A parallel method for solving the DC security constrained optimal power flow with demand uncertainties

Electrical Power and Energy Systems 102 (2018) 171–178

Contents lists available at ScienceDirect

Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

A parallel method for solving the DC security constrained optimal power flow with demand uncertainties Linfeng Yanga, Chen Zhangb, Jinbao Jianc,

T

⁎

a

School of Computer Electronics and Information, Guangxi Key Laboratory of Multimedia Communication and Network Technology, Guangxi University, Nanning 530004, China b School of Electrical Engineering, Guangxi University, Nanning 530004, China c College of Science, Guangxi University for Nationalities, Nanning, 530006, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Security constrained optimal power flow Uncertainty of electric power demand Interval optimization method ADMM

The security constrained optimal power flow (SCOPF) is a fundamental tool to analyze the security and economy of a power system. To ensure the safe and economic operation of a system considering demand uncertainties and to acquire economic and reliable solutions, in this paper, a parallel method for solving the interval DC SCOPF with demand uncertainties is presented. By using the interval optimization method, the uncertain nodal load can be expressed as interval variables and integrated into the DC SCOPF model, which is then formed as a large scale nonlinear interval optimization formulation. According to the theory of interval matching and selection of the extreme value intervals, the interval DC SCOPF problem can be transformed into two deterministic nonlinear programming problems and solved by alternating direction method of multipliers (ADMM) to obtain the range information of interval formulation variables. Using ADMM, the above two deterministic problems, which are large in scale because of the large number of preconceived contingencies, all can be split into independent subproblems corresponding to pre-contingency status and each individual post-contingency cases. These small-scale sub-problems can be solved in parallel to improve the computing speed. Compared with the Monte Carlo (MC) method, the simulation results of the IEEE 30-, 57- and 118-bus systems validate the effectiveness of the proposed method.

1. Introduction The security constrained optimal power flow (SCOPF) [1,2], which is an extension of the conventional optimal power flow (OPF) [3], aims at determining an optimal operating point for control variables that minimizes a given objective function subject to physical constraints and control limits and takes into account both the normal state and contingency constraints. It can ensure the safe and economic operation of the power grid in theory. 1.1. Deterministic SCOPF The traditional SCOPF, a large-scale optimal problem which might include nonlinear and non-convex items, discrete variables, etc., is computationally intensive because it considers a large number of contingencies [4]. Therefore, determining how to efficiently solve the deterministic SCOPF problem has become the research focus. The SCOPF problem has been widely classified into two classes: preventive security constrained optimal power flow (PSCOPF) [1], which assumes that the ⁎

post-contingency conditions can be met without redispatching, and Corrective security constrained optimal power flow (CSCOPF) [2], which permits post-contingency control variables such as generators’ active power and terminal voltage to be readjusted for removing any violations caused by the contingency. In the SCOPF literature, many proven methodologies have been proposed in order to efficiently deal with this problem [4]. These methods can be divided into two categories. One category is solving the SCOPF directly [5,6]. However, this direct method for handling largescale power systems with numerous contingencies in a centralized manner would result in a prohibitive memory and CPU times requirements [4]. The other category is reducing the size of the SCOPF problem [4,7]. Novel filtering techniques, relying on the comparison at an intermediate PSCOPF solution of post-contingency constraint violations among postulated contingencies, are proposed in [8] to accelerate the iterative solution of PSCOPF. In [7], the author used contingency filtering techniques to identify the binding contingency and limited the number of contingency. In addition, network compression method is used to reduce the size of each post-contingency model. Actually, all the

Corresponding author. E-mail address: [email protected] (J. Jian).

https://doi.org/10.1016/j.ijepes.2018.04.028 Received 25 December 2017; Received in revised form 20 March 2018; Accepted 24 April 2018 Available online 26 May 2018 0142-0615/ © 2018 Elsevier Ltd. All rights reserved.

Electrical Power and Energy Systems 102 (2018) 171–178

L. Yang et al.

selection of extreme interval, the interval optimal problem was translated into determinate quadratic programming problem, and solved by alternating direction method of multipliers (ADMM) to obtain the range information of interval formulation variables in parallel. Finally, the interval corrective security constrained optimal power flow results on the IEEE 30-, 57- and 118-bus systems are shown and the impact of the key parameters for the ADMM algorithm are described. The major contributions of this paper are summarized as follows: (1) By using interval mathematics, the uncertain parameters can be expressed as interval form, and be integrated into SCOPF model which then formed as a more concise nonlinear interval optimization formulation; (2) According to the theory of interval matching and selection of the extreme value intervals, the interval SCOPF problem can be transformed into two deterministic nonlinear programming problems for determining the range information of interval formulation variables; (3) The resulted problems are solved in parallel by using ADMM. The simulation results show that the proposed method is computationally efficient and reliable, and it is effective for the SCOPF problem with multiple interval variables. The remainder of this paper is organized as follows. Section 2 provides the formulations of the interval CSCOPF (ICSCOPF) problem. Section 3 presents method for solving the ICSCOPF problem based on ADMM. Section 4 contains numerical experiments, and Section 5 concludes the paper.

