A Decomposition Based Approach for Solving a General Bilevel Linear Programming

Transcription

1 A Decomposition Based Approach for Solving a General Bilevel Linear Programming Xuan Liu, Member, IEEE, Zuyi Li, Senior Member, IEEE Abstract Bilevel optimization has been widely used in decisionmaking process. However, there still lacks an efficient algorithm to determine an optimal solution of a bilevel optimization problem, especially for a large-size problem. To bridge the gap, this paper proposes an efficient decomposition algorithm for a general bilevel linear programming(gblp). The simulation results on large-size testing system demonstrate its correctness and efficiency. Index Terms bilevel linear programming, benders decomposition, general bilevel linear programming, power systems. s slack variable vector S shift factor matrix of the power grid U bus-generator incidence matrix V bus-load incidence matrix W bus-line incidence matrix X branch reactance matrix ρ, μ, μ Lagrangian vector x variable vector in the lower level y, y variable vector in the upper level Note that represents incremental change and symbols in bold represent vectors or matrices. U, U V, V U r, U r V r ε τ i, j k K h k max L k u z lower z upper c, d e, f D D A, b E, F, G h J L L max P P max NOMENCLATURE sets of extreme points sets of extreme rays reduced sets of extreme points reduced sets of extreme rays given value for gap given maximum percentage of load uncertainty subscript: index for lagrangian multipliers subscript: index for lines objective value the lower level optimization problem physical flow of line k rating of line k loading level of line k lower bound of the GBLP upper bound of the GBLP coefficient vector for the objective function in the upper level coefficient vector for the objective function in the lower level load vector false data injection vector into load measurements coefficient matrices in the upper level coefficient matrices in the lower level physical line flow vector load shedding vector line flow vector line rating vector generator output vector maximum generation output vector B I. INTRODUCTION ILEVEL optimization has attracted a lot of research attentions in recent years[1]-[7]. In general, a bilevel optimization problem can be casted into a decider-follower problem. In the upper level, a decision is determined. Then, the follower in the lower makes a best response according to the determined decision. For example, the power grid interdiction[7] is formulated as a bilevel max-min problem. In the upper level, the attacker determines the optimal set of lines to be attacked. In the lower level, the operator minimizes the load shedding given the set of attacked lines determined in the upper level. The most popular methods to solve a bilevel optimization problem are the Karush-Kuhn-Tucker (KKT) based approach and duality based method. In the KKT based approach, the lower level optimization problem is replaced by its KKT optimality conditions. To linearize the nonlinear complimentary constraints, additional binary variables will be introduced to form the so called big-m constraints, which limit the computation efficiency. In the duality based approach, the lower level is transformed into its dual problem [8]. However, the resulting nonlinear terms in the strong duality condition make the whole problem hard to solve, especially for a largesize system. Thus, it is necessary for us to develop a novel algorithm to get the optimal or suboptimal solution of a bilevel optimization problem. A little efforts [9]-[11] has been devoted to apply some decomposition techniques to solve a bilevel optimization problem. However, we notice that these methods can be only applied to a special class of the bilevel linear programming with the same objectives in the lower and upper levels. So far, how to solve a GBLP with different objectives in the lower and upper levels efficiently is still an open question. In this paper, we propose a benders decomposition based approach to

2 solve a GBLP with different objective functions in the lower and upper levels. Different from the traditional benders decomposition approach, the subproblem consists of two sequential LPs. In the first stage, the first LP is solved to get the optimal objective. Then the constraint of the objective function will be inserted into the second LP to get the dual variables used to generate benders cuts. The rest of this paper is organized as follows. Section II proposes an decomposition algorithm to solve a GBLP. Section III demonstrates the proposed model with the IEEE 118-bus system. Section IV concludes the paper. II. DECOMPOSITION METHODOLOGY OF GBLP In this section, we propose a decomposition algorithm to solve a GBLP. A. Decomposition Method for GBLP Consider the general bilevel linear program(gblp) (1)-(4) as follows.. (GBLP) min c T x + d T y (1) Ay b (2) min e T x + f T y (3) Ex + Fy G (4) where variables x, y are continuous variables. Note that the objectives in the lower and upper