only when the duality gap is zero. Then, what information can be obtained based on the (near) optimal multipliers if the duality gap is other than zero? The above problems are studied in this paper. Firstly, a UC example is given and it is proved that the duality gap for this example is positive. Then, a detailed analysis of the relationship between (near) optimal multipliers and the optimal cost of primal problem is established. The results obtained can be viewed as relevant addition to LR based methods for UC and are useful to lead to a better understanding of the nature of LR. The paper is organized as follows. The mathematical formulation of UC problem is given in section II. The LR based algorithm and the concept of duality gap is summarized in section III. Then, the UC example with positive duality gap and the interpretation of the multipliers are discussed in section IV and the paper is concluded in section V.

*The research presented in this paper is supported in part by National Natural Science Foundation of China (60736027), 863 High Tech Development Project (2007AA04Z154) and Program for New Century Talents of Education Ministry (NCET-08-0432) of China.

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2

PROBLEM FORMULATION

Suppose there are I electric power generating units and the time horizon is T , the UC problems considered in this paper are then formulated as, I

T

[

]

min J ( p, u ) = ∑∑ Ci ( pit ) + S i ( x it −1 , u it −1 ) , i =1 t =1

[ pit ] I ×T

[ xit ] I ×T

(1)

[uit ] I ×T

, and are generation levels, where discrete states and start-up/shut-down operations of different units in different time period respectively ([14]); Ci ( pit ) the fuel cost of unit i at time t ; S i ( xit , uit ) the cost associated with start-up/shut-down operations. The constraints of the problem are classified into two categories: system-wide constraints and individual unit constraints. System-wide constraints include: I

∑ pit = Pdt , t = 1,2,!, T ,

(2)

i =1 I

∑ r ( p ) ! P , t = 1, 2,!.T i =1

i

t i

t r

,

(3)

where Pdt and Prt are the system load demand and reserve requirement at time t ; ri ( pit ) the spinning reserve contribution of unit i at time t . Individual unit constraints include: Evolution of discrete state ⎧⎪x it + uit , if xit u it > 0 =⎨ else ⎪⎩u it , Minimum up/down time xit +1

(4)

⎧1, if 1 " xit < τ i ⎪ uit = ⎨−1, if − τ i < xit " −1 , ⎪any one of {1, −1}, else ⎩ (5) where τi and τ i are the minimum up time and minimum down time of unit i respectively. It can be seen from equations (4), (5) that xit is in fact the number of time periods that unit has been kept up (positive value) or down (negative value). Capacity limits

⎧⎪ p = 0, if x < 0 , ⎨ t t t p " p " p , if x > 0 i i ⎪⎩ i t i

where

t i

t p i

and

pit

pit − pit +1 " ∆i ,

if xit > 0, xit +1 > 0

(7) Minimum generation for the first and last period

if ( xit > 0 and x it −1 < 0) or ( xit > 0 and xit +1 < 0)

(8)

Constraints (7) and (8) may be effective only for some special units. Initial state constraint xi0 = xiinitial and pi0 = piinitial

(9) [ pit ] I ×T

[uit ] I ×T

and , Clearly, the decision variables are and therefore the total cost in equation (1) is formulated as J ( p, u ) . Besides, ri ( pit ) in equation (3) is defined as ⎧⎪min{ri , pi − pit } , if xit > 0 ri ( pit ) = ⎨ , ⎪⎩ 0 , else (10) where ri is the maximum allowable spinning reserve contribution. However, the particular form of ri ( pit ) is immaterial to the rest of this paper. The above formulation is basically that adopted in [14] and slightly different from those in [5, 11] (‘=’ in equation (2) is replaced by ‘ ≥ ’ in these references but it’s equivalent under most cases). For convenience, two different classes of feasible solutions are defined as Feasible solutions to the primal problem FP = {( p, u) | ( p, x, u ) satifies equations (2) to (9)} (11) Feasible solution to individual units FI = {( p, u ) | ( p, x, u ) satifies equations (4) to (9)} (12) Then it’s clear that FP ⊆ FI (13)

3

FRAMEWORK OF LR

For the UC problem formulated by equations (1)-(9), the dual objective function under Lagrangian Relaxation (LR) framework is defined as follows.

