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Fast Identification of Inactive Security Constraints in SCUC Problems Qiaozhu Zhai, Member, IEEE, Xiaohong Guan, Fellow, IEEE, Jinghui Cheng, and Hongyu Wu
Abstract—Security constrained unit commitment (SCUC) is one of the most important daily tasks that independent system operators (ISOs) or regional transmission organizations (RTOs) must accomplish in daily electric power market. Security constraints have long been regarded as difficult constraints for unit commitment problems. If the inactive security constraints can be identified and eliminated, the SCUC problem can be greatly simplified. In this paper, a necessary and sufficient condition for a security constraint to be inactive is established. It is proved that all inactive constraints can be identified by solving a series of small-scale mixed integer linear programming (MILP) problems. More importantly, an analytical sufficient condition is established and most of the inactive constraints can be quickly identified without solving MILP or linear programming (LP) problems. A very important feature of the conditions obtained is that they are only related to the load demands and parameters of the transmission network. Numerical testing is performed for three power grids and the results are impressive. Over 85% of the security constraints are identified as inactive and the crucial transmission lines affecting the total operating cost are among those associated with the remaining security constraints, providing useful information for transmission planning. Index Terms—Inactive constraints, linear programming, mixed integer programming, security constrained unit commitment.
I. INTRODUCTION
S
ECURITY constrained unit commitment (SCUC) is one of the most important daily tasks that an independent system operator (ISO) or regional transmission organization (RTO) must accomplish for clearing a daily electric power market, and one of the most important tools that a generation company has to use for optimizing bidding strategy [1]–[3]. The aim of SCUC is to minimize the total bid or operating cost while meeting all system-wide constraints such as system demands, reserve requirements, security related transmission constraints, and individual unit operating constraints such as Manuscript received June 10, 2009; revised October 29, 2009. First published March 25, 2010; current version published October 20, 2010. This work was supported in part by the National Natural Science Foundation (60921003, 60736027, 60704033) , in part by the Program for New Century Talents of Education Ministry (NCET-08-0432), in part by the 863 High Tech Development Plan (2007AA04Z154), and in part by the 111 International Collaboration Program, of China. Paper no. TPWRS-00440-2009, Q. Zhai, J. Cheng, and H. Wu are with the Systems Engineering Institute, SKLMSE Lab and MOE KLINNS Lab, Xi’an Jiaotong University, Xi’an 710049, China (e-mail:
[email protected];
[email protected];
[email protected]). X. Guan is with the Systems Engineering Institute, SKLMSE Lab and MOE KLINNS Lab, Xi’an Jiaotong University, Xi’an 710049, China, and also with the Center for Intelligent and Networked Systems, and TNLIST, Tsinghua University, Beijing 100084, China (e-mail:
[email protected]). Digital Object Identifier 10.1109/TPWRS.2010.2045161
minimum/maximum generation, minimum up/down times, ramping limits, and so on. Many optimization methods including Lagrangian relaxation (LR) [4]–[7], [13], Benders’ decomposition (BD) [2], [3], [8], [9], [14], branch and cut (BC) [1], [10] for general mixed integer linear programming (MILP), and meta-heuristics [11], [12] are applied to solve SCUC and security constrained economic dispatch as reported in the literature. A comprehensive bibliographical survey is given in [17] and different methods used in the unit commitment (UC) problem-solving techniques are summarized. The most successful methods include Lagrangian relaxation, general MILP methods such as branch and cut and Benders’ decomposition [1]–[14], [17]. Security constraints have long been regarded as difficult constraints for UC problems [2], [5]–[7], [13]. The number of security constraints is directly proportional to the product of the number of transmission lines and commitment horizon and is one of the main factors that affect the computational complexity. Consider a SCUC problem with 30 transmission lines, 20 generation units, and commitment horizon of 168 h (one week) as an example. The number of security constraints is . Dimension of the coefficient matrix corresponds to the with security constraints is a dc power flow model. If the problem is solved by the LR method, 10 080 multipliers must be introduced to relax the corresponding security constraints. All of the 10 080 components of the subgradient must be calculated in each iteration to update the multipliers, and with these security constraints, it is generally very difficult to obtain a feasible solution based on the dual solution [2], [5], [7], [13], [14]. When general MILP methods are applied, a series of LP problems usually needs to be solved. The calculation and manipulation involved with the constraint would be intenmatrix of dimension of sive and the issues of numerical stability and accumulated errors would be very challenging [15]. As a result, SCUC problems are generally much more complicated than the UC or generation scheduling problems without considering security constraints. Having appropriate security margin is one of the most important principles for designing and operating modern power grids. In general, most security constraints are inactive or redundant in operation, especially during off-peaks, and do not affect the solutions of the SCUC problems. If these inactive or redundant security constraints can be identified and eliminated, it is possible to greatly simplify the problems. The advantages of simplifying the large-scale SCUC problems are significant and obvious. For the general MILP methods, the problem scale can be greatly reduced if most security constraints are eliminated. In LR-based approaches, if many inactive constraints are eliminated before solving the problem, the number of multipliers will
