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Procedia Computer Science 6 (2011) 435-440

Complex Adaptive Systems, Volume 1 Cihan H. Dagli, Editor in Chief Conference Organized by Missouri University of Science and Technology

2011-Chicago, IL

Modified SPEA2 for Probabilistic Reliability Assessment

in Smart Grids

Hiroyuki Mori*, Hiroki Kakuta

Meiji University Tama-ku, Kawasaki, 214-8571, Japan

This paper proposes a new method for probabilistic reliability evaluation with multi-objective meta-heuristics (MOMH) in smart grids. Recently, the smart grids increase the degree of the uncertainties due to the renewable energy. The importance of the probabilistic reliability assessment of electric power systems has been recognized. Probabilistic reliability evaluation has been broadly used for power system operation and planning due to the capability of considering the various system uncertainties. This paper proposes a new method for probabilistic reliability evaluation with MOMH. It is useful for evaluating a set of the Pareto solutions systematically. As MOMH, SPEA2 (Strength Pareto Evolutionally Algorithm 2) is used to evaluate the Pareto solution due to the good performance of solution diversity which is quite different from the conventional multi-objective reliability evaluation. The effectiveness of the proposed method is successfully demonstrated to the IEEE reliability Test System (IEEE-RTS).

© 2011 Published by Elsevier B.V.

Keywords: Power System Reliability; Optimization; Optimization Methods; Pareto Optimization; Probabilistic Logic

1. Introduction

This paper proposes a multi-objective meta-heuristics method for power system reliability evaluation. A lot of players have joined power markets to maximize the profit through purchasing and selling electricity. As a result, smart grids become more competitive and deregulated. Since there are a lot of players that provide the service of electricity, the degree of the uncertainties increases under new circumstance due to incomplete information provided by unknown renewable energy, weather, economic social conditions, etc. Therefore, it is important to deal with power supply reliability efficiently. As the conventional deterministic methods on power system reliability assessment, N-1 and N-2 contingency analysis have been well-spread. However, they have a drawback that it does not reflect the probability of fault scenarios and energy not supplied quantitatively [1]. Under uncertain smart grids,

* Corresponding author. Tel.: +81-44-9347353; fax: +81-44-9347909; E-mail address: hmori@isc.meiji.ac.jp

1877-0509 © 2011 Published by Elsevier Ltd. doi:10.1016/j.procs.2011.08.082

it is necessary to focus on probabilistic reliability assessment so that planners understand power supply reliability appropriately. Looking over the conventional methods, they may be classified as follows:

a) State Enumeration Method [2]

b) Monte-Carlo Simulation (MCS) [3]-[6]

c) Meta-heuristics (MH) [7]-[12]

Method a) is a technique that successively lists up potential fault scenarios. It has a drawback that it is very time-consuming in large-scaled systems. Method b) is a technique that repeats the process of data sampling while the reliability index reaches at the specified accuracy. Although the method is applicable to large-scaled or complex power systems, it is not practical in a sense that it needs a lot of computation time. Method c) makes use of meta-heuristics (MH) that repeatedly employs simple rules or heuristics to obtain highly approximate solutions to a global minimum. In fact, MH gives better solutions within given time in comparison with other optimization methods. Since the method has advantage that it easily evaluates a set of fault scenarios that affect power system conditions significantly. Therefore, this paper focuses on Method c). As the typical MH on Probabilistic reliability evaluation, the following methods are given:

1) GA-based Method [7][8]

2) PSO-based Method [9][11]

3) EPSO-based Method [12]

GA (Genetic Algorithm) stems from the natural selection of biology and uses genetic operators such as crossover, mutation, reproduction etc. to evaluate better solutions. Ref. [7] revealed that Method 1) was better than MCS. Regarding Method 2), PSO (Particle Swarm Optimization) is one of swarm intelligence techniques that make use of multi-agent optimizers to find out better solutions. Ref. [11] pointed out that the PSO-based method outperformed the GA-based method. Method 3) is an extension of Method 2) in a way that PSO is replaced with EPSO (Evolutionally Particle Swarm Optimization). EPSO is an efficient technique that combines PSO with the evolutionary strategy to construct adaptive PSO. Ref. [12] reported that the EPSO-based method is superior to the PSO-based one. They have drawbacks that they require prior knowledge on each objective function and disregard the existence of the Pareto solutions. Recently, MOMH (Multi-objective Meta-heuristics) has been developed to focus on evaluating a set of the Pareto solutions systematically. Since it is based on GA to create a set of solutions, it has a drawback that it gets stuck in local minimum in the search process. The proposed method makes use of the solution diversity strategy with sharing function to overcome the drawback. The effectiveness of the proposed method is successfully demonstrated in the IEEE 24-bus system.

