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Procedía Engineering 30 (2012) 1146 - 1154

Procedía Engineering

www.elsevier.com/Iocate/procedia

International Conference on Communication Technology and System Design 2011

Particle swarm optimization based generation maintenance scheduling using probabilistic approach

K.Suresha, N.Kumarappanb , a*

aDepartment of EEE, Sri ManakulaVinayagar Engineering College, Pondicherry 605107, India _bDepartment of Electrical Engineering, FEAT, Annamalai University, Annamalainagar 608002, India_

Abstract

This paper presents maintenance scheduling (MS) of generating units using particle swarm optimization (PSO) based probabilistic levelized risk method. The PSO for the generator maintenance scheduling generates optimal and feasible solution and overcome the limitations of the conventional methods. The maintenance of generators are directly associated with the overall reliability of the power system. The objective function of this paper is to reduce the loss of load probability (LOLP) for a power system. The capacity outage probability table (COPT) is the initial step in creating the maintenance schedule using the probabilistic levelized risk method. Probabilistic techniques are widely used in power system reliability evaluation of long-term planning. Moreover, this paper proposes the PSO to construct the generation model COPT without utilizing the analytical model. A case study for the roy billinton test system (RBTS) and real power system model, Thailand power system shows that the developed algorithm can achieve a substantial levelization in the reliability indices over the planning horizon and demonstrates the effectiveness of the proposed approach.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Amrita University Keywords: Levelized risk method; Particle swarm optimization; Loss of load probability; Capacity outage probability table

1. Introduction

Preventive maintenance scheduling of generating unit is an important requirement which is a challenging task in a power system. It plays a vital role in minimizing the unexpected outages, extending equipment lifetime and extension operation planning. Mathematically, the unit maintenance scheduling problem is a multiple-constraint, non-linear and stochastic optimization problem. However, the MS problem is a constrained optimization problem [1]. Many kinds of intelligence computation methods such as artificial neural network, simulated annealing, expert system, fuzzy systems, tabu search and evolutionary optimization methods have been applied to solve the unit maintenance scheduling problem.

* K. Suresh. Tel.: +91 0413 2222551; fax: +91 0413 2641136. E-mail address: sureshkaliyamoorthy@yahoo.co.in.

1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.01.974

[3-5]. In recent years with rapid development of the evolutionary theory, GA and PSO have become very powerful optimization tool and obtained wide application in power engineering. In a global maintenance scheduling problem, the network constraints were included [4]. Generation adequacy of Mexico's national interconnected power system was analyzed [7]. The PSO algorithm had used to solve the operation planning problem in power system [10]. In power system MS, there are two categories of objective functions are used namely, either deterministic or probabilistic. The methods reported in the literature [35] are considered the deterministic based levelized reserve method. The main drawback of the deterministic approach is that it neglects the randomness of the available generating unit's capacity. The probabilistic reliability objective function removes the above defect by taking into account the random forced outage of the generating units [1-2]. The removal of generators create excessive risk to the power system. This paper emphasizes the PSO algorithm based probabilistic levelized risk method MS which is considered the random outage of generators. Moreover, the PSO is utilized to create the COPT which is the essential step for the probabilistic levelized risk method. Since the analytical method based COPT becomes tedious for the large scale power system problems to find the global optimum solutions for MS problem. Complexity of the MS problem increases dramatically with large number of generating units.

Nomenclature

LOLPi , LOLPj system risks in the maintenance time interval

t maintenance time interval

T number of intervals in the maintenance period p

LOLE p loss of load expectation in the maintenance period p

n number of days in a week

Pi probability of loss of load on day i

C available capacity on day i

Lr forecast peak load on day i

Pi(Ci<Li) probability of loss of load on day i obtained from the COPT

mkt maintenance state where 0 is for no maintenance in sub-interval t and 1 for under

maintenance in the sub interval t

Sk number of intervals for maintenance (the number of weeks or months)

tK starting interval for maintenance (the week or month)

k generating unit

S set of generating units involved in maintenance in the period under examination

Vr maintenance crew

Vrt maximum number of generating units that the maintenance crew can work

Simultaneously in interval t

2. Problem formulation

Probabilistic levelized risk method considers the probabilistic nature of the generating unit and daily peak load variation. The objective of probabilistic levelized risk method is that the risk (LOLP) is made more or less the same throughout the period under study. The principal objective of this method can be equated as

LOLR = LOLPi iet,jet,t= 1,2,3,......,T (1)

LOLEp = Y,Pi (C< < Li ~>days / period (2)

The reliability of the power system can be measured by the annual loss of load expectation (LOLEa) which is calculated as follows

LOLEa=YJLOLEp (3)

subject to the following constraints to obtain the feasible maintenance schedule in power system.