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ZHAI et al.: FAST IDENTIFICATION OF INACTIVE SECURITY CONSTRAINTS IN SCUC PROBLEMS
be greatly reduced and the scale of the dual problem is greatly reduced. For the decomposed SCUC and power flow decomposition techniques, the reduced security constraint set can help obtain an equivalent solution to the original problem, as will be discussed in Section III. Several issues should be resolved in order to eliminate the inactive security constraints: 1) what is the definition of redundant or inactive security constraint? 2) how does one identify an inactive security constraint? 3) could the inactive security constraints be identified efficiently without much computational efforts? This paper tries to answer the above questions. First, it is defined that a security constraint is inactive if it is redundant to the load balance and other security constraints and can be eliminated without changing the solution to the original SCUC problem. A necessary and sufficient condition for a security constraint to be inactive is established. It is proved that finding inactive constraints is equivalent to solving a series of small-scale and simpler MILP problems than the original one. More importantly, an analytical sufficient condition is established and most of the inactive constraints can be quickly identified by simple analytical calculations without solving MILP or LP problems. A very important feature of the conditions obtained for identifying an inactive security constraint is that they are only directly related to the system load demands and parameters of the transmission network. In fact, the inactive security constraints are determined by the feasible region defined by system load demands and transmission security constraints. If a security constraint is identified by the analytical sufficient condition as inactive, it is inactive for all possible unit commitment status if only the system load balance constraints are satisfied. This result is very interesting and important. Surprisingly, based on our numerical testing experiences, most inactive security constraints can be identified by this analytical condition. We also found that the system loads are the most important factor influencing the power flows over transmission lines. Numerical testing is performed for three standard power grids and the results are promising. Over 85% of security constraints are identified inactive analytically, and the crucial transmission lines affecting the total operating cost are among those associated with the remaining security constraints, providing useful information for transmission planning. The theoretical results obtained in this paper are also useful for congestion management in power system operation and selecting power source locations in power system planning. The paper is organized as follows. An MIP formulation of SCUC problem is given in Section II. Definition and identification criteria of inactive security constraints are presented in Section III. Numerical testing results are given and analyzed in Section IV and the concluding remarks follow. II. PROBLEM FORMULATION Consider a SCUC problem. Suppose there are thermal units, transmission lines, load buses in the system, and the time horizon is divided into time periods. For convenience of presentation, some notations are introduced as follows:
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time period index,
;
unit index,
;
transmission line index, load bus index,
; ;
matrix relating unit generations to power flows on transmission lines; li-component of
;
matrix relating bus loads to power flows on transmission lines; lk-component of
;
power flow limit on transmission line ; load demand on bus
at time ;
system load demand at time ; spinning reserve requirement at time ; maximum generation level of unit ; minimum generation level of unit ; maximum allowable change in generation level of unit between two consecutive time periods (the ramp rate limits); maximum allowable spinning reserve contribution of unit ; total start-up/shut-down cost of unit during the whole time horizon; power generation level of unit at time ; spinning reserve contribution of unit at time ; power generation cost of unit at time ; discrete decision variable of unit at time : “1” for up and “0” for down. The aim of SCUC is to determine the commitment and generation levels of all units over all periods such that the total operating cost is minimized. It is generally formulated as the following mixed-integer optimization problem: (1) and subject to the following constraints. A. System Load Balance
(2) B. System Spinning Reserve Requirements
(3)
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where the reserve contribution is given by
(4) C. Transmission Line Capacity (Security Constraints)
(9)
See (5) at the bottom of the page. D. Generation Capacity
(6) where the feasible region of the discrete decisions variable are (7)
It is seen that is a matrix and a matrix (vector). Then, (2) and (5)–(6) at time period can be reformulated as follows. : System
and in (7) is determined by the minimum up/down time constraints, must up/down constraints, etc. (the detailed descriptions can be found in [5] and [7]). E. Ramp Rate Constraints See (8) at the bottom of the page, where and in (8) correspond to the initial up/down state and generation level of unit . It should be noted that the formulation of the start-up/ shut-down cost is slightly different but equivalent to the general form in the literature. III. IDENTIFICATION OF INACTIVE SECURITY CONSTRAINTS The inactive security constraints are identified based on a detailed analysis of the structure of (2) and (5)–(6). To aid analysis, the following notations are introduced:
(10)
The system formulated by (10) is denoted by since it is uniquely determined by , , and . In , the decision variables are and , and , , and are parameters. The maximum/minimum generation and are assumed constants without losing of generality since the results would not be affected when the maximum/minimum generation levels vary with time. Other notations related to (10) are listed as follows: ji-component of -component of th row of
; ;
(a row vector);
matrix obtained by removing the th row of ; .. .
