A Distribute Parallel Approach for Big Data Scale ...

A Distribute Parallel Approach for Big Data Scale Optimal Power Flow with Security Constraints Lanchao Liu∗ , Amin Khodaei∗, Wotao Yin† , and Zhu Han∗ ∗ Electrical

and Computer Engineering Department, University of Houston, Houston, TX, 77004 of Computational and Applied Mathematics, Rice University, Houston, TX, 77005

† Department

Abstract—This paper presents a mathematical optimization framework for security-constrained optimal power flow (SCOPF) computations. The SCOPF problem determines the optimal control of power systems under constraints arising from a set of postulated contingencies. This problem is challenging due to the significantly large problem size, the stringent real-time requirement and the variety of numerous post-contingency states. In order to solve the resultant big data scale optimization problem with manageable complexity, the alternating direction method of multipliers (ADMM) is utilized. The SCOPF is decomposed into independent subproblems correspond to each individual precontingency and post-contingency case. Those subproblems are solved in parallel on distributed nodes and coordinated through dual (prices) variables. As a result, the algorithm is implemented in a distributive and parallel fashion. Numerical tests validate the effectiveness of the proposed algorithm.

I. I NTRODUCTION The deregulation of electric power grids offers the opportunity for electricity market participants to exercise least-cost or profit-based operations [1]. Despite of the market-driven tendency of the electric power business, security remains to be a significant concern of sustainable power system operations which cannot be compromised. The security-constrained optimal power flow (SCOPF) [2], [3] aims at minimizing the cost of system operation while satisfying a set of postulated contingency constraints. It is an important management task promising to optimal control of power systems securely. The SCOPF is an extension of the conventional optimal power flow (OPF) problem [4], whose objective is to determine a generation schedule that minimizes the system operating cost while satisfy the system operation constraints such as hourly load demand, fuel limitation, environmental constraints and network security requirements. It has been recognized [5] that the optimal control of the normal state may violate the system operation constraints after the occurrence of the event of a major disturbance or contingency, and thus jeopardize the security of power systems. To address this problem, the SCOPF is performed by considering both pre-contingency and postcontingency constraints to guarantee sustainable operations of the electric grid. An illustration of the SCOPF is shown in Fig. 1. The system security level is improved by taking into account a number of contingencies in a dedicate selected contingency list. The solution to the SCOPF should satisfy the so called N − 1 criterion, which requires the operational limits of the power system should not be violated in case of a single contingency (line and/or generator outage).

System Security Level Contingency List

Contingency Case N

Contingency Case 3 Contingency Case 2 Contingency Case 1

Base Case

Fig. 1. The system security level is improved by solving the SCOPF, which takes into account a number of contingency cases in a dedicate selected contingency list.

The SCOPF can be broadly classified as preventive and corrective, where control variables are restricted to their precontingency condition settings, and corrective, whose control variables are allowed to be rescheduled [6]. We will focus on the corrective model in this paper. The seminal paper [5] proposed the generalized Benders decomposition method to solve the corrective SCOPF. Since then, an extensive literature exists for SCOPF in power systems both for traditional operations and under market environments [7]–[10], [13]. The nested Benders decomposition method is utilized in [7] to solve the SCOPF for determining the optimal daily generation scheduling in a pool-organized electricity market, and is tested in an actual example of the Spanish power system. [8] embedded the SCOPF into the security-constrained unit commitment (SCUC) model, and designed an effective corrective/preventie contingency dispatch over a 24-hour period which balanced the economics and security in the restructured markets. An iterative approach is proposed in [9] to obtain the solution of SCOPF, which aims to efficiently identify a as small as possible superset of the binding contingencies to achieve the SCOPF optimum. [10] applied the Benders decomposition to decompose the traditional SCOPF, and the underlying computational complexity is analyzed in this approach. [13] solved the SCOPF by a non-decomposed method based on the compression of the post-contingency networks, which can reduce the size of the security constraints and relieve the computational burden in the problem. The SCOPF is computationally intensive due to the signifi-