2. Probabilistic Reliability Evaluation

Background — Power System reliability assessment is outlined in this paragraph. The power supply reliability implies the capability of the system to supply power energy without blackout. Power system reliability assessment consists of two basic aspects of adequacy and security. The former is related to static reliability in power system planning while the latter is concerned with dynamic reliability in power system operation. In this paper, adequacy that deals with network planning is discussed. The assessment of the adequacy is divided into three hierarchical levels (HL) [3]. This paper deals with HL II that considers the probabilistic behaviors of the generators and transmission lines to evaluate the adequacy in the transmission network. The problem to be solved may be classified into deterministic and probabilistic approaches. As deterministic one, N-1 and N-2 contingency criteria have been used because of the easiness of the implementation. However, they are not satisfactorily useful in a sense that they do not reflect the probabilistic characteristics of component forced outages. To overcome the drawback of deterministic approaches, the probabilistic one becomes more practical. This paper employs the probabilistic reliability evaluation to consider more realistic power supply reliability.

Problem Formulation — This section outlines the problem formulation of probabilistic reliability evaluation [6]. The formulation may be divided into the state enumeration method and MCS. The former is a method that lists up

potential fault scenarios to evaluate the reliability indices analytically. In general, it is very time-consuming in large scaled systems. The latter is one of the most popular methods that repeat the process of data sampling while the reliability index reaches at the specified accuracy. It has advantage to set complicated system conditions easily. However, due to its dependence on the number of sampling, it requires more computational time in evaluating the more accurate reliability indices. This paper employs the former that is capable of reducing computational time in sampling more critical system states if meta-heuristics is used. It should be noted that critical states deteriorate reliability indices such as high occurrence probability states and high load curtailed states. The problem formulation results in the minimization problem. The use of meta-heuristics contributes to the reduction of computational time in finding out more critical states. The mathematical formulation for finding out system state may be expressed as follows:

Objective function:

v = PSiLi ^ max Constraints:

Sf + g + r = d |f| * brpt 0 < St < n

0 < g < g 0 < r < d

where,

PSj=n GUT.

1 - FOR FOR 1 - PTt PT

if S ,

if St= 1

if St = 0

if St = 1

(8) (9)

PSj: state probability of sample j

G: state probability of generator i

T : state probability of transmission line i

ng: total number of generators

nt: total number of transmission lines

FOR: the failure probability of generator i

PTthe failure probability of transmission line i

S,: i-th element of the state

One system state consists of N generators in a system with GNg generator buses and LNl transmission lines. Now, suppose that a system state is sampled and it is examined whether it is a failure or not. If it is a failure state, the amount of the load curtailment in the system is determined by the power flow calculations. Probabilistic reliability evaluation needs a lot of the power flow calculations to minimize the energy not supplied. To reduce the computational effort, the DC power flow calculation is employed due to the numerical efficiency. The formulation for minimizing curtailed energy may be written as the following linear programming (LP) problem:

Objective function:

w=T/i ^ m

Constraints:

Sf + g + r = d

V\ ± brPi 0 < g < g 0 < r < d 9: unbounded

(11) (12)

The above problem is solved by the dual simplex method [18]. Eqn. (11) gives the DC power flow equation and Eqn. (12) denotes the constraints on the line flow limitation at each line. Eqn. (13) provides the upper and the lower bounds of generator output. Eqn. (14) denotes the lower and the upper bounds of the dummy generator output that contributes to the rescheduling of generators. This paper uses (10) to minimize the energy not supplied in sampled state. The reliability indices are calculated by finding out more critical system states. The index may be written:

F = x p,F,

where:

f : estimated value of reliability index Pi: probability of failure state i Fi: evaluated value in system state i ^ : set of all possible state

Probabilistic Reliability Indices — This section describes the indices for evaluating probabilistic power supply reliability. Unlike the deterministic reliability evaluation, probabilistic reliability evaluation calculates the degree of the reliability quantitatively. So for, a lot of studies on the reliability indices have been done [2]. Probabilistic reliability assessment evaluates the probability, the duration, and the frequency of the load curtailment, etc. In this paper, EENS (Expected Energy Not Supplied) is employed to focus on power supply reliability due to the capability of handling with the probability and the amount of the load curtailment.