∑ [Ci ( p it ) + S i ( xit −1 , u it −1 )]

L(λ, µ ) = min

( p ,u )∈FI

i ,t

T

I

I

t =1

i =1

+ ∑ [λ t ( Pdt − ∑ pit ) − µ t (∑ ri ( pit ) − Prt )] i =1

T

I

I

t =1

i =1

i =1

= min { J ( p, u ) + ∑ [λ t ( Pdt − ∑ pit ) − µ t (∑ ri ( pit ) − Prt )]}

(6)

( p ,u )∈FI

(14)

are the minimum and maximum

generation level of unit i respectively. Ramp rate constraints

pit = p ti and ri ( pit ) = 0,

where λ and µ are Lagrange multipliers defined as λ = [λ 1 , λ 2 , ! , λ T ], µ = [µ 1 , µ 2 , !, µ T ] (15) Clearly the dual objective function can be re-arranged as T

I

t =1

i =1

L(λ, µ ) = ∑ (λ t Pdt + µ t Prt ) + ∑ Li (λ, µ ) , (16)

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where Li (λ, µ) corresponds to optimal objective of the i-th dual sub-problem defined as follows. T

∑ [Ci ( pit ) ( p ,u )∈FI

Li (λ, µ ) = min +

t =1 t −1 t −1 S i ( xi , u i ) −

L∗ = max L(λ, µ)

Suppose that ( p ∗ , u ∗ ) and (λ ∗ , µ ∗ ) are optimal solutions to the primal problem and the dual problem respectively (the optimal solution may not be unique), then according the well-known duality theory it holds that,

L(λ ∗ ,µ ∗ ) = L∗ " J ∗ = J ( p ∗ , u ∗ )

and the fuel cost function is

⎧0.5 × p + 2 , if 2 " p " 4 . c( p ) = ⎨ 0 , else ⎩ Then J ∗ = 3.5 and p ∗ = 3 . The dual problem is defined as, L∗ = max L(λ ) = max{min[C ( p) + λ × (3 − p)]} λ

λ

p ,u

It is easy to verify that

(19) The difference of J ∗ − L∗ is the (inherent) duality gap in absolute terms and ( J ∗ − L∗ ) / L∗ or ( J ∗ − L∗ ) / J ∗ is the duality gap in relative terms. Therefore, if a feasible solution to the primal problem and a feasible solution to the dual problem satisfies

J ( p, u ) − L(λ,µ) "ε , J ( p, u ) (20) where ε is a small positive number, then a pair of near optimal solutions to both problems are obtained. Since the dual problem defined by equation (18) is essentially a convex problem (to maximize a concave function is equivalent to minimize a convex function) and is generally nondifferentiable, methods for nonsmooth and convex programming are usually adopted. Considering the simplicity of the feasible region of the dual problem, cutting plane methods for convex programming is not a good choice. Therefore, subgradients methods and bundle methods ([4-5, 12]) are two popular approaches for solving the dual problem. The basic idea of these methods is to generate a sequence of feasible multipliers (a new pair of multipliers is generated by adjust the current multipliers along a good direction) (λ k , µ k ) such that, lim (λ k , µ k ) = (λ ∗ , µ ∗ ) or lim L(λ k , µ k ) = L∗ k → +∞

(21)

4

⎧ p = 0 , if u < 0 (Individual Unit Constraint), ⎨ ⎩2 " p " 4 , if u > 0

(18)

µ!0, λ

(Systems Demand)

s.t. p = 3

λ t pit − µ t ri ( pit )]