methods belonging to the second category, including the above-mentioned methods, can be divided into four classes: iterative contingency selection schemes [8–10]; network compression [11]; hybrid methods which combine the two methods above [7]; decomposition methods [2,12–15]. The reader is referred to [4] for a detailed literature review on this topic. And, these methods are important because that the largescale CSOPF problems should be reduced in size before to be solved, and the reduced problems can be used as the starting point for our approach in this paper. Linear DC approximation of the (nonlinear, non-convex) AC power flow equation is frequently used in existing power systems [16]. DCSCOPF is an approximation of AC-SCOPF for obtaining the optimal active power dispatch solution of the entire power system. In [17], the author improved the solution techniques for the AC SCOPF problem of active power dispatch by using the DC SCOPF approximation within the SCOPF algorithm. In this paper, we are interested in DC power flow constraints for the problem. 1.2. SCOPF under uncertainty Modern power grids are characterized by increasing penetration of renewable energy sources and demand evolution, such as electric vehicles and energy storage [18,19]. This trend is expected to increase in the near future. Due to the privatization of the electricity market, the modern electricity market has changed. For example, some of the actions of the power participants are unpredictable [20]. The stochastic programming theory, robust optimization [21], fuzzy set theory [22] and interval analysis [23] are frequently-used methods for dealing with uncertainty problems. So far, the first two methods has been used to describe uncertainty in SCOPF successfully [4]. In [24], a robust AC SCOPF algorithm with three-stage feasibility checking problem is proposed to deal with the day-ahead power systems security planning under uncertainties. [25] propose a robust DC SCOPF algorithm which uses the mixed integer bi-level program optimization to compute the worst patterns of uncertain variables associated to each contingency. A multi stage stochastic programming model is established in [26] based on the uncertainty of the node load. The proposed model and the planning method are flexible, but the integer variable of the model increases. A new robust optimization method considering both safety and economy was given in document [13]. Aiming at circuit risk and system risk, using Lagrangian relaxation and Benders decomposition technology to solve the corrective risk-based SCOPF problem was proposed in [27]. Taking into account the uncertainty of renewable energy production, load consumption, and load reserve capacities, in [28] the authors formulated a chance constrained OPF to achieve minimum cost. However, most of these methods require probability distribution functions for uncertain items [29]. And these functions are difficult to obtain in the actual situation. According to the theory of interval matching and selection of the extreme value intervals, the interval optimization problem was translated into two determinate nonlinear programming problems and was solved to obtain the range information of the interval optimization problem [23,30]. The interval analysis model does not require the distribution parameters of uncertainty to be determined, as it requires only the upper and lower boundary information.

2. CSCOPF formulation 2.1. The OPF problem descriptions The standard AC-OPF determines the least-cost operation of power systems by dispatching generation resources to supply systems loads, while satisfying prevailing system-level and physical constraints. In practice, AC-OPF problems are typically approximated by a more tractable “DC-OPF” problem that focuses exclusively on real power constraints in linearized form [31]; these linear real power constraints are also used in this paper. Given that the operational costs of thermal units is commonly represented as a quadratic function of the generation level, the objective 2 is fi (PG,i ) = (ai + bi PG,i + ci PG, i ) , where ai , bi and ci are the generation cost coefficients of the generator i . PG,i is the active output of the generator i . The equality constraint of OPF formulation is the power flow equation of the system. N

PG,i−PD,i− ∑ (Bij θj ) = 0, i = 1,…,N j=1

(1)

where Bij is the susceptance between the i -th and j -th buses. θj is the voltage angle at node j . PD,i is the real power demand at node i . N is the total number of nodes in the system. The inequality constraints of the OPF model include the constraints of state variables and the limits of each physical quantity describing the power system, i.e.,

1.3. Paper contributions and organization

P G,i ⩽ PG,i ⩽ PG,i, i ∈ SG

(2)

P ij ⩽ Pij ⩽ Pij

(3)

where PG,i and P G,i are the maximum and minimum active power output of generator i respectively. SG represents the set of generators in the grid. Pij and P ij are the maximum and minimum active power flow, respectively, on the line between node i and j respectively. Pij = Bij (θi−θj ) represents the active power flow on line between node i and j , in the direction from node i to node j . The constraint (2) is the upper and lower bound for the active power output of the generator i . The constraint is (3) power flow limit on each line. For the sake of convenience, let x denote the vector of state

In this work, we propose a novel approach based on the interval optimization method to solve CSCOPF with demand uncertainties problem in parallel. Above all, according to the interval optimization algorithm, the uncertain load variables are expressed as interval variables and lead into the model of corrective security constrained optimal power flow. Thus, an interval optimization formulation was constructed for corrective security constrained optimal power flow with demand uncertainties. Next, based on the theory of direct interval matching and 172

Electrical Power and Energy Systems 102 (2018) 171–178

L. Yang et al.

∼ x −∼ y = [xlow −yup ,x up−ylow ]

variables (i.e., voltage angle at all buses), and let u denote the vector of control variables (e.g., generators active power); we can abbreviate the OPF problem as follows,

min (x ,u) f (x ,u)

∼ x ·∼ y = [min(xlow ·ylow ,xlow ·yup ,x up ·ylow ,x up ·yup ), max(xlow ·ylow ,xlow ·yup ,x up ·ylow ,x up ·yup )]

(4)

∼ x /∼ y = [xlow ,x up]·[1/ yup ,1/ ylow ],if 0 ∉ ∼ y

g (x ,u) = 0 s.t. ⎧ ⎨ ⎩ h (x ,u) ⩽ 0

∼ f (x ) = [flow (x ),fup (x )]

2.2. The basic CSCOPF problem

The uncertain demand can be expressed as interval form. Thus, the power flow Eq. (1) of the system can also be expressed as an interval function. ∼ Replacing PD,i in the Eq. (1) with the interval variable PD,i , Eq. (1) turns into interval function (13). n