.. .
.. .
.. .
..
..
.
.
.. .
.. .
matrix (vector) obtained by removing the th row of ; set containing all feasible solutions . That is, to the system if and only corresponds to a feasible if . solution to system
(5)
(8)
ZHAI et al.: FAST IDENTIFICATION OF INACTIVE SECURITY CONSTRAINTS IN SCUC PROBLEMS
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Theorem 1: Consider the following MILP problem:
(15)
Fig. 1. Illustration of inactive constraints.
Definition 1: The following constraint (11) is defined inactive to system following equation is satisfied:
is inactive Then the constraint if and only if . to system , then for any feasible Proof: 1) Sufficiency If to [i.e., a feasible solution solution to system (15)], the following equation is satisfied:
if and only if the (16) (12)
With the above definition, a security constraint is inactive if and only if the feasible region of the problem is unaffected by eliminating/adding the constraint (i.e., the feasible region itself keeps unchanged). In other words, an inactive constraint is in fact redundant to the system per Definition 1. Since the feasible region of a system with more constraints is naturally a subset of a system with less constraints, namely, (13) Equation (12) is thus equivalent to combination of (13) and the following equation: (14) With (14), we have the following equivalent definition. Definition 2: The constraint (11) is inactive to system if and only if (14) is satisfied. It should be noted that the above definition of inactive constraint is different from that in some optimization literatures, where an active/inactive constraint is defined for a specific feasible solution. In this paper, however, a constraint is defined inactive if it is redundant versus all other constraints. If a security constraint is inactive, it is unnecessary to consider it when solving the corresponding SCUC problem. That is, the inactive security constraints have no influence on the feasible region of the problem and can thus be eliminated as illustrated in Fig. 1. Although some inequality constraints are not bounded in a particular solution to the problem, eliminating these constraints would change the solution. These constraints are not defined as inactive in this paper. If most of the security constraints are inactive and can be identified quickly, then the efficiency of any solution method will be greatly improved. The main purpose of this paper is to establish some efficient methods to identifying most of the inactive security constraints. The following theorem is the basis of such methods.
Since the feasible region of (15) is (16) implies that any feasible solution to is also a feasible solution to system
, , i.e., (17)
is inactive Thus the constraint to system according to Definition 2. 2) Necessity If the constraint is inactive, (14) is satisfied. Any feasible solution to (15) is also a feasible solution to . Since is a constraint in system , holds for any feasible to (15). Therefore, the maximum value solution of among all feasible solutions to , i.e., holds. (15) is also less than Theorem 1 gives a necessary and sufficient condition for a security constraint to be inactive. However, it is not directly useful since a series of MILP problems needs to be solved and it is not less complex than the original SCUC problem (1)–(8). We will further develop simple sufficient conditions based on Theorem 1. Theorem 2: Consider the following LP problem:
(18)
, then constraint is inactive to system . Proof: By comparing the MILP problem (15) with the LP problem (18), it is seen that any feasible solution to problem (15) also corresponds to a feasible solution to problem (18). Clearly since the feasible region of (18) covers that of If
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(15). Therefore, implies that based on Theorem 1, the constraint
and
is inactive to system . If Theorem 2 is applied to identify an inactive constraint, only an LP problem needs to be solved rather than an MILP as required in Theorem 1. However, it is still inconvenient since the scale of the LP may still be very large. A more convenient sufficient condition is established directly based on Theorem 2 as follows. Theorem 3: Consider the following LP problem:
Fig. 2. Illustration of (21)–(22).