levels could be different, that is c T x e T x w where w is a scalar. This characteristic poses a huge challenge for developing an efficient algorithm for solving it. Note that for a fixed y, the lower level optimization problem becomes a linear programming. Rewrite the lower level optimization problem (3)-(4) as (SP): min e T x + f T y (5) Ex G Fy μ (6) Since the term f T y is a constant, we have (LP1) K = min e T x (7) Constraint (6) As LP1 is a LP whose optimal value can be determined at one of its extreme points. Note that the goal of the bilevel optimization problem is to determine a best y that minimizes the objective value of LP3: (LP3) min c T x + f T y (8) min e T x (9) Constraint (6) Note that the bilevel optimization problem has a special structure in which only variable x is transmitted to the upper level for a given y. For a given y, the GBLP can be solved by solving two sequential LPs: LP1 and LP2. (LP2): min c T x (10) e T x = K ρ (11) Ex G Fy μ (12) where K is the objective value of the lower optimization problem for a given y. The dual problem of (10)-(12) is max ρk + (μ ) T (G Fy ) (13) [e E T ] [ ρ μ ] c (14) Let S denote the feasible region of the dual problem, S = {(ρ, μ ) [e E T ] [ ρ μ ] c} (15) The region S has two good chatteristics: 1) It is independent with the value of the variable y in the upper level and the value of K; 2) It is a fixed polygon with a finite number of extreme points and extreme rays. Let U, V denote the sets of all the extreme points and extreme rays of the feasible region S, respectively. U = {u 1, u 2,, u n } V = {v 1, v 2,, v q } According to [12], the optimization problem is equivalent to (16)-(17): max { ρ i K + (μ i )T (G Fy ) (ρ i, μ i ) U } (16) ρ j K + (μ j )T (G Fy ) 0 (ρ j, μ j ) V (17) Constraint (17) holds for ensuring the bound of the dual problem (13)-(14). Similarly, for a fixed y, the optimization problem (5)-(6) is a LP. Thus, its dual problem is written as max(μ) T (G Fy ) (18) E T μ c (19) Let T denote the feasible region of (18)-(19) T = {(μ) E T μ c} (20) It can be seen that the region T is also independent with the value of the variable y in the upper level and has a finite number of extreme points and extreme rays. U = {u 1 1, u 2 1,, u m 1 } V = {v 1 1, v 2 1,, v p 1 } Since the strong duality condition can be satisfied, we have

3 K = max(μ) T (G Fy ) (21) Introducing (21) into (16), which gives max ρ i (max(μ) T (G Fy )) + (μ i )T (G Fy ) (22) It can be verified that (22) is equivalent to max{ρ i (μ j T (G Fy )) + (μ i )T (G Fy ) = max (ρ i μ j T + (μ i )T ) (G Fy ) (ρ i, μ i ) U, μ j U} (23) Constraint (17) Note that LP2 is feasible as long as LP1 is feasible, so constraint (17) can be replaced by (μ j ) T (G Fy ) 0 μ j V (24) Thus, the optimization problem GBLP is equivalent to (25): (DMP): min z (25) z (ρ i μ j T + (μ i )T )(G Fy) + d T y (ρ i, μ i ) U, μ j U (26) B. Algorithm Constraints (2), (24) As discussed, the original GBLP can be transformed into a single level optimization problem DMP if we can enumerate all its extreme points and extreme rays. In practice, however, it is very difficult to enumerate all the extreme points and extreme rays for a feasible region. In particular, the size of problem becomes large. To overcome the computation difficulty, Benders in [12] developed an iterative algorithm to determine the optimal solution by adding a subset of extreme points and extreme rays. Define the restricted master problem(rmp) as follows: (RMP): min z (27) z (ρ i μ j T + (μ i )T )(G Fy) + d T y (ρ i, μ i ) U r, μ j U r (28) (μ j ) T (g Ex ) 0, (μ j ) V r (29) Constraints (2) where U r, U r, V r are the reduced sets of U 2, U 1, V 1. Thus, the RMP determines an upper bound for the optimal objective of the dual problem DMP. Theorem 1: Suppose that y, z is the optimal solution of the RMP. If g and ρ, (μ ) are the optimal objective value of the subproblem and corresponding dual variables, then we have 1) if z > g, then ρ, (μ ) is the new extreme point that will be added into the master problem; 2) if z = g, then y, z is the optimal solution of the DMP; 3) if the subproblem is unbounded, then ρ, (μ ) is the new extreme ray that will be added into the master problem. Initial z lower, z upper, ε, y Solve LP1 to get K Feasible? Solve LP2 and update z lower Solve RMP and update z upper Converged? Stop Yes Yes Add infeasibility cuts No Add Optimality cuts No Fig.1 Flowchart of decomposition algorithm for GBLP Proof: Since g and ρ, (μ ) are the optimal objective value of the subproblem and corresponding dual variables, we have g = ρ K + (G Fy ) T (μ ) + d T y (30) To ensure the bound of the subproblem, constraint (31) must hold, ρ K + (G Fy ) T (μ ) + d T y 0 (ρ, μ ) V (31) z is the optimal solution of the RMP, so z = min{ρ K + (G Fy ) T (μ ) + d T y } (32) Combining (30) and (32), we have f T y + z f T y + min {ρ K + (G Fy ) T (μ ) + d T y } So, If z > g,then f T y + g (33) z g (34) z > g = ρ K + (G Fy ) T (μ ) + d T y (35) Together with (32), which gives min{ρ K + (G Fy ) T (μ ) + d T y } > ρ K + (G F y ) T (μ ) + d T y (36) This indicates that (ρ, μ ) U (37) It is a new extreme point. If z = g, then we have g min{ρ K + (G Fy ) T (μ ) + d T y } ρk + (G Fy ) + d T y (38)