(17) The sub-problem is only relevant to unit i . The Dual Problem (DP) to the primal one can then be formulated as

k → +∞

literature to the best our knowledge. In order to clarify this point, a counterexample is given here. Example 1 Consider a simple UC problem as follows, min J ( p, u) = C ( p)

IMPORTANT NOTES ON LR

4.1 Note on the Duality Gap The criterion of equation (20) can be used only when the theoretical duality gap ( J ∗ − L∗ ) / J ∗ is zero or very small. Is it holds for UC problems? Though it’s known that a positive duality gap may be possible for nonconvex programming, no counterexamples to UC are reported in

⎧3λ , if λ " 1 L (λ ) = ⎨ ⎩4 − λ , if λ ! 1 So we have L∗ = 3 and λ ∗ = 1 , and a positive duality gap exists. The authors believe that if the fuel-cost functions are replaced by bidding curves in deregulated electric power markets, the duality gap will be still larger. In [11], it is proved that the relative duality gaps goes to zero with the number of units increasing to infinity (an alternative method for estimating the duality gap is given in [13]). This conclusion itself is important but one must keep in mind that the number of units cannot be too large (e.g., larger than 1000) for practical systems. In fact, an upper bound to the absolute duality gap given in [11] is

J ∗ − L∗ " 2aT + 2bT 2 , if I > 2T , (22) where a = max{S i ( −τ i ,1)} and b = max{C i ( p )} . If we i

i

i

consider a UC within a week and the time period is one hour, then I > 2T means that there are more than 300 units to be scheduled. Apart from this, the right-hand of equation (22) is very large and does it imply the possibility of a large duality gap (in absolute terms)? The condition of I > 2T in equation (22) can be omitted by using the method given in [13] (need a little revise) but the upper bound is still larger than bT 2 and therefore we must admit that the absolute duality gap may be very large even if the relative duality gap do tend to zero with the scale of the problem increasing. It is the absolute duality gap, rather than the relative one, which we care much. Though, it must be emphasized that all the numerical results in literature shows the duality gap is far smaller than the upper bound given above. In our experiences, the relative duality gap is generally less than 1% and tends to be smaller for large-scale problems. So some new methods for a better estimation of the duality gap should be explored.

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If the effect of CPU time is not considered and LR is only used to give a lower bound to J ∗ then general methods for mixed integer programming (MIP) are better choices. Since the basic idea of such methods (e.g., branch and bound, cutting plane, or their mixture) is enumeration, near optimal solution within any required error limitation can be found after a long time computation. 4.2 Economical Interpretation of the Optimal Multipliers It is well-known that for smooth mathematical programming, under some strong conditions, the optimal multipliers is exactly the derivatives of the perturbation function (optimal primal objective as a function with the right-hand of constraints). For convex programming, if the theoretical duality gap is zero, then the optimal multipliers correspond to the subgradient of the perturbation function. For general MIP problems, however, few has been stated in literature. In deregulated electric power markets, the interpretation of the multipliers is viewed as an important problem that requires fundamental research ([8], Chapter 1). For UC problem, if the theoretical duality gap is other than zero, the optimal multipliers may hardly have any importance. The following example shows this point.

Let J" ∗ be the optimal value of the perturbed UC and (λ ∗ , µ ∗ ) the optimal solution to the dual problem, then we have the following conclusion. Theorem 1 If J ∗ = L∗ , then J" ∗ ! J ∗ + λ ∗ ( P"d − Pd )T + µ ∗ ( P"r − Pr )T (26) Proof By the assumption and definition we have, J ∗ = L∗ = L(λ ∗ , µ ∗ ) = min {J ( p, u ) + ( p ,u )∈FI T

I

I

t =1

i =1

i =1

∑ [λ *t ( Pdt − ∑ pit ) − µ *t (∑ ri ( pit ) − Prt )]} According to equation (25), the right-hand

"

{J ( p, u ) +

min

( p ,u )∈FP ( P"d , P"r )