cantly large problem size, the stringent real-time requirement and the huge number of various post-contingency states. Additionally, the anticipated smart grid data deluge, generated by the sensing and measurement devices and reinforced by communication and information technologies, makes the problem more complicated. To ensure the efficient and secure operations of the entire grid, we propose a distributive and parallel approach to solve the SCOPF by utilizing the alternating direction method of multipliers (ADMM). The original SCOPF problem is decoupled and divided into independent subproblems correspond to each individual pre-contingency and post-contingency cases. Each subproblem is of the approximately the same size and optimized at a local computing node. Computing nodes are coordinated thorough delicately designed dual (price) variables. Numerical tests on IEEE buses validate the performance of the proposed algorithm. The remaining of this paper is organized as follows. Section II presents the formulation of the SCOPF problem. The proposed algorithm is described in Section III and numerical tests are given in Section IV. Finally, Section V concludes the paper. II. P ROBLEM F ORMULATION Before presenting the formulation of the optimization problem, it is useful to recall a general formulation of the conventional SCOPF problem compactly described as follows: minimize

x0 ,...,xC ;u0 ,...,uC

subject to

f 0 (x0 , u0 )

(1)

g0 (x0 , u0 ) = 0,

(2)

0

0

0

(3)

c

c

c

(4) (5)

h (x , u ) ≤ 0, g (x , u ) = 0, hc (xc , uc ) ≤ 0, |u0 − uc | ≤ ∆c ,

c = 1, . . . , C,

(6)

where f 0 is the objective function, which aims to maximize the total social welfare or equivalently minimize the offerbased energy and production cost. xc is the vector of state variables, which includes magnitude and voltage angle at all buses, and uc is the vector of control variables, which can be generators’ real power or terminal voltage. The superscript c = 0 corresponds to the pre-contingency configuration, and c = 1, . . . , C corresponds to different post-contingency configurations, respectively. ∆c is the maximal allowed adjustments of the control variables between the normal and contingency states. In the conventional SCOPF problem, the equality constraints gc , c = 0, . . . , C represent the system nodal power flow balance over the entire gird, and inequality constraints hc , c = 0, . . . , C stand for the physical limits on the equipments, such as the operational limits on the branch currents and bounds on the generators’ power output. Constraints (2)-(3) stand for the economic dispatch and enforce the feasibility of the pre-contingency state. Constraints (4)-(5) stand for the security-constrained dispatch and enforce the feasibility of the

TABLE I N OTATION DEFINITIONS G N B θc ∈ R|N | Pg,c ∈ R|G| fig Pg,0 i Bcbus ∈ R|N |×|N | Bcf ∈ R|B|×|N | Pd,c ∈ R|N | Ag,c ∈ R|N |×|G| Fmax Pg,c Pg,c ∆c

Set of generators Set of buses Set of branches Vector of voltage angles Vector of real power flows Generation cost function Displaceable real power of each individual generation unit i for pre-contingency configuration Power network system admittance matrix Branch admittance matrix Real power demand Sparse generator connection matrix, whose element (i, j) element is 1 if generator j is located at bus i and 0 otherwise Vector for the maximum power flow Upper bound of real power generation Lower bound of real power generation Pre-defined maximal allowed variation of power outputs

post-contingency state. Constraint (6) stands for the securityconstrained dispatch with rescheduling, which couples control variables of pre-contingency and post-contingency states and prevents unrealistic post-contingency corrective actions. Note that there are some variations on the objective function and constraints of the SCOPF problem, and we focus on the conventional formulation in this paper. Following the standard approach of formulating SCOPF problem, the objective here is to minimize generation cost while safeguard power system sustainability. For the sake of simplicity and computational tractability, constraints (2)-(5) are modeled with the linear DC load flow, and we assume the list of contingencies is given. Assuming a DC power network modeling and neglecting all shunt elements, the standard SCOPF problem above can be simplified to the following optimization problem: X g g,0 (7) minimize fi (Pi ) θ 0 ,...,θC ;Pg,0 ,...,Pg,C

subject to

i∈G

B0bus θ0 + Pd,0 − Ag,0 Pg,0 = 0, (8) Bcbus θc + Pd,c − Ag,c Pg,c = 0, |B0f θ0 | − Fmax |Bcf θc | − Fmax g,0 g,0