∼ PG,i−PD,i− ∑ (Bij θj ) = 0

(5)

j=1

(13)

g to represent Eq. (13), the ICSCOPF Next, using interval function ∼ formulation is as follows: ∼0 min x 0 ,… ,xC ;u0 ,… ,uC f (x 0,u 0)

where C is the total number of preconceived contingencies in the system; c = 0 corresponds to pre-contingency state which actually corresponds to the traditional OPF described in the last subsection, whereas c = 1,…,C corresponds to the c th post-contingency state; Δc is the vector of maximal allowed adjustment of the control variables between pre-contingency and the c th post-contingency state. Thus, the first and second constraints in (5) impose the feasibility of the precontingency and corrected post-contingency states. The third constraint limits the maximal amount of adjustment of the control variables between the pre-contingency and the c th post-contingency state.

(14)

(x c ,uc ) ∈ ∼ χc , c = 0,…,C s.t. ⎧ ⎨ − ⩽ | u u | Δc , c = 1,…,C 0 c ⎩ ∼ g c (x c ,uc ) = 0 ⎫ χc = ⎧(x c ,uc )| c where ∼ . ⎨ h (x c ,uc ) ⩽ 0 ⎬ ⎩ ⎭ 2.5. Interval CSCOPF problem transformations In this paper, the system demand PD,i in the interval CSCOPF (ICSCOPF) formulation is an interval variable. Thus, the ICSCOPF is an interval nonlinear programming problem. Interval algorithm is firstly put forward to solve the error and uncertain problem in scientific calculation. The purpose is to provide the upper and lower bounds of the computational results cause by errors and uncertainties. [30] improved the interval algorithm and put forward the theory of Interval Nonlinear Optimization. According to the literature [30], the interval matching is as follows:

2.3. Demand uncertainty Unexpected fluctuation and prediction error of demand lead to certain error between predicted demand and actual demand. Therefore, the system demand in the SCOPF problem is uncertain. Actually, some uncertain items, such as power output, often can be taken as negative demand [33], therefore this paper only considers the uncertainty load. An interval variable defines the range that an uncertain variable may take in terms of its lower and upper boundaries. In the following formula (6), the lower boundary xlow of interval variable ∼ x represents the minimum possible value of an uncertain variable, whereas the upper boundary x up represents the maximum possible value.

∼ x ⩾∼ y ⇔ xlow ⩾ ylow and x up ⩾ yup

(15)

x ,∼ y are a pair of interval variables, where ∼ The ∼ x = [xlow ,x up] and ∼ y = [ylow ,yup ]. y are comparable and satisfy relation (15), the If the intervals ∼ x and ∼ condition (xlow ⩾ ylow and x up ⩾ yup ) should be satisfied. If the conditions ((xlow ⩾ ylow and x up ⩾ yup ) or (xlow ⩽ ylow and x up ⩾ yup ) ) are satisfied, then they are incomparable. In other words, only interval shifted relative to each other are comparable, with the intervals shifted to the left being smaller, and if one interval covers another, then they are incomparable [30].Similarly, if the interval functions are comparable, then condition (16) should be satisfied

(6)

Here, the system demand uncertainties can be expressed as interval variables. (7)

∼ ∼ {f (x ) = [flow (x ),fup (x )] ⩾ F (x ) = [Flow (x ),Fup (x )] , ∀ x ∈ Ω}

where P low,D,i represents the minimum value for the demand of the system node i , and Pup,D,i represents the maximum value for the demand of the system node i . Interval variables have four arithmetic operations [32]. Addition, subtraction, multiplication, and division of an interval are defined as:

∼ x +∼ y = [xlow + ylow ,x up + yup ]

(12)

2.4. ICSCOPF problem

c ⎧ g (x c ,uc ) = 0, c = 0,…,C c s.t. h (x c ,uc ) ⩽ 0, c = 0,…,C ⎨ ⎩|u 0−uc | ⩽ Δc , c = 1,…,C

∼ PD,i = [P low,D,i,Pup,D,i]

(11)

The four arithmetic operations of the interval function are the same as those of the interval variables.

The traditional OPF (4) includes only the steady-state operating limits (1)–(3)and does not take the possible contingencies under consideration. The CSCOPF is an extension of the traditional OPF problem, which should consider the contingencies while minimizing the system operating cost. The CSCOPF formulation studied in this paper, whose aims is to achieve the best balance between economy and security against the preconceived contingencies by exploiting the corrective control actions. The model of CSCOPF is a nonlinear programming that can be compactly described as follows [2]

∼ x = [xlow ,x up] = {x ∈ R|xlow ⩽ x ⩽ x up},

(10)

If the parameter of the function is uncertain, then the function can also be expressed in an interval form, as given by (12):

where g (·) = 0 includes all equality constraints and h (·) ⩽ 0 includes all inequality constraints.

min x 0 ,… ,xC ;u0 ,… ,uC f 0 (x 0,u 0)

(9)

⇔ {flow (x ) ⩾ Flow (x ) and fup (x ) ⩾ Fup (x ) ,∀ x ∈ Ω}

(16)

Theorem 1. If the interval functions ∼ x1,∼ x2,…,∼ x n can be compared and the x1 is the minimum, then it is sufficient and necessary that conditions interval ∼ (17) be satisfied.