3. Proposed Method

This paper proposes multi-objective meta-heuristics (MOMH) for Probabilistic Reliability Evaluation in consideration of the solution diversity. It is important to carry out probabilistic reliability evaluation and find out system states that deteriorate reliability indices. The conventional methods such as GA, PSO and EPSO have several drawbacks that they require prior knowledge on each objective function and disregard the existence of the Pareto solutions. Recently, MOMH has been developed to focus on evaluating a set of the Pareto solutions systematically. It is based on GA to create a set of solutions. However, GA has a drawback that it gets stuck in local minimum in the end of the search. The proposed method makes use of SPEA2 of MOMH with the solution diversity strategy to employ sharing function to overcome the drawback. As the objective function in the search process, this paper uses the state probability and energy not supplied explained in Section 2 [12]. The algorithm of the proposed method may be described as follows:

Step 1: Set the initial conditions.

Step 2: Calculate the fitness of each solution and carry out environmental selection. Step 3: Go to Step 5 if the termination conditions are satisfied, otherwise go to Step 4. Step 4: Apply the genetic operation to current solution set and construct new solution set. Step 5: Calculate reliability indices with the obtained system states.

4. Simulation

Simulation conditions — The proposed method is applied to the IEEE Reliability Test System (IEEE RTS). It has 41 transmission lines and the 32 generators. Therefore, the number of system states is 9.4x1021. The outage rates of the components follow data of IEEE RTS. This paper evaluates EENS as the probabilistic reliability index. The proposed method is compared with the conventional methods. For convenience, the following methods are defined.

Method A: MCS Method B: SPEA2

Method C: SPEA2 with solution diversity strategy (proposed method) Table 1 shows the parameters of each method that were determined by the preliminary simulation. As a selection

strategy in Methods B and C, this paper makes use of tournament strategy. All computations were performed on UNIX Server Fujitsu PRIMEPOWER 1500 (SPARC 64V, 8CPU, 1.89GHz, SPEC int 2000: 108, SPEC fp 2000: 126).

Simulation Results — Fig. 1 shows the convergence characteristics of each method. It can be seen that reliability index EENS of Method A converges to a certain value as the computational time increases sufficiently. That is because it needs a lot of iterations to realize more critical system states due to the low occurrence probability in MCS. On the other hand, Methods B and C evaluate high accurate reliability index quickly. The proposed method calculate reliability index more efficiently than Method B. Method C required 13% computational time of Method B.

Table 1 Parameters of Each Method

Methods Parameters

A No. of Maximum Sampling 100,000

Coefficient of Variation 0.01

No. of Population 500

B No. of Generation 1,000

Crossover Rate 1

Mutation Rate 0.01

No. of Population 500

No. of Generation 1,000

C Crossover Rate 1

Mutation Rate 0.01

Taboo Radius 1.00E-08

(1 'X)? 'J.(XI1 (1 00} 0.0«; OOOl 0 - McOiodA ...... V.1, ,1 N/V /w—^ —dl""<: , > j f-1 r 1-:--^L.-L-i-l--——-

r-...........................................

« I« m su KO 1^0 nu K.U iau zoo ::<:■ 2iu

Computation Time | f?

Fig. 1 Convergence Characteristics of Each Method

Method A Method B Method C

Fig. 2 Computational Time of Each Method

5. Conclusion

This paper has proposed a new method for probabilistic reliability evaluation with multi-objective meta-heuristics (MOMH) in consideration of the solution diversity. It is important to find out more critical system state that deteriorates reliability indices. The proposed MH-based method is superior to MCS in finding out more critical states in terms of efficiency although the conventional MH-based methods do not provide better results due to the drawbacks that they require prior knowledge on each objective function and disregard the existence of the Pareto solutions. Since the conventional MOMH methods are based on GA to create a set of solutions, they have a drawback to get stuck in local minimum in the end of the search. The proposed method makes use of the solution diversity strategy with the sharing function to overcome the drawback. The effectiveness of the proposed method was successfully demonstrated in the IEEE 24-bus system. The simulation results have shown that the proposed method required less computational effort while satisfying the accuracy of the solution in comparison with the conventional methods. Therefore, the proposed method allows system planners to realize more efficient evaluation of probabilistic reliability appropriately.

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