3) If (21)–(22) are satisfied, the optimal solution to the LP problem in (19) is given by the following equations: if (23) if and the value of the optimal objective function is given as
(19)
(24)
If
, then constraint is inactive to system . Proof: The proof is trivial since the problem in (19) is almost the same as that in (18), except that one constraint dropped , and therefore, the in (19). Clearly conclusion holds. The conclusion of Theorem 3 is interesting and gives the reason why the security constraint is inactive. It means that is with the system load balance constraint satisfied, an upper bound on the maximum possible power flow over a is not greater than the transmistransmission line. If sion capacity of the corresponding line, the security constraint is naturally inactive. Conclusions of Theorems 1 and 2 can be interpreted similarly. Theorems 2 and 3 give two sufficient conditions for a security constraint to be inactive. However, a series of LP problems, although much simpler with only one equality constraint than the original MILP problem, need to be solved. It is desirable directly to develop an analytical condition to obtain without solving the LP problem. be a permutation of such that Let
where are defined in (20) and (22), respectively. Proof: Conclusion 1) is equivalent to the condition of sufficient supply and it can be easily proved by contradiction. Based on (21), (22) is obvious and is illustrated in Fig. 2. Now consider the solution given by (23). If (22) is satisfied, then it is clear that (25) According to (23), we then have (26) Comparing with (19), (26) suggests that the solution given by (23) is a feasible solution to LP problem (19). We now prove that it is also an optimal solution to (19). Consider the dual problem to (19): (27)
(20) We have the following theorem. Theorem 4: Consider the LP problem defined by (19). 1) The LP problem has feasible solution if and only if
where (28) (29) (21)
2) If inequality (21) is satisfied, then there exists an integer number such that
According to the duality theory [18], the optimal objective value of (27) is no less than that of (19), i.e., (30)
(22)
Let (31)
ZHAI et al.: FAST IDENTIFICATION OF INACTIVE SECURITY CONSTRAINTS IN SCUC PROBLEMS
Then the following result can be obtained:
(32) The constraint on (32) is given by (29). According to (20), we have the following equation: if if if
(33)
Comparing (33) with (32), we know that the solution given by (23) is the optimal solution to (32). Substitute the solution into (32) and according to (26), we have (34) Equation (34) reveals that the value of the dual objective is identical with the value of the primal objective at at , a feasible solution to the original LP problem (19). Therefore, is the optimal solution to the dual problem is the optimal solution to the primal and problem according to the duality theory [18]. By substituting in (23) into the primal objective function, it is seen that the value of the primal optimum of LP problem (19) is given by (24). Combining Theorems 3 and 4, we have the following theorem. Theorem 5: If
(35)
, then the constraint is inactive to system and can thus be eliminated. Theorem 5 gives a fast approach to identifying and eliminating inactive security constraints analytically. The index “ ” in (35) can be efficiently obtained as illustrated in Fig. 2. Based on this theorem, the inactive security constraints can be identified and eliminated one by one along the commitment horizon. It should be noted that Theorem 5 gives a sufficient condition but not a necessary condition to eliminate all inactive constraints. Theorem 2 can be further applied to eliminate some of the remaining inactive constraints by solving LP problem (18) with the reduced security constraint set after applying Theorem 5. holds for some
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It should be noted that not all inactive constraints can be identified based on (35) since it is a relaxed LP problem to (15), and the influences of the temporal constraints (such as minimum up/down time and ramping limits) are not considered in (15). It is very difficult to determine whether a remaining constraint is inactive after eliminating the redundant constraints by (35) since we must compare the feasible regions of the original problem and the one after eliminating this constraint. Alternatively, we can solve (15) exhaustively with and without the one from the set of remaining constraints to see if the solution changes. This is computationally difficult since the number of the remaining constraints may still be very large. On the other hand, the percentage of the inactive constraints identified among the entire constraint set is a good indirect indication on how effective the analytical method by Theorem 5 is. Based on our