(8) 173

Electrical Power and Energy Systems 102 (2018) 171–178

L. Yang et al.

∼ xlow,1 ⩽ mini ∈ n∼ xlow,i and ∼ x up,1 ⩽ mini ∈ n∼ x up,i

⎧u 0−uc + u 0c = Δc , c = 1,…,C c = 1,…,C s.t. 0 ⩽ u 0c ⩽ 2Δc ⎨ (x c ,uc ) ∈ χlow,c , c = 0,…,C ⎩

(17)

Formula (17) reduces the selection of the extremal interval to the usual selection of two extremal numbers. According to Eq. (17), we have the extremum condition of the in∼ terval function: if the interval functions f (x ) = [flow (x ),fup (x )] take the minimum value at the point x ∗ of its domain of G, then it is necessary and sufficient that the minimum value be taken by its boundary function flow (x ) and fup (x ) at the same point [30], as follows:

The u 0 in u 0−uc + u 0c = Δc ,c = 1,…,C is considered as a partitioned block, and −uc + u 0c is considered as another block, then, the problem (26) is a two-block large scale quadratic programming (QP) problem which is suitable for ADMM. Thus, the scaled augmented Lagrangian is

∼ ∼ {f (x ∗) = minx ∈ G f (x )} ∼ ∼ ∼ ∼ ⇔ {flow (x ∗) = minx ∈ G flow (x ) and fup (x ∗) = minx ∈ G fup (x )}

C 0 Lρ (x 0,…,x C ;u 0,…,uC ;u 01,…u 0C ;v1,…vC ) = f low (x 0,u 0) +

c=1

(18)

Algorithm 1. IO-ADMM for the lower boundary ICSCOPF

|u 0−uc | ⩽ Δc , c = 1,…,C s.t. ⎧ (x ,u ) ∈ χlow,c , c = 0,…,C ⎨ ⎩ c c

Initialization: x = 0,u =

> 0,ρc > 0,ε pri > 0,ε dual > 0

C

0 (x 0k + 1,u 0k + 1) = argmin (x 0,u0) ∈ χ low,0 = f low (x 0,u 0 ) + ∑c = 1

ρc ||u 0−uck 2

+ u 0kc−Δc

+ vck ||22

|u 0−uc | ⩽ Δc , c = 1,…,C s.t. ⎧ (x ,u ) ∈ χup,c , c = 0,…,C ⎨ ⎩ c c

for each post-contingency state c = 1,…,C in parallel x c ,uc and u 0c -update:

g c (x c ,uc ) = 0 ⎫ where define sets χlow,c = ⎧(x c ,uc )| low , c = 0,…,C . The set ⎨ hc (x c ,uc ) ⩽ 0 ⎬ ⎩ ⎭ χup,c is similar to χlow,c . Now, we can use deterministic algorithm to solve the boundary problems (19) and (20) that are similar to the conventional CSCOPF

(xck + 1,uck + 1,u 0kc+ 1) = argmin (x c,uc ) ∈ χ low,c ,0 ⩽ u0c ⩽ 2Δc +

ρc ·‖u 0k + 1−uc 2

+ u 0c−Δc

vck ‖22

vc -update: vck + 1 = vck + u0k + 1−uck + 1 + u0kc+ 1−Δc end for

3. ADMM for ICSCOPF formulation

r k + 1 = ||u0k + 1−uck + 1 + u0kc+ 1−Δc ||2 s k + 1 = ‖ρc ·(u0kc+ 1−u0kc )‖2

3.1. ADMM for ICSCOPF

if r k + 1 ⩽ ε pri and s k + 1 ⩽ ε dual break. end if end for return x 0 and u 0 .

In this section, we introduce a distributed algorithm to solve the ICSCOPF by decomposing it to a set of simpler and parallel sub-problems corresponds to the pre-contingency case and each post-contingency case. The approach is based on the ADMM [34,35], whose general form is described as follows:

minx ,z f (x ) + g (z )

At each iteration, we decompose the lower boundary ICSCOPF into C sub-problems with approximately the same size of the smaller scale OPF problem, which is suitable for parallel implementation. Note that, ADMM is introduced for solving convex problem, and the problem (26) is a QP problem. Thus, the proposed IO-ADMM can converge to the global optimal solution

(21)

s.t. Ax + Bz = c 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 ||Ax + Bz− c+ v||22 2

3.2. IO-ADMM algorithm steps

(22)

In this paper, the ICSCOPF model is established and transformed into two deterministic conventional nonlinear optimization problems. The implementation steps are as follows

where ρ > 0 is the penalty parameter and v is 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:

x k + 1 = argmin x f (x ) +

ρ ‖Ax + Bz k− c+ v k‖22 2

(23)

z k + 1 = argminz g (z ) +

ρ ||Ax k + 1 + Bz− c+ v k||22 2

(24)

v k + 1 = v k + Ax k + 1 + Bz k + 1−c

1. Using the above IO-ADMM to calculate the lower boundary problem (19), the lower boundary solution of the problem is xlow . 2. As in step 1, using the IO-ADMM to calculated the upper boundary problem (20), the upper boundary solution of the problem is x up . If x is the optimal x = [xlow ,x up] is not empty, then, ∼ xlow and x up satisfy ∼ solution of the interval problem (14). Otherwise, the problem (14) does not have a solution.