2.1. Time constraint

Generators must be scheduled at the certain intervals. In addition to that, one more consideration is continuity of maintenance activity. Maintenance must be completed in a continuous maintenance intervals once started.

tk+Sk-l

Z '"kt=Sk keS (4)

2.2. Maintenance crew constraint

Normally, two generating units cannot be scheduled for maintenance together in the same power plant and at the same time, i.e. Vrt=1. Only a few power plants with considerable maintenance resources can allow Vrt>1. However, the following constraint must still be met

Tmkt*Vrt <5>

3. Probabilistic levelized risk method

The effective load carrying capacity (Ce) and the equivalent load (Le) are the two significant factors to be calculated in the probabilistic levelized risk method. The COPT is formed; the risk characteristic coefficient 'm ' is computed. The 'm' is defined as the corresponding change of the generating unit's outage capacity in MW when the system's risk or P(X) changes by a factor of e. So

XB (6)

where P(XA) is the probability of outage at point A, P(XB) is the probability of outage at point B, XA is the outage at point A, XB is the outage at point B. Effective load carrying capacity is the actual capacity used for meeting the load demand. The reserve capacity of the system should be increased when a generating unit is added so as to maintain the system's risk at a constant level in power system. Ce = C - m * In(p + q exp(c / m)) (7)

where p is the availability of a machine, q is the unavailability or forced outage rate of a machine, C is the capacity of the generator being added. In the levelized risk method equivalent load is used in place of the maximum load in the maintenance time interval. The equivalent load can be computed as

Le=Lm+ m * /n(£ (exp(Z, -Lm) hn)Tp) (8)

where Lm is the maximum load of the interval under study, Lj is the daily maximum load of the interval under study, Tp is the number of days in an interval.

3.1. Capacity outage probability table

The generation model consists of a table which contain states of capacity unavailable due to outages in the ascending order [2], [7]. Analytical approaches require mathematical modelling to generate the system states which will add more burdens on the process of creating the COPT. The total number of probable outage system capacity outcome (states) of generators which is 2N [6]. The outage state of the generating units in the states array are represented as follows

states array =

staten statev - statew "

state 01 state ■■■state2N

state 2 „j state 2 „j state 2„N ^

The "capacity in service" and the "capacity out of service" of the generating units are calculated for each system state from the system states array using the following equations

state capacity inf (MW) = ^ statejk *capk (10)

state cap outi (MW) =TIC-state capacity ini (11)

where capk is the capacity of unit k, state k is the state of unit k in the system state i, TIC is the total installed capacity in MW, N is the number of generating units. State probability is calculated as follows

state probability=YYprobk (12)

probk = FORk . if stateik=0

=1-FORk if state ik=1 (13)

where probk is the state probability of unit k, FORk is the forced outage rate (FOR) of unit k. The total probability of the collected states "cumulative probability" is calculated as follows

cumulative probabilitym=cumulative probabilitym.I+ ^state probability, (14)

where M is the total number of states at the end of iteration process.

4. Implementation of proposed maintenance scheduling in power system

Step (1): Get the generator data viz the number of generators, their rated capacity and FOR. Step (2): Get the load data for 52 weeks which includes the daily maximum load data. Step (3): Create the COPT using the PSO algorithm.

Step (4): Select cumulative probability values (corresponds to outage capacities of generator) close to

P(Xa)= 0.1 and P(Xb)= 0.0001. Step (5): Compute the value of PSO based m' using Eq. (6). according to the data in the COPT. Step (6): Calculate the values of effective load carrying capacity of each generator using Eq. (7). Step (7): Find the equivalent load using Eq. (8).

Step (8): Take the first generator and calculate the minimum sum of the Ce and Le. Step (9): Schedule the generator by searching the intervals with minimum sum of equivalent load and effective load carrying capacity on the load curve and schedule the generator until the maintenance intervals are exhausted for that particular generator.

Step (10): Repeat steps 8 and 9 for all the generators for MS.

Step (11): If the MS is not feasible within the limits of the generation system, go to step 3.

Otherwise terminate the program.

Step (12): Obtain the reliable and optimal maintenance schedule for the generators.

