Pergamon Computers Elect. Engng Vol. 22, No. 5, pp. 343-351, 1996

Copyright 0 1996 Elsevier Science Ltd

PII: SOO45-7906(96)000@2 Printed in Great Britain. All rights reserved

0045-7906/96 SI5.00 + 0.00

A NEW APPROACH TO EVALUATE RELIABILITY AND PRODUCTION COST OF A POWER GENERATING SYSTEM

A. K. AZAD and R. B. MISRA Department of Electrical Engineering, IIT, Kharagpur -721 302, India

Abstract-This paper describes a computationaly efficient approach to evaluate the production cost (PC), expected served energy (ES), expected unserved energy (USE) and loss of load probability (LOLP) of a power generating system consisting of different types and sixes of unit. This approach uses hourly loads or any suitable time interval for system demand for a given period. The generating units are convolved with the sampled demand in an economic order of merit. The joint probability density function (JPDF) concept is utilized for the convolution process of hourly system loads and generating unit outages and the binomial equation has been successfully implemented in the reduction of computations. The proposed approach can simulate multistate representation of generating units as well. The proposed approach is applied to a modified IEEE Reliability Test System (IEEE-RTS). The PC, ES, USE and LOLP of the system are evaluated and the proposed approach is found efficient exact and easy to implement as compared to other existing methods. Copyright 0 1996 Elsevier Science Ltd

Key words: Power system planning, reliability, LOLP, production costing, unserved energy and expected generation.

1. INTRODUCTION

The prediction of the generation production cost and system reliability are important aspects of generation system planning. This paper deals with the evaluations of expected energy generation, operating or fuel cost, expected unserved energy and loss of load probability in a power generating system.

The LOLP is obtained as probability that the equivalent load equals or exceeds the installed capacity in the system. A number of papers [l-l l] have contributed to the advancement and improvement of probabilistic techniques in the area of reliability evaluation and production costing of power generating systems.

The most important development for the evaluation of production costing was suggested by Booth [l] using the probabilistic simulation. This method evaluates the expected energy generation of each unit and the probability that the available generating capacity will be insufficient to meet the demand or LOLP. The method takes into consideration the random failures of generating units in terms of the Forced Outage Rate (FOR). The fuel costs are evaluated from the knowledge of the expected energy generation of all units.

For the economic operation of a power system the usually accepted dispatch strategy for minimum cost is one that commits units in order of increasing average marginal or incremental costs known as merit order of loading. In probabilistic simulation the units are committed in their merit order to meet the demand. As units are committed or simulated by a convolution process which gives rise to an equivalent load curve and takes into account the effect of random failures of the committed units. The equivalent load may be viewed as an augmented load caused by the random outages of the generating unit.

In this paper, the unit capacities are discretized by a common factor. The zeroth and first moment of the hourly loads are calculated and named as the random variables (RVs) in the range of unit capacity discretization. The LOLP and expected unserved demand are evaluated from zeroth and first moment of the equivalent load after committing the generating unit outages with loads. This method uses the hourly loads and the frequency distribution of demand thus obviating the use of the load duration curve (LDC). In addition the numerical errors in calculating the area under the LDC [4] are also avoided. The new approach avoids the inherent errors present in the evaluation of unserved demand and LOLP when using a numerical convolution formula Gram-Charlier expansion [3,4] and Fourier method [lo]. The method is capable of simulating the

343

344 A. K. Azad and R. B. Misra

concept of the multistate representation of generating units. Therefore the approach proposed in this paper is an efficient method to evaluate the production cost expected energy generation unserved energy and LOLP in which the approximations of the method of moments [3-61 are removed.

The results obtained for IEEE-RTS [ 121 and other practical existing systems are compared with those obtained using existing methods. The proposed approach is found to be efficient exact and simple.

2. PROPOSED APPROACH

The probability density function (PDF) of hourly system demands is considered in this approach. This method is based on the zeroth and first moment of hourly system demand. Before committing any generating unit outages the zeroth and first moment of demand are evaluated from the hourly loads of the system. The sum of the first moments is the expected demand of the system. The LOLP and expected unserved energy are evaluated from the equivalent load moments after committing all the generating units in the merit order of loading for economic operation. The mathematical relationship to evaluate LOLP expected unserved energy expected served energy and production costing are given. The mathematical model of proposed convolution process is also described in the following sections.