4 Together with (31), we can see that (y, z ) is a feasible point of the dual problem. Consider MP is relaxed to RMP, z RMP z MP (39) Thus, (y, z ) is the optimal solution of the dual master problem. Finally, if the subproblem is unbounded. That is, the primal problem is infeasible. In this case, ρ K + (G Fy ) T (μ ) + d T y < 0 (40) As all the existing extreme rays in the RMP satisfy the following constraint ρ K + (G Fy ) T (μ ) + d T y 0 (41) Thus, It is a new generated extreme ray. (ρ, μ ) V (42) Theorem 2: The optimal solution of the DMP can be determined by solving the RMP a finite number of times. Proof: The total number of the combinations of extreme points or extreme rays in U, U, V, V is N = (m + p)(n + q) According to Theorem 1, at each iteration, one new combination of extreme points or extreme rays will be added into the master problem. Thus, the program will stop in a finite number of iterations. It should be pointed that the algorithm can find the optimal solution in N iterations in the worst case. Luckily, the iterations in practice is far less than N, which garatte the efficiency of the algorithm. This will be verified in the test cases. According to the analysis above, the entire algorithm can be described as follows: Step 1: Initial the tolerance ε, z lower =, z upper = + and the initialy = y 0 ; Step 2: For the given y, solve the optimization problem (43): min 1 T s (43) Ex s G Fy π (44) s 0 The optimization problem is used to the feasibility of y. If the objective of the optimal problem is zero, then y is feasible. In this case, go to Step 3. If the objective of the optimal problem is greater than zero, then the subproblem (5)-(6) is infeasible. In this case, we need to introduce an infeasibility cut into the master problem. π T (G Fy ) 0 (45) Step 3: Solve LP1 to get the value of K. (LP1) K = min e T x (7) Constraint (6) Step 4: Solve LP2. Suppose the optimal solution is [μ j ρ i μ i ] T, then an upper bound of the BLP is determined z lower = (ρ i μ j T + (μ i )T )(G Fy ) + d T y (46) Step 5: If the following constraint is satisfied, stop; z upper z lower z lower ε (47) Otherwise, add the following cut into the master problem z (ρ i μ j T + (μ i )T )(G Fy) + d T y (48) Step 6: Solve the master problem to update the variable y. The upper bound of BLP is the objective value of z upper = min z (49) Constraints (2), (28)-(29) and then go to Step 2. The advantage of the proposed algorithm is to reduce the execute time by solving a small number of LPs. III. CASE STUDY In this section, we test the proposed decomposition algorithm using the IEEE 118-bus system[13].the testing model is described in section A. Simulations are carried out on a 3.4GHz personal computer with 8GB of RAM. The model and algorithm are implemented in MATLAB. A. Modeling The security constrained economic dispatch(sced) is performed every 5-15 minutes based on the forecasted loads and received real-time measurements. However, the high integration of renewables and malicious false data attacks significantly increases the uncertainty of forecasted loads and real-time measurements. It has been show that the uncertainty of these data could lead to the overloading of some lines. If these line overloads cannot be mitigated immediately, severe consequences, such as cascading failures, might occur. To ensure the secure operation of a power system, it is very essential for an operator to identify these lines whose power flows could exceed the flow limits after SCED. This is done by solving the GBLP below: max u (50) u = h k /L k max (51) h = S (U P V (D J)) (52) 1 T D = 0 (53) τd d D d τd d (0 < τ < 1) (54) min c g T P + c d T J (55) 1 T P = 1 T (D J) (56) L = S U P S V (D + D J) (57) P min P P max (58) L max L L max (59)

5 0 J D + D (60) Constraint (52) gives the power flows of line k after attacks. Note that the injected false data D is not included since it is not a physical load. Without loss of generality, the maximum allowable attacking amount at a bus is set to 50% of its load. The injected false data D is summed to zero (53). Constraint (56) represents the lower power balance equation. The line flow under the interruption of false data D is determined by constraint (57). Constraints (58)-(60) represent the lower and upper bound constraints of generation, line flow, and load shedding, respectively. The upper level of the above bilevel problem simulates the attacker s attacking strategy, which is to determine the injected false data D that maximizes the potential flow of line k (51). The lower level simulates the system operator s operation strategy, which is essentially an SCED problem that minimizes the total generation cost and load shedding cost (55). B. Results The GBLP (1)-(4) is tested on the IEEE 118-bus system[13], which consists