P

g,c

P

≤P

g,c

≤P

(9)

≤ 0,

(10)

≤ 0,

(11)

≤

Pg,0 ,

(12)

≤

Pg,c ,

|Pg,0 − Pg,c | ≤ ∆c , i ∈ G, c = 1, . . . , C,

(13) (14) (15)

where the notation is given in Table I. The solution to (7) ensures the economic dispatch while guaranteing the power system security, by taking into account a set of postulated contingencies. The major challenge of the SCOPF is the size of the problem, especially for large systems with numerous contingency cases to be considered. Directly solving the SCOPF by simultaneously imposing all

post-contingency constraints will result in prohibitive memory requirements and huge CPU burden. To achieve efficient and secure operations of the entire electrical grid, a distributed and parallel optimization algorithm is designed to solve the SCOPF in large-scale power systems in the next section. III. P ROPOSED A LGORITHM In this section, a distribute algorithm is proposed to solve the SCOPF by decomposing it to a set of simpler and parallel subproblems corresponds to the base case and each contingency case, respectively. The approach is based on the ADMM [14], [15], whose general form is described as follows: minimize

f (x) + g(z)

(16)

subject to

Ax + Bz = c,

(17)

x,z

where x ∈ Rn , z ∈ Rm and c ∈ Rp , matrices A ∈ Rp×n and B ∈ Rp×m . Functions f and g are closed, convex and proper. The scaled augmented Lagrangian can be expressed as: ρ Lρ (x, z, µ) = f (x) + g(z) + kAx + Bz − c + µk22 , (18) 2 where ρ > 0 is the penalty parameter and µ is the scaled dual variable. Using the scaled dual variable, x and z are updated in a Gauss-Seidel fashion. At each iteration k, the update process can be expressed as: ρ x[k + 1] = arg min f (x) + kAx + Bz[k] − c + µ[k]k22 , 2 x ρ z[k + 1] = arg min g(z) + kAx[k + 1] + Bz − c + µ[k]k22 . 2 z Finally, the scale dual variable is updated by: µ[k + 1] = µ[k] + Ax[k + 1] + Bz[k + 1] − c.

Pg,0 − Pg,c + pc = ∆c , 0 ≤ pc ≤ 2∆c , c = 1, . . . , C.

(19) (20) (21) (22)

The above optimization problem can be solved distributively using the ADMM. The scaled augmented Lagrangian can be calculated as: Lρ (Pg,0 , . . . , Pg,C ; p1 , . . . , pC ; µ1 , . . . , µC ) =

X i∈G

fig (Pg,0 i )+

C X ρc c=1

2

Input: Bcbus , Bcf , Ag,c , Pd,c , Pg,c , Pg,c , ∆c ; Initialize: θc , Pg,c , pc , µc , ρc , k = 0; while not converge do Pg,0 -update: P Pg,0 [k + 1] = arg minPg,0 i∈G fig (Pg,0 i ) PC ρc g,0 g,c c + c=1 2 kP − P [k] + p [k] − ∆c + µc [k]k22 subject to Constraints (8),(10), and (12). Pg,c -update, distributively at each computing node: c Pg,c [k + 1] = arg minPg,c ,pc ρ2 kPg,0 [k + 1] − Pg,c + pc − ∆c + µc [k]k22 subject to Constraints (9),(11),(13), and (22), µc [k + 1] = µc [k] + Pg,0 [k + 1] − Pg,c [k + 1] + pc [k + 1] − ∆c . Adjust penalty parameter ρc is necessary; k = k + 1; end while return θc , Pg,c ; Output θ c , Pg,c ;