4.1. Proposed PSO algorithm for outage system state probability and risk characteristic co-efficient

PSO is a population based optimization tool which is motivated from the simulation of the behaviour of social systems such as fish schooling and birds flocking [8]. It is easy to implement and there are few parameters to control which is used to solve the large scale power system problems [10].

4.1.1. Algorithm

Step (1): Choose the input parameters for PSO. Risk characteristic co-efficient, outage state capacity of generators are taken as control variables. Select the COPT of the generators. Set the time counter t=0, create the initial population for 'm' values, all possibilities of system outage capacities of the power system. Generate 'n' particles by randomly selecting a value with uniform probability over the search space between maximum and minimum outage capacities of generator [xmin,xmax]. For example if there are N outage capacities in the COPT, the ith particle of risk characteristic co-efficients are represented as follows: Xmi=( Xm1, Xm2,

Xm3,..............Xm N).

Step (2): Generate random initial velocity of all particles over the search space [vmin,vmax]. Maximum velocity of a particular dimension is given by the equation V k max=(x max-xmin )/Na where xmax, xmin are the maximum and minimum position of the particles , Na is the number of iteration.

Step (3): Evaluate the fitness of each particle according to the objective function using Eq. (6) and (10).Calculate system state capacity out of service using Eq. (11). Calculate the system state probability using Eq.(12) and (13). The cumulative probability of system capacity outage states are calculated using Eq. (14).

Step (4): Set evaluated fitness values as pbest values for each particle, identify gbest value among all pbest value that corresponds to the particle shown by xmi (0)= [xmi t (0), ..., xmi N(0)], where xmi(0) is the initial particle position.

Step (5): Update the time counter t=t+1.Update the velocity using the gbest and the pbest of each particle, the ith particle velocity in the jth dimensions are updated. Based on the updated velocity, each particle changes its position. If a particle violates its position limits in any dimension, set its position at the proper limit.

Step (6): Update the position of the each particle.

Step (7): Evaluate the fitness function for each particle. If the current value is better compared to the previous pbest of the particle, the previous value is replaced by the current value. Otherwise the current value is replaced by the previous value.

Step (8): If stopping criterion is reached terminate the program, the position of particles represented by gbest, the optimal solution. Otherwise, the procedure is repeated from step 5.

5. Numerical results and discussion

In order to verify the usefulness of the proposed method which is tested on RBTS and the real power system model in Thailand have been considered for the case study. The RBTS has 11 generating units, ranged from 5 MW to 40 MW. The system peak load is 185 MW and the total generation is 240 MW. The detailed system data is given in [9]. The PSO based MS using probabilistic levelized risk method is implemented in Matlab 7.0. The corresponding PSO parameters are itermax=1000, n=330, k = 0.7298,c1 =

2.05 and c2 =2.05, Wmax and Wmin are set at 0.9 and 0.2 respectively. COPT has been built using the PSO. The total number of states for the RBTS is 211 (2048) states. The COPT for RBTS is shown in Table 1. The initial population of PSO algorithm for the first iteration which is generated randomly. The PSO search the state space to test out the most possible failure states of the generator and store them in a state array using Eq. (9). The state probabilities of each outage capacity of the units are calculated using Eq. (12). The "capacity in service" and the "capacity out of service" are calculated for each system outage state of the generators recovered in the system state array. Moreover, the unrepeated states are added to a system state array which creates new population for PSO algorithm. The repeated trivial states will be truncated from the COPT. From Table 1 the fitness values for the repeated states (230 MW, 235 MW and 240 MW) are assigned a very small value and appears as zero, it is revealed that the state probability of system outage states 230 MW, 235 MW and 240 MW are identical. It depicts that all of the system outage capacity states of generator are recovered at the end of iterations. However, it is not practical to incorporate all of the outage of generator system states in the COPT for the large scale systems. Iterations are performed until the cumulative probability value reaches 1 and all the significant states with high probabilities are recovered. The cumulative probability approaches 1.0 corresponding to the maximum outage capacity of the generator in the power system.