numerical testing results in Section IV, nearly 85% security constraints are possibly identified as inactive by using Theorem 5. The method is therefore effective. Fast identification of inactive security constraints would be useful for applying the traditional decomposition-based techniques or the indirect method [5] to solve the SCUC problem. The basic idea of this technique is to initially relax (ignore) all security constraints and identify the transmission lines that are either overloaded or near-overloaded. The violated constraints are put back into the UC problem as the constraint set for recalculating the dispatches. It is interesting to compare the proposed method in this paper and the decomposition method mentioned above. If the proposed method is applied to solve the SCUC problem, most of the inactive constraints are possibly eliminated in advance and only one UC problem needs to be solved with a reduced set of security constraints. The inactive constraints are identified analytically and processed for only one time. On the other hand, if the decomposition approach is applied, a series of UC problems need to be solved with a variable, possibly expanding, set of security constraints. Since the redundant constraints cannot be distinguished from the unbounded active constraints, all remaining constraints not in the constraint set must be checked to see which should be added into the next UC problem. This means many redundant constraints are repeatedly checked in iterations. The necessary number of iterations for obtaining the optimal solution in the decomposition approach is theoretically unknown due to existence of the unbounded active constraints. These two methods can also be combined. In fact, the analytical conditions for identifying the inactive security constraints given by Theorem 5 can be applied first. Most of the inactive constraints are possibly excluded then and a much reduced security constraint set is thus obtained. When the decomposition procedure starts, only the constraints in this reduced set need to be checked. The number of unbounded active constraints is possibly reduced in each iteration with the reduced constraint set. This may help speed up convergence of the decomposition procedure. The model used in this paper requires the sensitivity matrices and that need inverse calculation of large conducting matrix Y as most SCUC models do in the literature. These matrices are calculated in advance once for the same grid configuration.
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TABLE I TESTING RESULTS OF EXAMPLE 1
TABLE II CPU TIME FOR SOLVING EXAMPLE 1
An approximate approach can be applied to reduce the dimensions of the sensitivity matrices with the similar idea of the above decomposition technique. That is, all security constraints are initially relaxed and the overloaded or near-overloaded branches are identified. The new SCUC problem incorporating these reduced constraintsissolvedwithreducedcomputationaleffort. However, to check if a branch is overloaded or near-overloaded, the original Y-matrix inverse is implicitly needed to obtain power flows. Missing unbounded active constraints may also be encountered similarly as above. Our method for eliminating redundant constraints analytically can avoid this issue. IV. NUMERICAL TESTING RESULTS The SCUC problem formulated in this paper is solved based on general MILP method with some nonlinearity in (1)–(8) converted into linear model as in [20]. The numerical testing is performed with ILOG CPLEX 8.0 package on a 2.0-GHz PC of 1 G RAM for three standard power systems, and the testing results are promising. Firstly, it is found that most of the security constraints are inactive and can be identified by Theorem 5. Secondly, the computational time of solving the SCUC is greatly reduced after the inactive constraints are eliminated, especially for large-scale problems. Furthermore, the crucial transmission lines affecting the total operating cost are also quickly identified. Example 1: This example is the IEEE 24-bus system [16] with 34 lines and 26 units. The scheduling horizon ranges from one day to one week. Based on Theorem 5, it is found that only the security constraints on one transmission line could be active and need to be considered. Basic results are given in Table I. It is also found that the remaining security constraints all correspond to only one transmission line in one direction, i.e., line 7–8 in Fig. 3. By checking the original data, it is found that the total capacity of units at bus 7 is 300 MW while the transmission capacity of line 7–8 is 200 MW. The comparison of the computational times for solving the original SCUC problem with that after eliminating the inactive constraints is listed in Table II. It is seen that the CPU time is greatly reduced with the inactive constraints eliminated. Example 2: This example is a 31-bus system with 39 lines and 16 units [5]. By using the method in this paper, it is found