(25)

We notice that the lower boundary ICSCOPF problem (19) can be reformulated by introducing auxiliary variables u 0c in a form suitable for ADMM as 0 min x 0;u0 f low (x 0,u 0)

(GenPmin + GenPmax) ,M 2

for iteration k = 0,1,…,M x 0 and u 0 -update: (20)

Lρ (x ,z ,v ) = f (x ) + g (z ) +

(27)

The ADMM scheme for the lower boundary ICSCOPF problem is defined as follows

(19)

0 min x 0;u0 f up (x 0,u 0)

ρc ||u 0−uc + u 0c 2

−Δc + vc ||22

According to formula (18), the ICSCOPF problem (14) would be reduced to two definite upper and lower boundary problems (19) and (20) 0 min x 0;u0 f low (x 0,u 0)

∑

We note that the convexity is required for the ICSCOPF models presented in Section 2 and ADMM solution method presented in this Section. So, direct application of the proposed method to deal with non-

(26) 174

Electrical Power and Energy Systems 102 (2018) 171–178

L. Yang et al. 198

convex AC SCOPF cannot guarantee convergence.

The upper total cost The lower total cost

197.8

4. Numerical results and analysis

197.6

Total cost ($/MW)

In this section we present some numerical results to test the efficiency and effectiveness of the proposed ADMM for the boundary ICSCOPF method. The machine on which we perform all of our computations is an Intel i7-4790U 3.6 GHz Lenovo desktop with 8 GB of RAM, running MS-Windows 7 and Matlab 2014a. CPLEX 12.6.2 is used to solve quadratic programming for the solver in our numerical experiments. Unless otherwise specified, we used parameter values ρ = 0.01, ∊abs = 10−2 , ∊rel = 10−3 , ε pri = p ∊abs + M = 1000 , k+1 k+1 rel ∊ max{‖u 0 ;…;uCk + 1 ‖2 ,‖u01 ;…;u0kC+ 1 ‖2 ,‖Δ1 + u1k + 1;…;ΔC + uCk + 1 ‖2 } and ε dual = d ∊abs + ∊rel ‖v1k + 1;…;vCk + 1 ‖2 . The factors p and d are the constraints number of the primal and dual variable dimension. Four classical test systems are used in the formulation of the ICSCOPF problem, IEEE 14 bus, IEEE 30 bus, IEEE 57 and IEEE 118 bus, whose structures and characteristics are summarized in Table 1. We artificially generated the list of contingencies; the corresponding numbers of scenarios are presented in the fifth column. The last column gives the number of variables. In our experiments, a contingency is related to a transmission line outage; each generator is allowed to readjust up to 5% of its maximal generator active output capacity. This paper studies only the uncertainty of the load power of the system nodes. It is assumed that a node load power fluctuation range per systems is ± 10%, i.e., the load power variation interval [0.9PD,1.1PD]

196.4

|B|

|C|

Variables

IEEE14 IEEE30 IEEE57 IEEE118

14 30 57 118

3 6 4 16

20 41 78 179

12 33 35 159

282 1422 2336 24,984

0

500

1000

1500

2000

2500

3000

3500

4000

times k

Fig. 1. Number of sample points in the MC method effect on the result of ICSCOPF.

value is $197.252 and the lower boundary value is $197.099, as obtained by the MC method, which is the same as that obtained by the IO method. The other reason is that the IO-ADMM algorithm is based on the most extreme mode of operation to consider the system, which is good for the operation of the staff. To study the influence of the number of interval parameters on the IO-ADMM, the number of interval parameters of each test system is gradually increased, and the objective value obtained by IO-ADMM is compared with the of the MC. The effect of the increase in the number of interval parameters on the system objective value is shown in Table 3. In this table, row “Num” represent the number of interval parameters of the system; row “IO_upper” and “ IO_lower” represent the upper and lower boundary solution of the problem by the proposed IOADMM, respectively; row “MC_upper” and “ MC_lower” represent the upper and lower boundary solution of the problem by MC, respectively; IO upper − MC upper reports the relative error of the upper boundary ε up = MC upper MC lower − IO lower

reports the relative error of the lower solution; ε low = IO lower boundary solution. In Table 3, with increasing number of interval parameters, the range of the objective value is also increasing, however, the relative error ε up and ε low remain approximately 2%. For the IEEE 14 and IEEE 30 system, the relative errors are less than 0.2%. For the IEEE 57 and IEEE 118 systems, when the number of interval parameters is less than 15, the relative errors are less than 1% or approximately 1%. In other words, the proposed IO-ADMM has “good performance” for our experiments, i.e., for systems with few interval parameters, IO-ADMM can achieve better interval results. In order to show the relation between the number of interval parameters and the DC ICSCOPF results, Fig. 2 gives the results for branch between bus 23 and bus 24 in IEEE 118 system with different Table 2 Test systems ICSCOPF results.