The optimization variables Pg,0 , Pg,c , pc are arranged into two groups {Pg,0 }, {Pg,c , pc } and updated iteratively. The variables of each group are optimized in parallel on distributed computing nodes, and coordinate by the dual variable µc during each iteration. At the k th iteration, the Pg,0 -update solves the base scenario with square regularization terms enforce by the coupling constraints and expressed as: X g g,0 Pg,0 [k + 1] = arg min fi (Pi )+ Pg,0

The idea of ADMM for optimization in power systems has been used by Ross Baldick [11] and by Stephen Boyd [12]. The optimization problem (7) cannot be readily solved using ADMM immediately, since the constraint (14) couples the precontingency and post-contingency variables together, and the inequality form makes the problem even more complicated. To address these challenges, the optimization problem (7) can be reformulated by introducing a slack variable pc ∈ R|G| : minimize (7) subject to Constraints (8)-(13),

Algorithm 1 Distributed SCOPF

kPg,0 − Pg,c + pc − ∆c + µc k22 . (23)

C X c=1

i∈G

c

ρ kPg,0 − Pg,c [k] + pc [k] − ∆c + µc [k]k22 2

subject to Constraints(8), (10), and (12).

(24)

The Pg,c -update solves a number of independent optimization subproblems correspond to post-contingency scenarios and can be calculated distributively at the cth computing nodes as: Pg,c [k + 1] = arg min Pg,c ,pc c

ρc g,0 kP [k + 1] − Pg,c + 2

p − ∆c + µc [k]k22 subject to

Constraints(9), (11), (13), and (22),

(25)

where the scaled dual variable dual variable is also updated locally at the cth computing utility as: µc [k + 1] = µc [k]+ Pg,0 [k + 1]− Pg,c[k + 1]+ pc[k + 1]− ∆c. (26) At the k th iteration, the original problem is divided into C + 1 subproblems with approximately the same size. The computing node handles Pg,0 needs to communicate with all computing nodes solving (25) during the iterations. The results of Pg,0 -update {Pg,0 } will be scattered among the

TABLE II C HARACTERISTICS OF TEST CASES |N | 57 118 300

|G| 7 54 69

|B| 80 186 411

Number of contingency cases 50 100 100

computing nodes for Pg,c -update. After Pg,c -update, the computed {Pg,c , pc , µc } will be collected to calculate the precontingency control variables. The subproblem data are iteratively updated so that in the end the block-coupling constraints (21) are satisfied. Note that since each of subproblems is a smaller scale OPF problem, existing techniques for OPF can be applied with minor modifications. ADMM is a primal-dual algorithm where each computing node c solves its own subproblem (25), the variations to constraint (21) are systematically penalized at certain prices through the scaled dual variable to each individual subproblem. Remark that in the ADMM framework for distributed computing the dual variables, or price, are not uniformly set for all nodes, which will require costly synchronization. For the convex optimization problem, ADMM converges to the optimum geometrically [16], and the convergence rate will be significantly enhance by using the warm start technique [17]. IV. N UMERICAL T ESTS In this section, the numerical tests are given to evaluate the performance of the proposed algorithm. Three classical test systems are used the formulate the SCOPF problem: IEEE 57 bus, IEEE 118 bus and IEEE 300 bus, whose structures and characteristics are summarized in Table II [18]. Two kinds of contingencies are considered in numerical tests: branch outage and generator failure. The contingencies are artificially generated and the number of contingencies considered are listed in Table II. We follow the physical limits on the equipments of test systems and assume every active generator is able to reschedule up to 50% of its maximum real power capacity. The numerical tests are implemented via MATLAB 7.10 on a PC with an Intel Q8200 2.33GHz and 8GbB memory. The basic OPF solver is the same for all test systems. The performance of the convergence and computing time of the proposed algorithm are investigated in the following parts. The numerical tests results are averaged out across a total number of 500 Monte Carlo implementations. A. Convergence Performance We first consider the convergence issue of the proposed algorithm. Since the number of contingencies and the optimal value for each test system are different, the relative error is used here to demonstrate the results. Suppose r[k] is the result of the value of objective function at the k th iteration, and r∗ is the optimal solution. The relative error e is defined as ∗ |. The convergence performances are shown in e = | r[k]−r r[0]−r ∗ Fig. 2. It is shown that after a moderate number of iterations, the proposed algorithm converges to the optimal values in different cases. From Fig. 2, we can see that the test system