2.1. Loss of load probablity

Consider a power system consisting of n generating units such that the installed capacity of ith generating unit is ci and its ( equivalent ) forced outage rate or unavailability is q1 where i = 1,2,. .n. Let X, be an independent discrete random variable (RV) that represents unavailable capacity or the outage capacity of unit i with a distribution of

Xi = c, with probability = q,

= 0 with probability = 1 - qi.

Let L denote the system peak load. Then the loss of load probability (LOLP) index is measured

by

LOLP = Pr(X, + X, + . + X, > c, + c2 + . . . + c, - L) (1)

In case of a random load L, having m points, i = 1,2,...m the

LOLP,=Pr(X,+X,+ . . . +X,>c,+c,+ . ..c.-L,). (2)

Hence,

LOLP = (l/m)*(LOLP, + LOLP, + . . + LOLP,) . (3)

Therefore in general,

s IC + P,

LOLP = f(X) dX (4) IC

where, f(X) is the equivalent load PDF IC = c, + c* + . . . + c, = installed capacity P, = peak load of the system.

2.2. Expected energy generation

The evaluation of unserved energy is essential to evaluate the expected energy generation of a particular generating unit. The expected energy generation by a given generating unit is obtained by evaluating the difference of unserved energies before and after the commitment of that generating unit. Initially, the system demand is sampled in terms of zeroth and first moment before calculating any unserved energy. The procedure of evaluating the expected unserved energy, energy generation, LOLP are as follows.

Reliability and production cost of a power generating system 345

Consider that (k - 1) generating units have been committed. At this stage the expected unserved energy of the system before committing the kth generating unit is given by

USEk_ = T&m(X) - Cfii(x)) (5) i=l

l=CJAC,N= iC,IAC+ 1andC,= &Zk (6) k=l k-L

where, M&I’), m,i(X)Ck = zeroeth & first moment of ith block of RV X (equivalent load) C, = total capacity of committed generating unit n = total number of generating units n, = total number of committed generating units N = total number of discrete blocks AC = step size of discretization T = time period under consideration.

After the commitment of the kth generating unit the unserved energy USE,, may be evaluated using equation (5), where the parameter of the kth generating unit must be included. Then the expected energy generation by the kth generating unit is given by

ESk = USEk_ - USE, . (7)

The total expected energy generation is given by

TES= iES,. k=l

(8)

The production cost of the kth generating unit is given by

PC&. = &‘Es, (9)

where & = incremental COSt Of the Unit. The total production cost is then as follows

(10)

The expected unserved energy and energy balance (EBE) and LOLP after committing all the generating units of the system are given by

USE = m,,(X) - TCAP*m&X) (11)

EBE = USE, _ - (TES + USE) (12)

LOLP = Wz&x) (13)

where total capacity of the system generation TCAP = k&Ck.

2.3. Convolution procedure

The equivalent load concept [l] is based on the convolution of PDF of generating unit outages and that of system demand. The proposed approach used for evaluating the unserved energy is an extension of Azad et al. [ 131. Considerf,(l) andj@) represents the PDF of generating unit outage and that of system demand, respectively. The PDF of equivalent load, f&J is obtained by convolving the PDF of each generating unit outage and that of system demand, i.e.

ML,) = f,(W#) (14)

where ‘*’ denotes the convolution.

346 A. K. Azad and R. B. Misra

The system LOLP is expressed as

LOLP = Pr{L, > IC}

where installed capacity IC = cl + c2 + . . . + c,.

(15)

Let X is a discrete random variable (RV) of system load which represents n values in ascending sequence (X, ,X2,X,, . . . , X,) at equal interval with the corresponding zeroth moment or PDF of mo,(Xl),mo2(X2),mo~(X,) ,..., m&X,) and first moment of mll(Xl)m12(X2)ml,(X3) ,..., m,,(X,,). The another discrete RV Y represents unit outage having m values in ascending sequence ( YI, Y,, Y+.., Y,,,) at equal interval with the corresponding zeroth moment or pdf of mo,( Y,),m,,( Y2),m,,( Y&..,m,,( Y,) and first moment of m,,( Y,)m,,( Y,),m,,( Y&..,m,,( Y,,,). Then the convolution of these two discrete RVs, + Z = X + Y can be performed as follows [13]:

X= iX,andm,,= r=,

,~,%i(X) = 1, MIX= ,~,mli(X)

Y= iY,andm,,= ,=I

jt,moj(q) = 1, ml, = ,$,mV(Y,) .