of 118 buses and 186 lines. The uncertainty of load power injection is assumed to be 50% of its actual data due to the interruption of cyber attacks. And the minimum outputs of generators are set to zeros. Without loss of generality, we only report the simulation results of the selected ten lines below. TABLE 1 TESTING RESULT FOR THE IEEE 118-BUS SYSTEM Line u gap Iterations t(s) % % % % % % % % % % In Table 1, the second column gives the line overloading level for each line. The third column represents the existing gap between the primal and dual problems. And the last two columns give the iterations and run times for the decomposition algorithm. It can be seen that lines 11-12, would be overloaded due to the uncertainty of forecasted loads and received measurements after SCED. These line are identified as high-risk lines that need to be strongly monitored for ensuring the secure operation of a power system. And the optimal solution of the GBLP can be determined fast. For example, the average run time for the IEEE 118-bus system is 0.1s. However, the KKT based approach did not converge for most lines after 5 hours. The small gap guarantees the required accuracy of the obtained solution. In addition, the algorithm converges in a small number of iterations. For instance, the maximum iterations for the IEEE 118-bus system is 10, far less than the iterations in the worst case. This verifies the computation efficiency of the proposed decomposition algorithm. IV. CONCLUSIONS AND FUTURE WORK This paper proposes an efficient decomposition algorithm to solve a GBLP. This is achieved by decomposing the GBLP into one master problem and a subproblem. The master problem is solved to update the decision variable in the upper level. And the subproblem is solved by solving two sequential LPs to generate the necessary cut added into the master problem. The simulation results on the large-size testing systems demonstrate its efficiency of the proposed decomposition algorithm. In the next work, we are going to develop a decomposition approach for bilevel mixed integer linear programming that includes some integer variables in the lower level. The difficult part is the strong duality condition might not be satisfied due to the existing integer variables. REFERENCES [1] L. P. Garces, A. J. Conejo, R. Garcia-Bertrand and R. Romero, "A bilevel approach to transmission expansion planning within a market environment", IEEE Trans. Power Syst., vol. 24, no. 3, pp , [2] Y. Yuan, Z. Li, and K. Ren, Modeling load redistribution attacks in power systems, IEEE Transaction on Smart Grid, vol. 2, no. 2, pp , Jun [3] K. C. Almeida and F. S. Senna, "Optimal active-reactive power dispatch under competition via bilevel programming", IEEE Trans. Power Syst., vol. 26, no. 4, pp , [4] X.Liu, Z.Li, X.Liu, and Z.Li Masking Transmission Line Outages via False Data Injection Attacks, IEEE Trans. IEEE Transactions on Information Forensics and Security, vol. 11, no. 7, pp , [5] T. Li and M. Shahidehpour, "Strategic bidding of transmission-constrained GENCOs with incomplete information", IEEE Trans. Power Syst., vol. 20, no. 1, pp , [6] S. Fliscounakis, P. Panciatici, F. Capitanescu and L. Wehenkel, "Contingency ranking with respect to overloads in very large power systems taking into account uncertainty, preventive, corrective actions", IEEE Trans. Power Syst., vol. 28, no. 4, pp , [7] J. M. Arroyo and F. D. Galiana, On the solution of the bilevel programming formulation of the terrorist threat problem, IEEE Trans. Power Syst., vol. 20, no. 2, pp , May [8] A. L. Motto, J. M. Arroyo, and F. D. Galiana, A mixed-integer LP procedure for the analysis of electric grid security under disruptive threat, IEEE Trans. Power Syst., vol. 20, no. 3, pp , Aug [9] A. Delgadillo, J. Arroyo, and N. Alguacil, Analysis of electric grid interdiction with line switching, IEEE Trans. Power Syst., vol. 25, no.2, pp , May [10] J. Salmeron, K. Wood, and R. Baldick, Analysis of electric grid security under terrorist threat, IEEE Trans. Power Syst., vol. 19, no. 2, pp , May [11] C. Losada, M. Scaparra, and J.R. Hamley Optimizing system resilience: A facility protection model with recovery time, European Journal of Operational Research., vol. 217, no.2, pp , Mar [12] J. F. Benders, Partitioning procedures for solving mixed-variables programming problems, Numer. Math., vol. 4, no. 1, pp ,Dec [13] R. D. Zimmerman, C. E. Murillo-Sánchez, and R. J. Thomas, MATPOWER: Steady-state operations, planning and analysis tools for power systems research and education, IEEE Transactions on Power Systems, vol. 26, no. 1, pp , Feb Xuan Liu received the B.S. and M.S degrees from Sichuan University, China, in 2008 and 2011, and Ph.D. degree from Illinois Institute of Technology, USA, in 2015, respectively. He is currently a senior research associate at the Robert Gavin Electric Innovation Center, Illinois Institute of Technology. His research interests include smart grid security, operation and economics of power systems.