TABLE III TESTING RESULTS OF EXAMPLE 2
TABLE IV CPU TIME FOR SOLVING EXAMPLE 2
that over 87% security constraints are inactive. The testing results are given in Table III. The comparison of the computational times for solving the original SCUC problem with that after eliminating the inactive constraints is listed in Table IV. The computational times for the two cases are similar for short commitment horizons, but the difference is significant for the horizon of seven days. This suggests that the method presented in this paper is especially valuable for large-scale problems since it utilizes more structure characteristics of the SCUC problem. Based on our numerical testing experience, about 20% of the remaining security constraints are active for the optimal solution obtained for Example 2. The other 80% may include a large number of unbounded active constraints that have influence on the optimal solution. This can be explained by the following simple example. Consider the following optimization problem: (36) demand security constraint
(37) (38) (39) (40) (41)
The constraints and solution structure of the above problem is illustrated in Fig. 4. It can be seen that the optimal solution to (36)–(41) is as follows: (42) Therefore, the security constraint (38) is not bounded at the optimal solution. However, if the security constraint is removed, then the optimal solution turns out to be (43) In this case, (38) is an unbounded active constraint since it has influence on the feasible region. In solving Example 2, it is found that some of the remaining security constraints are either
ZHAI et al.: FAST IDENTIFICATION OF INACTIVE SECURITY CONSTRAINTS IN SCUC PROBLEMS
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TABLE V TESTING RESULTS OF EXAMPLE 3
TABLE VI CPU TIME FOR SOLVING EXAMPLE 3
Fig. 3. Transmission grid of Example 1.
Fig. 4. Illustration of an unbounded active constraint.
Example 3: This example is the IEEE 118-bus system with 179 lines (seven double-lines among the original 186 lines are aggregated as single lines) and 54 units [19]. By using the method in this paper, nearly 87% security constraints are identified as inactive. The testing results are given in Table V. It is seen that the number of inactive constraints identified by Theorem 2 is about 10% of that identified by Theorem 5. The results suggest that the reduction of the inactive constraints by the analytical condition can be further improved. The comparison of the computational times for solving the original SCUC problem with the problem after eliminating the inactive constraints is listed in Table VI. It is observed that the computational time for solving the problem with the reduced constraint set is always less than that for solving the original problem. The speedup ratio increases steadily with the commitment horizon. This suggests that the method presented in this paper is more beneficial for larger problems. V. CONCLUSIONS
Fig. 5. Load demand versus the number of remained security constraints.
active or unbounded active. This result suggests that the crucial transmission lines affecting the total operating cost are among those associated with the remaining security constraints. This would provide useful information for transmission planning. Another interesting result is that the number of the remaining security constraints in different time period is closely related to system load profiles. In Fig. 5, it is seen that there are less remaining security constraints during off-peak periods. It is also found that the transmission lines associated with the remaining security constraints during peak load periods are different from those in the off-peak periods.
Most security constraints are possibly inactive in SCUC problems. The computational requirements and numerical error would be greatly reduced if the inactive constraints are eliminated. We develop an analytical condition to quickly identify inactive security constraints in SCUC problems. Numerical testing results for three standard power grids show that most of the inactive security constraints are identified, and the method established in this paper is effective. It is also found that the crucial transmission lines affecting the total operating cost are among those associated with the remaining security constraints. This would provide useful information for transmission planning. In our future work, we will combine the current and previous work to develop the analytical conditions for determining the feasibility of SCUC problems and propose an efficient and systematic method for obtaining feasible solutions to SCUC problems.