Then,

n+m n+FTl “+lFi

Z = c Zk and moz = 1 mok(Zk) = 1, mz = c m&2

(16)

(17)

(18) k-l k-l k=l

where, Z, = Xi -t Y, and mOk(Zk) = x,,+ Y,=z,modXi)'moj(~i),

mlk(zk) = C mo,( Ww(X> + m~j(Yj)*moi(Xi)l. r,+ y,=q

Using the above approach, hourly loads of a power system are convolved with the generator capacity outages to calculate the LOLP and unserved energy of that system. The LOLP value can easily be found by examining the (IC + 1)th step of equivalent load. The unserved energy expected energy generation and production costs are evaluated using equations (5~( 12). Since all probability values for demands less than base load are zero the number of steps of equivalent load are reduced to (IC - base load + 2) and this reduces the computational complexity. Another way of reducing the volume of computation is to convolve all identical units in a pulse train. It may be achieved as follows [14]:

Let the ith group of identical units have ni generating units and capacity of ci MW each. Let q, be the outage probability then the convolved distribution will have nonzero probabilities for MW values 0, ci,2c ,,..., (ni - 1) c, an n,c,, and corresponding probabilities are given by: d

Pr(X = kciMW) = ()(qi)k(l - qJ’- k (19)

where, () = ni!/[k!(ni - k)!] and k = 0,1,2 ,.., n,. Another way to reduce computation, i.e. improve speed is to increase largest common factor.

It is possible when all the generating capacity is an integral multiple of any real number. The following example clarifies the new approach to evaluate the LOLP production cost expected

energy generation and expected unserved energy of a power system.

3. EXAMPLE

The outage function of a unit is represented by the probabilities of its having two or more capacity outage states and consists of one impulse at the origin and one or more impulses at different values of capacity on the x-axis as shown in Fig. l(a). The load PDF consists of impulses at different load points on the x-axis as shown in Fig. l(b).

The computational steps of the proposed approach are described with the help of an example of a power system in what follows.

Reliability and production cost of a power generating system 341

Cal r ‘ip’ (1111 Tg’ i’ T41

Cl L, L3 L4

Fig. 1. (a) Outage PDF of ith unit. (b) PDF of load.

Consider hourly loads of a small power system as shown in Table 1 and the generating units data given in Table 2. Let the step size of demand and generating unit capacities be 5 MW and 10 MW, respectively, using their largest common factor. PDFs of all generating unit outages of the system are convolved with load moments of the system to obtain the moments of equivalent load. The capacity outages size in the evaluation of the equivalent load as in the previous section is (IC - base load + 2) MW.

The sampling of load in terms of zeroth and first moment is shown in Fig. 2(a). With a base load of 15 MW and 5 MW step the load moments of Fig. 2(a) can also be represented by Fig. 2(b). The unserved energy before committing any generating unit is evaluated using equation (5) as

USE, _ = 4.(15 + 20 + 25 + 30)/4 = 90 MWh . (20)

3.1. Convolution of first generator group

The capacity of first generator is 10 MW forced outage rate is 0.2 and number of unit is two. That is cI = 10 MW, n, = 2 and FOR = 0.2. Now using equation (19) for identical generator group the generator outage PDF of first generator group is evaluated as shown in Fig. 3.

The discrete RV, X of Fig. 2(b) and discrete RV, Y of Fig. 4 are convolved using the equations (16H18) to generate the moments of an equivalent load as shown in Fig. 4. The equivalent load moments of Fig. 4 can also be represented as new values of moments at a 5 MW step as shown in Fig. 5.

The unserved energy after committing the first generator group is evaluated using equation (5) from the equivalent load as shown in Fig. 5.