Table 1 Test systems characteristics. |G|

197

196.6

To verify the effectiveness of IO-ADMM algorithm, the boundary information obtained by Monte Carlo simulation [36] is used as the standard of comparison. Assuming that the known load interval parameters are subject to normal distribution, the Monte Carlo (MC) method is used to calculate the ICSCOPF, with the maximum and minimum values of the calculation results set as the upper and lower bounds of the optimization results. The effect of MC sampling number on the optimization result of the IEEE 14 system when considering the 4 parameters change rate of ± 10% is shown in Fig. 1. From Fig. 1, we know that when the number of sampling exceeds 2500 times, the objective function of the upper and lower boundary values are stabilized; as a result, this paper chooses 3000 samples of the MC method to calculate the results as a comparison standard. The examples considered in this paper are as follows: for the IEEE 14 and IEEE 30 systems, it is assumed that the load power fluctuation range of one node of the system is ± 10%, and, for the IEEE 57 and IEEE 118 systems, it is assumed that the load power fluctuation range of the five nodes of the system is ± 10%. The number of interval variables considered is 5. The objective function values for deterministic CSCOPF and ICSCOPF for different test systems are shown in Table 2. The results in Table 2 show that the results of the IO-ADMM method are close to the results of the MC method. Actually, the results of our method always contain the results of the MC method. One reason for this result is that, for the MC method, the number of times of sampling is not sufficient, resulting in the MC method results being too small; when the number of samples reaches 5000, for the IEEE 118 system, the upper boundary

|N |

197.2

196.8

4.1. The ICSCOPF analysis

Case

197.4

175

Total cost

IEEE14 ($/MW)

IEEE30 ($/MW)

IEEE57 ($/MW)

IEEE118 ($/MW)

CSCOPF The upper boundary by IO The lower boundary by IO The upper boundary by MC The lower boundary by MC

197.176 197.252

86.182 86.243

269.532 271.472

146.418 147.668

197.099

86.122

267.609

145.174

197.251

86.241

271.204

147.348

197.102

86.124

267.983

145.465

Electrical Power and Energy Systems 102 (2018) 171–178

L. Yang et al.

Table 3 The influence of numbers of interval parameters.

IEEE14 System

Num

1

3

5

8

11

15

30

40

IO_upper ($/MW) IO_lower ($/MW) MC_upper ($/MW) MC_lower ($/MW) ε up

197.25 197.10 197.25 197.10 0 0

197.75 196.60 197.73 196.66 0.0001 0.0003

197.82 196.53 197.69 196.63 0.0007 0.0005

197.97 196.39 197.77 196.56 0.001 0.0009

198.10 196.29 197.74 196.53 0.0018 0.0012

– – – – – –

– – – – – –

– – – – – –

86.24 86.12 86.24 86.12 0 0

86.27 86.10 86.26 86.10 0.0001 0

86.60 85.77 86.54 85.83 0.0007 0.0007

86.74 85.64 86.62 85.79 0.0014 0.0018

86.78 85.60 86.61 85.76 0.002 0.0019

– – – – – –

– – – – – –

– – – – – –

270.10 268.97 270.10 268.97 0 0

270.56 268.51 270.51 268.59 0.0002 0.0003

271.47 267.61 271.20 267.98 0.001 0.0014

274.38 264.85 273.63 265.99 0.0027 0.0043

278.76 260.89 275.51 263.12 0.0118 0.0085

280.25 259.60 276.27 263.07 0.0144 0.0134

– – – – – –

– – – – – –

146.77 146.08 146.75 146.09 0.0001 0.0001

147.13 145.70 147.08 145.79 0.0003 0.0006

147.67 145.17 147.35 145.46 0.0022 0.002

148.56 144.28 147.74 144.86 0.0056 0.004

149.45 143.41 147.74 144.48 0.0116 0.0075

150.35 142.52 148.81 144.23 0.0103 0.012

151.33 141.57 149.04 144.11 0.0154 0.0179

153.23 139.74 149.01 143.67 0.0283 0.0281

ε low IEEE30 System

IO_upper ($/MW) IO_lower ($/MW) MC_upper ($/MW) MC_lower ($/MW) ε up

ε low IEEE57 System

IO_upper ($/MW) IO_lower ($/MW) MC_upper ($/MW) MC_lower ($/MW) ε up

ε low IEEE118 System

IO_upper ($/MW) IO_lower ($/MW) MC_upper ($/MW) MC_lower ($/MW) ε up

ε low

0.05

MC upper MC lower IO upper IO lower

-0.05

0

Voltage phase angle [rad]

Active power flower [p.u]

0

IO upper IO lower MC upper MC lower

-0.05 -0.1 -0.15 -0.2

-0.1 -0.15 -0.2 -0.25 -0.3

-0.25

-0.35

5

10

15

20

25

30

35

40

45

50

-0.4

55

interval Number

0

10

20

30

40

50

60

Node number i

Fig. 2. Results for different number of interval variables.

Fig. 3. Range of voltage phase angle.

4.2. IO-ADMM calculate time analysis

number of interval parameters. As can be seen in this figure, when the interval variables are no more than 40, the active power area increases as the numbers of interval parameters increased. And the results of IOADMM gradually and slight deteriorate. When more than 40 interval parameters are considered, the computation results tend to be stable. Figs. 3 and 4 show a comparison of the results of interval SCOPF for IEEE 57 system considering five interval parameters. Without loss of generality, choose the first five load nodes in the system to perform load disturbance. Fig. 3 depicts the bounds for the voltage phase angle. Fig. 4 shows the bounds for active power line flows. Note that the numerical results are similar to the MC method, indicating that the interval algorithm is valid.

For the IEEE 57 test system with one interval parameter, IO-ADMM calculates the lower boundary problem of ICSCOPF. The convergence of IO-ADMM is shown in Figs. 5 and 6. Figs. 5 and 6 show the procedures of the primal and dual residuals converging to primal and dual feasibility tolerances, respectively. According to these figures, the dual residual is basically less than the duality of the feasibility. This result may be due to the fact that our penalty ρ is small. The small values of penalty ρ tend to reduce the dual residual (s k + 1 = ρ ATB(z k + 1−z k ) ) at the expense of reducing the penalty on primal feasibility, which may result in a large the primal residual. Thus, we can observe from Fig. 5, the

176

Electrical Power and Energy Systems 102 (2018) 171–178

L. Yang et al. 1

Table 4 Calculation time of each systems. Calculation times (s)

Active power flows [p.u]

0.5

0

-0.5

-1

-2

0

10

20

30

40

50

60

70

80

Line Number i Fig. 4. Interval range of branch active power.