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REFERENCES [1] B. F. Hobbs, M. H. Rothhopf, R. P. Oneill, and H. Chao, The Next Generation of Electric Power Unit Commitment Models. Norwell, MA: Kluwer, 1999. [2] Y. Fu and M. Shahidehpour, “Fast SCUC for large-scale power systems,” IEEE Trans. Power Syst., vol. 22, no. 4, pp. 2144–2151, Nov. 2007. [3] J. M. Crespo, J. Usaola, and J. L. Fernández, “Security-constrained optimal generation scheduling in large-scale power systems,” IEEE Trans. Power Syst., vol. 21, no. 1, pp. 321–332, Feb. 2006. [4] A. Cohen and V. Shenkat, “Optimization-based methods for operations scheduling,” Proc. IEEE, vol. 75, no. 12, pp. 1574–1591, Dec. 1987. [5] J. J. Shaw, “A direct method for security constraints unit commitment,” IEEE Trans. Power Syst., vol. 10, no. 3, pp. 1329–1339, Aug. 1995. [6] S. Al-Agtash, “Hydrothermal scheduling by augmented Lagrangian: Consideration of transmission constraints and pumped-storage units,” IEEE Trans. Power Syst., vol. 16, no. 4, pp. 750–756, Nov. 2001. [7] X. Guan, S. Guo, and Q. Zhai, “The conditions for obtaining feasible solutions to security-constrained unit commitment problems,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1746–1756, Nov. 2005. [8] B. Lu and M. Shahidehpour, “Unit commitment with flexible generating units,” IEEE Trans. Power Syst., vol. 20, no. 2, pp. 1022–1034, May 2005. [9] J. F. Benders, “Partition procedure for solving mixed-variables programming problems,” Numer. Math., vol. 4, no. 3, pp. 238–252, 1962. [10] X. Guan, Q. Zhai, and A. Papalexopoulos, “Optimization based methods for unit commitment: Lagrangian relaxation versus general mixed integer programming,” in Proc. 2003 IEEE Power Eng. Soc. General Meeting, Toronto, ON, Canada, Jul. 2003. [11] Z. Gaing, “Particle swarm optimization to solving the economic dispatch considering the generator constraints,” IEEE Trans. Power Syst., vol. 18, no. 3, pp. 1187–1195, Aug. 2003. [12] I. G. Damousis, A. G. Bakirtzis, and P. S. Dokopoulos, “Network-constrained economic dispatch using real-coded genetic algorithm,” IEEE Trans. Power Syst., vol. 18, no. 1, pp. 198–205, Feb. 2003. [13] S. J. Wang, S. M. Shahidehpour, D. S. Kirschen, S. Mokhtari, and G. D. Irisarri, “Short-term generation scheduling with transmission constraints using augmented Lagrangian relaxation,” IEEE Trans. Power Syst., vol. 10, no. 3, pp. 1294–1301, Aug. 1995. [14] H. Ma and S. M. Shahidehpour, “Unit commitment with transmission security and voltage constraints,” IEEE Trans. Power Syst., vol. 14, no. 2, pp. 757–764, May 1999. [15] G. L. Nemhauser and L. A. Wolsey, Integer and Combinatorial Optimization. Chichester, U.K.: Wiley, 1988. [16] “IEEE reliability test system,” IEEE Trans. Power App. Syst., vol. PAS-98, no. 6, pp. 2047–2054, 1979. [17] N. P. Padhy, “Unit commitment—A bibliographical survey,” IEEE Trans. Power Syst., vol. 19, no. 2, pp. 1196–1205, May 2004. [18] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd ed. New York: Wiley, 1993. [19] M. Shahidehpour, H. Yamin, and Z. Li, Market Operations in Electric Power System. New York: Wiley, 2002. [20] M. Carrion and J. M. Arroyo, “A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem,” IEEE Trans. Power Syst., vol. 21, no. 3, pp. 1371–1378, Aug. 2006.
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Qiaozhu Zhai (M’07) 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.
Xiaohong Guan (M’93–SM’95–F’07) received the B.S. and M.S. degrees in control engineering from Tsinghua University, Beijing, China, in 1982 and 1985, respectively, and the Ph.D. degree in electrical engineering from the University of Connecticut. Storrs, in 1993. He was a Senior Consulting Engineer with PG&E from 1993 to 1995. He visited the Division of Engineering and Applied Science, Harvard University, Cambridge, MA, from January 1999 to February 2000. Since 1995, he has been with the Systems Engineering Institute, Xi’an Jiaotong University, Xian, China, and was appointed as the Cheung Kong Professor of Systems Engineering in 1999, and the Dean of School of Electronic and Information Engineering in 2008. Since 2001, he has been the Director of the Center for Intelligent and Networked Systems, Tsinghua University, and served as the Head of Department of Automation, 2003–2008. His research interests include scheduling of power and manufacturing systems, bidding strategies for deregulated electric power markets, and security of complex network systems.
Jinghui Cheng received the B.S. degree in information and communication engineering and the M.S. degree in the System Engineering Institute from Xi’an Jiaotong University, Xi’an, China, in 2006 and 2009, respectively. His research interest is mainly on optimization of large-scale systems.
Hongyu Wu received the B.S. degree in energy and power engineering from Xi’an Jiaotong University, Xi’an, 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.