Table 1. COPT results for RBTS

Outage Cumulative Outage Cumulative Outage Cumulative

capacityfM^) probability capacityfM^) probability capacityfM^) probability

0 1.000000000000000 85 0.000271985961167 170 0.000000001828552

5 0.187140385743580 90 0.000231552205827 175 0.000000000851997

10 0.170718979394965 95 0.000190501650260 180 0.000000000832367

15 0.154047071279746 100 0.000189676471580 185 0.000000000039499

20 0.153711940537938 105 0.000030226637111 190 0.000000000023483

25 0.083353402522989 110 0.000027005512527 195 0.000000000007224

30 0.081932052110152 115 0.000023735251019 200 0.000000000006897

35 0.080489019584235 120 0.000023669513782 205 0.000000000000294

40 0.080460012432953 125 0.000002446752258 210 0.000000000000161

45 0.011190283753061 130 0.000002018016308 215 0.000000000000026

50 0.009790898254492 135 0.000001582740169 220 0.000000000000023

55 0.008370165708533 140 0.000001573990455 225 0.000000000000001

60 0.008341606820807 145 0.000000110586595 230 0.000000000000000

65 0.002513155213010 150 0.000000081023774 235 0.000000000000000

70 0.002395411629758 155 0.000000051009983 240 0.000000000000000

75 0.002275871916808 160 0.000000050406661

80 0.002273468986538 165 0.000000002790435

Most of these states have low probability of occurrence which means that they are unusually to occur. They do not influence significantly on the power system reliability evaluation. COPT results for the Thailand power system is shown graphically in Fig 1. From Fig 1. it is noticed that the cumulative probability of generators are obtained for all outage capacities of generator ranging from 0-4950MW. The cumulative probabilities of the collected states are evaluated. Obviously, the cumulative probability starts with zero; the cumulative probability varies between 0-1. Fig 2.shows the LOLP values for maintenance time interval for case study 2. The effective load carrying capacity is shown in Fig 3. From Fig 3.it is clear that each generator contributes a specific amount of its capacity for the entire system's reserve in power system. Fig 4. shows the equivalent load in the maintenance time interval. The Ce, Le varies with change

in the risk characteristic co-efficient. The obtained optimal 'm' is 10.445 MW by Eq. (6) in 302 iterations using the PSO which cannot normally be analyzed by hand calculations for large scale systems.

Fig 1. Cumulative outage probability for case study 2

Fig 2. LOLP values for RBTS

Fig 3. Effective load carrying capacity for RBTS Fig 4. Equivalent load for RBTS

The calculated LOLEa value is 15.5 days per year using Eq. (2) and (3). It is observed that the expected number of days (approximately 16 days) in a year in which the daily peak load exceeds the available generation (capacity deficiency). The second case study considered the real power system model, Thailand power system [11] in Thailand. Electricity Generating Authority of Thailand (EGAT) is responsible for the long term planning based on the forecasted demand through the year 2025.

Table 2. Maintenance schedule for RBTS

Generating Maintenance Generating Maintenance

unit_intervals_unit_intervals

"1 38-39 7 31-32

2 12-13 8 40-41

3 35-36 9 12-13

4 10-11 10 34-35

5 38-39 11 8-9

6 14-15

Table 3. Maintenance schedule for Thailand power system

Generating unit Maintenance intervals

1 15-18

3 10-13

4 33-36

5 42-45

6 19-21

7 38-41

The total installed capacity of the generation is 4950 MW. The forecasted peak demand for the year 2015 is 20981 MW. The LOLE value obtained is 23.5643 hours per year. The obtained optimal maintenance schedule for the RBTS and Thailand power system are given in Table 2 and Table 3 respectively. None of the generating units are maintained in the maintenance time interval 50-52 for both the case studies. Since the obtained LOLP (risk) values and equivalent loads are maximum in that particular intervals. The risk gradually increases when the load is increased. It is found that all the generators are continuously scheduled in the specified maintenance time intervals which satisfy the maintenance constraints according to Eq. (4) and (5). The obtained results indicated that the leveling of risk (more or less same) is achieved over the planning horizon incorporating the uncertainties in power system. The optimal MS is obtained through minimization of LOLE, which maximizes (reserve capacity) the overall reliability of the power system. The proposed probabilistic MS model effectively incorporates the stochastic nature of generating units and explores the solution from the population of points not from a single point.

6. Conclusion

In this paper the probabilistic levelized risk method has presented in conjunction with the PSO technique to determine the risk characteristic co-efficient. Moreover, this paper uses the exploitation of PSO to explicitly create the COPT which is the fundamental prerequisite of power system reliability assessment. The proposed method is validated for RBTS and Thailand power system to create the power system generation model which is used to decide how much generation adequacy is needed to assure reliability level to meet the future energy demand. The PSO starts with the feasible solutions generated based on heuristics by avoiding the awkward computational effort required by the conventional methods with increase in variables for large scale systems. It is envisaged that the proposed algorithm is suitable for the practical power system in capacity generation planning and its potential to solve the MS problem while considering the random failures in the maintenance model for power system utilities.

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