3 jjr k jj 2 " pr i

Iterative accuracy

2.5

2

1.5

1

0.5

0

0

2

4

6

8

10

12

14

16

18

0.2 jjsk jj 2 " dual

0.14

Iterative accuracy

εr

IEEE14 IEEE30 IEEE57 IEEE118

13.26 47.42 145.51 945.77

0.27 1.17 2.83 6.19

49 40 51 152

In this paper, a distributed method for solving the interval securityconstrained optimal power flow with demand uncertainties was proposed under the framework of ADMM. According to the theory of direct interval matching and selection of the extreme value intervals, the interval DC SCOPF optimization problem was translated into two determinate nonlinear programming problems, and solved in parallel by ADMM to obtain the range information of interval formulation variables. The proposed IO-ADMM method can help the system operator to improve system security while optimizing the system’s economic performance. The simulation results show that the proposed method can obtain high-quality solutions for DC ICSCOPF problems and is suitable for large-scale electricity market application.

Fig. 5. iterative values of ‖r k‖2 and ε pri for inner iterations.

0.16

IO-ADMM_times (s)

5. Conclusion

20

iterations k

0.18

MC_times (s)

primal residuals are relatively large at the beginning, but they gradually decrease with the iteration of the algorithm. The total number of iterations for this IO-ADMM is approximately 20. Therefore, our proposed IO-ADMM has good convergence for our experiments, i.e., it can converge to a point with good objective value. Table 4 shows the run-time comparison of ICSCOPF calculations using the MC and the IO-ADMM methods for different test systems. Column “ε r ” represents the IO-ADMM efficiency ratio. The efficiency MC times ratio ε r is defined as ε r = IO − ADMM times . The MC method require a large number of sampling points (1000 in this paper) to be calculated, and the IO-ADMM method need solve the calculations of the ADMM nonlinear programming problem twice (no parallel), i.e., the calculation efficiency has a greater advantage. As can be seen in Table 4, IO-ADMM has a strong computing power, and in the absence of parallel environment, the acceleration ration is more than 40. The proposed IO-ADMM can be executed in parallel by using Matlab Parallel Computing Toolbox with 4 workers; the calculation time and speedup are listed in Table 5. In this table, column “ pspu ” represents the parallel speedup; column “ ppe ” represents the parallel efficiency. The results of 1, 2, and 4 workers show that, as the number of workers increases, the calculating time reduces dramatically, but the parallel efficiency decreases. Because of the communication between an excessive number of processors and low efficiency allocation requiring excessive time, the parallel efficiency is a decreasing function of the CPU number.

MC upper Pij MC lower Pij IO upper Pij IO lower Pij

-1.5

Case

0.12 0.1 0.08

Acknowledgement

0.06

This work was supported by the National Natural Science Foundation of China (51767003, 11771383, 51407037, and 61661004), the Guangxi Natural Science Foundation (2016GXNSFDA380019, 2014GXNSFBA118274, and 2014GXNSFAA118391), Guangxi Key Laboratory of Power System Optimization and Energy Technology Foundation (15-A-01-11).

0.04 0.02 0

0

2

4

6

8

10

12

14

16

18

20

iterations k

Fig. 6. iterative values of ‖s k‖2 and ε dual for inner iterations.

177

Electrical Power and Energy Systems 102 (2018) 171–178

L. Yang et al.

Table 5 Results of H-ADMM in parallel environment. Case.

IEEE14 IEEE30 IEEE57 IEEE118

1 worker

2 workers

4 workers

Ctime (s)

pspu

ppe

Ctime (s)

pspu

ppe

Ctime (s)

pspu

ppe

0.51 1.35 3.16 6.32

1 1 1 1

100.0% 100.0% 100.0% 100.0%

0.5 0.94 2.03 3.57

1.02 1.44 1.56 1.77

2.0% 30.4% 35.8% 43.5%

0.32 0.80 1.87 2.62

1.59 1.69 1.69 2.41

37.3% 40.7% 40.8% 58.5%

References [18]