T

I

I

t =1

i =1

i =1

∑ [λ* t ( Pdt − ∑ pit ) − µ* t (∑ ri ( pit ) − Prt )]} By the definition of FP ( P"d , P"r ) (see equation (24)),

"

min"

T

{ J ( p , u ) + ∑ [λ* t ( Pdt − P"dt ) − µ * t ( P"rt − Prt )]}

( p ,u )∈FP ( Pd , P"r )

t =1

= J" − λ ( P"d − Pd ) − µ ( P"r − Pr )T ∗

T

∗

∗

Example 2 If the system demand in Example 1 is 3 + ε , where −1 < ε < 1 , it can be seen that the optimal value of the primal objective is 3.5 + 0.5 × ε . The derivative of this

Now the conclusion is at hand.

value with respect to ε is 0.5 but λ ∗ = 1 .

the marginal cost in a sense.

Meanwhile, it should be noted that even if there is no duality gap for a given UC problem, we could only get the near optimal multipliers rather than the exact optima in numerical computation. So, the importance of the multipliers should never be exaggerated. Another question is: if there is no duality gap then which information can be got from the optimal multipliers? To answer this question, some notations must be introduced. 1 2 T 1 2 T ⎪⎧ Pd = [ Pd , Pd ,! , Pd ] , Pr = [ Pr , Pr ,!, Pr ] ⎨" 1 2 T 1 2 T ⎪⎩ Pd = [ P"d , P"d ,! , P"d ] , P"r = [ P"r , P"r ,!, P"r ] (23) P" and P" can be viewed as the system demand and reserve d

r

requirement with some perturbations. With the perturbations, the feasible solution to the primal UC problem must change and they are characterized by the following set. FP( P"d , P"r ) = (24) {( p, u) | ( p, u) is feasible to the perturbed UC} Then we still have (see equation (12)) FP ( P" , P" ) ⊆ FI d

(25)

r

Q.E.D. ∗

∗

∗

∗

Equation (26) implies that if J = L then (λ , µ ) is still For example, if Pdt is

increased to Pdt + ε at some time period t and other components of Pd and Pr keep unchanged, then we have J" ∗ ! J ∗ + ε ⋅ λ∗ t (27)

For this reason, it may be more precise to call (λ ∗ , µ ∗ ) a lower bound on shadow prices. Since the optimal multipliers may not be unique (the counterexample is easy to construct), the ‘maximum’ of the optimal multipliers is of crucial importance (the best lower bound on shadow prices). It is interesting to note that such relation does not hold for Example 2 where the duality gap is positive. That is, the following inequality does not hold. (×) 3.5 + 0.5 × ε ! 3.5 + ε Therefore, it is interesting to investigate the interpretation of the optimal multipliers when the duality gap is great than zero. Theorem 1 is thus generalized and an interesting conclusion is obtained as follows. Theorem 2 No matter what the duality gap is, we always have J" ∗ ! L∗ + λ ∗ ( P" − P )T + µ ∗ ( P" − P )T d

d

r

r

(28) Proof According to the proof of Theorem 1, we have the following result

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L∗ " J" ∗ − λ ∗ ( P"d − Pd )T − µ ∗ ( P"r − Pr )T (29) Besides, we have the following equation

efforts. The estimation is very useful in determining the step size for updating the multipliers ([10, 14]). 4.4 Estimation on the Range of the Profit of Thermal Producer in Electric Power Market It is very important for electric energy producers to estimate the profits in deregulated electric power markets ([16]). In this aspect, we found that a fast approach for obtaining the possible range of electric energy producers’ profits can be established based on the LR framework for UC. This will be very useful in making the bidding strategy or evaluating the rationality of market prices. In fact, suppose [ p" it ]I ×T , [u"it ]I ×T , λ and µ (see (15)) are