IEEE case57 IEEE case118 IEEE case300

1

0.8

Relative Error

Case IEEE 57 bus IEEE 118 bus IEEE 300 bus

1.2

0.6

0.4

0.2

0

−0.2

Fig. 2.

0

5

10

15

20 25 Number of Iterations

30

35

40

45

Convergence performance of proposed algorithm on test systems.

IEEE 57 bus has the fastest convergence rate. This is mainly due to the small scale of the test system as well as the number of the contingencies in the system. A large system leads to a large scale optimization problem, and the numerous number of contingencies considered will make the problem scale even larger. Note that after very few iterations, the optimization result are very close to the optimal value, which means that the proposed algorithm are able to yield a good approximation to the optimal value in a short time. B. Computing Time Performance In this part, we compare the computing time of the proposed algorithm with the centralized approach to SCOPF. The computing time to obtain the optimal solution are considered in both cases. The computing time to achieve an approximate solution with a relative error e = 1% is also considered for the distributed case. To better illustrate numerical results, the speedup factor is defined as Sp = Tc /Tp , where Tc is the computing time of the centralized approach, and Tp is the computing time of the distributed approach. The results of the computing time performance are presented in Table III. It is shown in Table III that the proposed distributed approach obtains the same optimum as the centralized approach, and can achieve a speedup factor Sp = 1.4 ∼ 2.4. Remark that if only an approximated result is needed, the speedup factor can even be improved to Sp = 4.4 ∼ 4.8 by the proposed distributed algorithm. The speedup factor for the smallest test system, IEEE 57 bus, is the smallest. This is due to the communication overhead between different computing nodes during the simulations. A larger Sp can be achieved on a large-scale test system because the communication overhead is negligible compared with the computing time of the optimization subproblem handled by each computing nodes. The performance of computing time for the test systems with different numbers of contingency cases are also in-

1

2

10

10 Centralized Algorithm Proposed Algortihm

Centralized Algorithm Distributed Algorithm

Computation time

Computation Time

Computation Time

Centralized Algorithm Proposed Algorithm

2

10

1

10

0

10 1h

20

30 Numer of Contingencies

40

50

(a) Computing time for case IEEE 57 bus Fig. 3.

20

40

60 Numer of Contingencies

IEEE 57 bus IEEE 118 bus IEEE 300 bus

487.53 1606.73 9567.12

20

40

60 Number of Contingencies

80

100

(c) Computing time for case IEEE 300 bus

Computing time for test systems with different number of contingency cases

DIFFERENT TEST SYSTEMS

Centralized Cost Time

100

(b) Computing time for case IEEE 118 bus

TABLE III C OMPUTING TIME PERFORMANCE OF THE PROPOSED ALGORITHM ON

Cases

80

Distributed ADMM Time Cost (e = 1%) 5.22 487.53 3.55 492.40 36.03 1606.73 18.92 1622.79 221.67 9567.12 95.74 9662.80 Cost