[1] Alsac O, Stott B. Optimal load flow with steady-state security. IEEE Trans Power Apparatus Syst 1974;PAS-93(3):745–51. [2] Monticelli A, Pereira MVF, Granville S. Security-constrained optimal power flow with post-contingency corrective rescheduling. IEEE Trans Power Syst 1987;2(1):175–80. [3] Dommel HW, Tinney WF. Optimal power flow solutions. IEEE Trans Power Ap Syst 1968;PAS-87(10):1866–76. [4] Capitanescu F, Ramos JLM, Panciatici P, et al. State-of-the-Art, challenges, and future trends in security-constrained optimal power flow. Electr Power Syst Res 2011;81(8):1731–41. [5] Qiu W, Flueck AJ, Tu F. A new parallel algorithm for security constrained optimal power flow with a nonlinear interior point method. IEEE power engineering society general meeting. 2005. p. 1–7. [6] Zhang R, Dong ZY, Xu Y, et al. Hybrid computation of corrective security-constrained optimal power flow problems. IET Gener Transm Distrib 2014;8(6):995–1006. [7] Platbrood L, Capitanescu F, Merckx C, et al. A generic approach for solving nonlinear-discrete security-constrained optimal power flow problems in large-scale systems. IEEE Trans Power Syst 2014;29(3):1194–203. [8] Capitanescu F, Glavic M, Ernst D, et al. Contingency filtering techniques for preventive security-constrained optimal power flow. IEEE Trans Power Syst 2007;22(4):1690–7. [9] Capitanescu F, Wehenkel L. A new iterative approach to the corrective securityconstrained optimal power flow problem. IEEE Trans Power Syst 2008;23(4):1533–41. [10] Jiang Q, Xu K. A novel iterative contingency filtering approach to corrective security-constrained optimal power flow. IEEE Trans Power Syst 2014;29(3):1099–109. [11] Karoui K, Crisciu H, Szekut A, et al. Large scale security constrained optimal power flow. In: Proc. 16th power syst. computation conf., Glasgow, U.K.; 2008. p. 14–8. [12] Wang Q, Mccalley JD, Zheng T, et al. A computational strategy to solve preventive risk-based security-constrained OPF. IEEE Trans Power Syst 2013;28(2):1666–75. [13] Zhang W, Xu Y, Dong ZY, et al. Robust security-constrained optimal power flow using multiple microgrids for corrective control under uncertainty. IEEE Trans Ind Inf 2017;13(4):1704–13. [14] Phan D, Kalagnanam J. Distributed methods for solving the security-constrained optimal power flow problem. In: Innovative smart grid technologies. IEEE; 2012. p. 1–7. [15] Phan D, Kalagnanam J. Some efficient optimization methods for solving the security-constrained optimal power flow problem. IEEE Trans Power Syst 2014;29(2):863–72. [16] Overbye TJ, Cheng X, Sun Y. A comparison of the AC and DC power flow models for LMP calculations. In: Proceedings of the 37th annual Hawaii international conference on system sciences; 2004. p. 1–9. [17] Marano-Marcolini A, Capitanescu F, Martinez-Ramos JL, et al. Exploiting the use of

[19] [20] [21] [22] [23] [24] [25]

[26]

[27]

[28]

[29]

[30] [31] [32] [33]

[34]

[35] [36]

178

DC SCOPF approximation to improve iterative AC SCOPF algorithms. IEEE Trans Power Syst 2012;27(3):1459–66. Roald L, Oldewurtel F, Parys BV, et al. Security constrained optimal power flow with distributionally robust chance constraints; 2015. Available from: 1508.06061. Lubin M, Dvorkin Y, Backhaus S. A robust approach to chance constrained optimal power flow with renewable generation. IEEE Trans Power Syst 2015;31(5):3840–9. Verbic G, Canizares CA. Probabilistic optimal power flow in electricity markets based on a two-point estimate method. IEEE Trans Power Syst 2006;21(4):1883–93. Ben-Tal A, EI Ghaoui L, et al. Robust optimization. Princeton University Press; 2009. Miranda V, Saraiva JT. Fuzzy modelling of power system optimal load flow. IEEE Trans Power Syst 1992;7(2):843–9. Moore RE, Kearfott RB, et al. Introduction to interval analysis. SIAM; 2009. Capitanescu F, Fliscounakis S, Panciatici P, et al. Cautious operation planning under uncertainties. IEEE Trans Power Syst 2012;27(4):1859–69. Fliscounakis S, Panciatici P, Capitanescu F, et al. Contingency ranking with respect to overloads in very large power systems taking into account uncertainty, preventive, and corrective actions. IEEE Trans Power Syst 2013;28(4):4909–17. Ding T, Hu Y, Bie Z. Multi-stage stochastic programming with nonanticipativity constraints for expansion of combined power and natural gas systems. IEEE Trans Power Syst 2018;33(1):317–28. Wang Q, Mccalley JD, Zheng T, et al. Solving corrective risk-based security-constrained optimal power flow with Lagrangian relaxation and Benders decomposition. Int J Electr Power Energy Syst 2016;75:255–64. Zhang Y, Shen S, Mathieu J. Distributionally robust chance-constrained optimal power flow with uncertain renewables and uncertain reserves provided by loads. IEEE Trans Power Syst 2016:1. Murillo-Sanchez CE, Zimmerman RD, Anderson CL, et al. Secure planning and operations of systems with stochastic sources, energy storage, and active demand. IEEE Trans Smart Grid 2013;4(4):2220–9. Levin VI. Nonlinear optimization under interval uncertainty. Cybernet Syst Anal 1999;35(2):297–306. Carpentier J. Contribution to the economic dispatch problem. Bull Sac France Elect 1962;3(4):836–45. Wang Z, Alvarado FL. Interval arithmetic in power flow analysis. IEEE Trans Power Syst 1992;7(3):1341–9. Zhao C, Wang J, Watson JP, et al. Multi-stage robust unit commitment considering wind and demand response uncertainties. IEEE Trans Power Syst 2013;28(3):2708–17. Boyd S, Parikh N, Chu E, et al. Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Mach Learn 2010;3(1):1–122. Chen G, Li CJ, Dong ZY. Parallel and distributed computation for dynamical economic dispatch. IEEE Trans Smart Grid 2017;8(2):1026–7. Billinton R, Allan RN. Monte Carlo simulation. Reliability evaluation of engineering systems. Springer US; 1992. p. 63–75.