J ∗ = J ∗ − L∗ + L∗ (30) Replace the last item in (30) by using (29) we have J ∗ " J ∗ − L∗ + J" ∗ − λ ∗ ( P" − P )T − µ ∗ ( P" − P )T d

d

r

r

(31) Rearrange the items in (31) and (28) is at hand. Q.E.D. The conclusion given in Theorem 2 is very interesting since it implies that the optimal multipliers still give some lower bound information on the primal optimal cost even if the duality gap is positive. If the conclusion is applied for Example 1, we have the following result. (32) 3.5 + 0.5 × ε ! 3 + ε It should be noted that (32) holds only when −1 " ε " 1 for otherwise the perturbation UC problem will be infeasible. In fact, the value of J" ∗ can be used only when the perturbation UC problem is feasible. 4.3 Fast Estimation on the Lower Bound of the Optimal primal Cost Another interesting result is that a fast approach for estimating the lower bound of the optimal primal cost can be established based on Theorem 2. The basic idea is to construct a set which contains a group of system demands and the corresponding optimal multipliers and optimal dual costs (the data can be obtained based on historical UC cases), and then obtain the best lower bound estimation for the current scheduling case based on (28). More clearly, we can get the following set of system demands based on the information of historical UC cases. H = { ( Pd , Pr )}

(33) Then, for each pair of system demand, the optimal multipliers and dual objective are denoted as follows (these values have been obtained when solving the historical UC cases under LR framework). (λ∗(Pd , Pr ) , µ (∗Pd , Pr ) ) , L∗(Pd , Pr ) (34) The subscripts are added since the optimal multipliers and dual objectives are different for different system demands. Now, suppose P"d and P"r are the system demand of the current UC case and J" ∗ the optimal primal cost, the following result can be obtained directly based on (28). J" ∗ ! max L∗ + λ∗ ( P" − P )T + µ ∗ ( P" − P )T ( Pd , Pr )∈H

{

(Pd , Pr )

(Pd , Pr )

d

d

(Pd , Pr )

r

r

}

(35) In general, a fairly good lower bound estimation can be obtained based on (35) with nearly no extra computation

the results of market clearing computation (i.e., [ p" it ]I ×T is the awarded generation level and λ , µ are the market prices), the profit of unit i (denoted by f i ) can then be determined by the following equation. T

fi = − { ∑ [Ci ( p" it ) + Si ( x"it −1 , u"it −1 ) − λ t p" it − µ t ri ( p" it )] } t =1

(36) It is clear that ([ p" it ]I ×T , [u"it ]I ×T ) ∈ FI (37) Now, let Ri (λ, µ) = max

( p , u )∈FI

T

∑ [C ( p ) t =1

i

t i

+ S i ( xit −1 , u it −1 ) − λ t pit − µ t ri ( pit )] (38) Then the following equation can be obtained based on (17), (36)-(38). − Ri (λ, µ) " f i " − Li (λ, µ) (39) It is seen that the range of profit of unit i can thus be obtained. In (39), Li (λ, µ) can be obtained efficiently based on the method established [15] and Ri (λ, µ) can be obtained similarly. It should be noted that Li (λ, µ) and Ri (λ, µ) are not dependent on the specific bidding strategy, i.e., the range given in (39) is valid for all possible bidding strategies. The result is therefore very useful in evaluating the quality of bidding strategies or evaluating the rationality of market prices.

5

CONCLUSION

LR based methods are among the most successfully approaches for UC problem. The duality gap can be used to evaluate the quality of the feasible solution obtained based on the dual solution. However, there are some UC problems for which the duality gap may be positive rather than zero. Besides, if the duality gap is zero then the optimal multipliers can give the lower bound on incremental optimal cost when the system demand is changed. If the duality gap is not zero, on the other hand, the optimal

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multipliers can be used to derive some lower bound on the primal optimal cost rather than the incremental optimal cost. The LR framework can also be utilized in the market environment. The conclusions obtained in this paper can thus be viewed as relevant addition to LR based methods for UC and are useful to lead to a better understanding of the nature of LR.