Time 1.18 7.93 52.87

vestigated and the results are given in Fig. 3. The number of contingencies are increased by 20% each time and the computing time is recorded. It is shown in the figure that with the increase of the number of contingency cases in the SCOPF, the computing time of the centralized algorithm increases much faster than the proposed algorithm. Thus, the proposed distribute algorithm is more scalable and stable than the centralized approach. V. C ONCLUSION The SCOPF has become an essential tool for operators of power systems for operation planning and real time operation of their system. The significantly large problem size, the stringent real-time requirement and the variety of numerous postcontingency states make the SCOPF a challenging problem to solve. In this paper, we propose a distributive parallel approach based on the ADMM to deal with the resultant big data scale optimization problem with manageable complexity. Specifically, The original SCOPF problem is decoupled and divided into independent subproblems of approximately the same size correspond to each individual pre-contingency and post-contingency case. Subproblems are optimized in a parallel fashion on distributed nodes and dual (price) variables are designed delicately for coordination. Numerical tests on IEEE buses validated the effectiveness of proposed algorithm. R EFERENCES [1] M. Shahidehpour, W. F. Tinney, and Y. Fu, “Impact of security on power system operation,” Proceedings of the IEEE, vol. 93, no. 11, pp. 20132025, Nov. 2005.

[2] O. Alsac and B. Scott, “Optimal load flow with steady-state security,” IEEE Transaction on Power Apparatus and System, vol. 93, no. 3, pp. 745-751, May 1974. [3] M. V. F Pereira, A. Monticelli, and L. M. V. G. Pinto, “Securityconstrained dispatch with corrective rescheduling,” IFAC Symposium on Planning and Operation of Electric Energy System, Rio de Janeiro, Brazil, Jul. 1985. [4] A. J. Wood and B. F. Wollenberg, Power Generation Operation and Control. Wiley, New York, 1996. [5] A. Monticelli, M. V. F. Pereira, and S. Granville, “Security-constrained optimal power flow with post-contingency corrective rescheduling,” IEEE Transactions on Power Systems, vol. 2, no. 1, pp. 175-180, Feb. 1987. [6] F. Capitanescu, J. L. M Ramos, P. Panciatici, D. Kirschen, A. M. Marcolini, L. Platbrood, and L. Wehenkel, “State-of-the-art, challenges, and future trends in security constrained optimal power flow,” Electric Power System Research, vol. 81, no. 8, pp. 1731-1741, Aug. 2011. [7] J. Mart´ınez-Crespo, J. Usaola, and J. L. Fern´andez, “Security-constrained optimal generation scheduling in large-scale power systems,” IEEE Transactions on Power Systems, vol. 21, no. 1, pp. 321-332, Feb. 2006. [8] Y. Fu, M. Shahidehpour, and Z. Li, “AC contingency dispatch based on security-constrained unit commitment,” IEEE Transactions on Power Systems, vol. 21, no. 2, pp. 897-908, May 2006. [9] F. Capitanescu and L. Wehenkel, “A new iterative approach to the corrective security-constrained optimal power flow problem,” IEEE Transcations on Power Systems, vol.23, no. 4, pp. 1533-1541, Nov. 2008. [10] Y. Li and J. D. McCalley, “Decomposed SCOPF for improving efficiency,” IEEE Transcation on Power Systems, vol. 24, no. 1, pp. 494-495, Feb. 2009. [11] R. Baldick, B. H. Kim, C. Chase, and Y. Luo, “A fast distributed implementation of optimal power flow,” IEEE Transactions on Power Systems, vol. 14, no. 3, pp. 858-864, Aug. 1999. [12] M. Kraning, E. Chu, J. Lavaei, and S. Boyd, “Dynamic network energy management via proximal message passing,” Foundations and Trends in Optimization, vol. 1, no. 2, pp. 1-54, 2013. [13] K. Karoui, H. Crisciu, A. Szekut, and M. Stubbe, “Large scale security constrained optimal power flow,” Power System Computation Conference (PSCC), Glasgow, Scotland, May 2008. [14] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundation and Trends in Machine Learning, vol. 3, no. 1, pp. 1-122, 2010. [15] D. Bertsekas and J. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods (2nd ed.) Athena Scientific, 1997. [16] W. Deng and W. Yin, “em On the global and linear convergence of the generalized alternating direction method of multipliers,” Rice CAAM technical report 12-14, 2012. [17] J. Nocedal and S. J. Wright, Numerical Optimization (2nd ed.), Springer, New York, 2006. [18] R. D. Zimmerman, C. E. Murillo-Snchez, 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. 12-19, Feb. 2011.