REFERENCES [1] A.

Renaud, “Daily Generation Management at Electricite de France: From Planning Towards Real Time,” IEEE Trans. On Automatic Control, 1999, 38(7): 1080-1093. [2] N. P. Padhy,“Unit Commitment – A Bibliographical Survey,” IEEE Trans. on Power Systems, 2004, 19(2): 1196-1205. [3] Cohen and V. Sherkat, “Optimization-Based Methods for Operations Scheduling,” Proceedings of IEEE, 1987, 75(12): 1574-1591. [4] X. Guan, S. Guo, and Q. Zhai. “The Conditions for Obtaining Feasible Solutions to Security-Constrained Unit Commitment Problems,” IEEE Transactions on Power Systems, 2005, 20(4): 1746-1756. [5] J. J. Shaw, “A Direct Method for Security Constraints Unit Commitment,” IEEE Trans. on Power Systems, 1995, 10(3): 1329-1339. [6] Y. Fu, M. Shahidehpour. “Fast SCUC for Large-Scale Power Systems,” IEEE Transactions on Power Systems, 2007, 22(4): 2144-2151. [7] M. Carrion, J. M. Arroyo. “A Computationally Efficient Mixed-Integer Linear Formulation for the Thermal Unit Commitment Problem,” IEEE Transactions on Power Systems, 2006, 21(3): 1371-1378. [8] B. F. Hobbs, M. H. Rothhopf, R. P. Oneill, H. Chao. The Next Generation of Electric Power Unit Commitment Models. Kluwer Academic Publishers, 1999. [9] Nemhauser G. L. and Wolsey L. A., Integer and Combinatorial Optimization, Wiley, Chichester, UK, 1988. [10] M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiely & Sons, Inc., 1993(2nd Edition).

[11] D. P. Bertsekas, G. S. Lauer, N. R. Sandell, T. A.

Posbergh, “Optimal Short-Term Scheduling of Large-Scale Power Systems,” IEEE Trans. on Automatic Control, 1983, 28(1): 1-10. [12] P. B. Luh, D. Zhang and R. N. Tomastik. “An Algorithm for Solving the Dual Problem of Hydrothermal Scheduling,” IEEE Transactions on Power Systems, 1997, 13(2): 593-600. [13] D. P. Bertsekas and N. R. Sandell, Jr., “Estimates of the Duality Gap for Large-Scale Seperable Integer Programming Problems,” Proc. Of IEEE Conf. Decision and Contr., Miami Beach, FL, Dec. 1982: 782-785. [14] Guan X, Luh P B, Yan H. An Optimization-Based Method for Unit Commitment. International Journal of Electric Power & Energy Systems, 1992, 14(1): 9-17. [15] Qiaozhu Zhai, Xiaohong Guan, and Feng Gao, “Production scheduling with hybrid dynamics and constraints,” 43rd IEEE Conference on Decision and Control, December 2004, Atlantis, Paradise Island, Bahamas, 0476-THA01.6, 2004. [16] J. M. Arroyo and A. J. Conejo, “Optimal response of a thermal unit to an electricity spot market,” IEEE Transactions on Power Systems, vol. 15, no. 3, pp. 1098–2000, Aug. 2000. Qiaozhu Zhai received the B.S. and M.S. degrees in applied mathematics and the Ph.D. degree in systems engineering from Xi’an Jiaotong University, Xi’an, China, in 1993, 1996, and 2005, respectively. He is currently an Associate Professor with the Systems Engineering Institute of Xi’an Jiaotong University. His research interests include optimization of large-scale systems and integrated resource bidding and scheduling in the deregulated electric power market. Hongyu Wu received his B.S. degree in energy and power engineering from Xi’an Jiaotong University, China in 2003. He is pursuing the Ph.D. degree at the System Engineering Institute, Xi’an Jiaotong University. His research interests include optimization of large-scale systems and scheduling in the deregulated electric power market.

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