USE, = 4 x [(24 + 28.8 + 12.6 + 14.4 + 1.8 + 2.0)/4 - 20

x (0.96 + 0.96 + 0.36 + 0.36 + 0.04 -t 0.04)/4] = 29.2 MWh. (21)

The expected energy generation and production costs of the first equations (8) (10) is

ES , = 90 - 29.2 = 60.8 MWh

PC, = 60.8 x 6.0 = 364.8 US%

3.2. Convolution of second generator

generator group evaluated using

The second generator of capacity cZ = 30 MW and FOR = 0.1 is a single unit therefore the generator outage PDF of second generator is having only two values, i.e. m&Y = 0 MW) = 0.9, m&Y = 30 MW) = 0.1, m,,(Y = 0 MW) = 0 MW and mlz(Y = 30 MW) = 30 MW. The moments

No. of units

2 1

Table 1. Hourly load data

Hour System load (MW)

1 I5 2 20 3 30 4 25

Table 2. Generation data

Small system generation data Cap (MW) FOR Av. inc. cost 1, (S/MWh)

10 0.2 6.0 30 0.1 9.0

348 A. K. Azad and R. B. Misra

(4 (b) X

:“/4 20 2s 30 x 1s 20 2s 30

mo, l/4 l/4 l/4

ml, 15/4 20/4 25/4 3014

Fig. 2. (a) Moments of load. (b) Moments of load (values to be divided by 4).

m,, Fig. 3. Outage moments of first generator group.

Y moY mlY 0

0 . 64

0

10

0. 32

10

20

0 .04

20

15 20 2s 30

0.64 0 .64 0.64 0.64

9.6 12.8 16 19.2

2s 30 3s 40

0.32 0.32 0 .32 0.32

8. 9.6 11.2 12.8

3s 40 45 SO

0.04 0.04 0.04 0.04

1.4 1.6 1.8 2.

Fig. 4. Equivalent load boxes after convolving system load with first unit group (values to be divided by 4).

of second generator are again convolved with equivalent load of Fig. 5 similar to that of the first generator group. The moments beyond the installed capacity are accumulated in the last block, i.e. 55 MW. Fig. 6 shows the equivalent load moments convolving all generating units.

z 15 20 25 30 35 40 4s 50

moz 0.64 0.64 0.96 0.96 0.36 0.36 0.04 0.04

m,, 9.6 12.8 24 28.8 12.6 14.4 1.8 2.0

Fig. 5. Equivalent load moments (values to be divided by 4).

Z 1s 20 25 30 35 40 4s 50 5s

moz 0.576 0.576 0.864 0.864 0.324 0.324 0.1 0.1 0.272

mIz 8.64 11.52 21.60 25.92 11.34 12.96 4.5 5.0 16.52

Fig. 6. Equivalent load moments after convolving all units (values to be divided by 4).

Reliability and production cost of a power generating system 349

Type of unit

Gil

Unit sixe (MW)

10

Table 3. A prototype generation [12]

Number of unit Forced outage rate (FOR)

5 0.02

Avg. inc. cost L (S/MWh)

25.R75 Oil 20 4 0.10 37.500

Hydro 50 6 0.01 0.0 Coal 80 4 0.02 13.494 Oil 100 3 0.04 20.853

Coal 150 4 0.04 10.704 Oil 200 3 0.05 20.730

Coal 350 1 0.08 10.883 Nuclear 400 2 0.12 5.450

Total 3400 MW 32 units.

Table 4. Expeacd energy generation and production costs

Type of unit Unit cap (MW) No. of unit Expected energy generation (GWh) Production cost (MS)

Hydro 50 6 648.- O.OOOO@IO Nl&ar 400 2 1537.536OOOOQ 8.59790131

Coal 150 4 1124.785036321 2.55260101 Coal 350 1 417.25453335 4.75670168 Coal 80 4 270. IS%768 1 4.02051631 oil 200 3 153.20737545 3.04423055 Oil 100 3 9.96805652 0.20015858 Oil 10 5 0.56621445 0.01616995 Gil 20 4 0.56096771 0.02103629

Step size (MW) LOLP (%)

Table 5. Results for different step sire

Unserved energy (GWh) Exacted enerw (GWh) Production costs (MS)

1 0.280078 0.794966 4162.68534 33.20924 5 0.280078 0.794449 4162.68586 33.20931 10 0.280078 0.792987 4162.68772 33.20879

The unserved energy after committing all the generating units is evaluated using equation (5) from the equivalent load as shown in Fig. 6.

USE* = 4.(16.32/4 - 50. x 0.27214) = 2.92 MWh . (22)

The expected energy generation and production costs of the last generating unit evaluated using equations (8)-(10) are

ES2 = 29.20 - 2.92 = 26.28 MWh

PC, = 26.28 x 9.0 = 236.52 US%.

The total expected energy generation, production cost, unserved energy, energy balance and LOLP of the system can be evaluated using equations (8)-(15) as

TES = 60.8 + 26.28 = 87.08 MWh

TPC = 364.8 + 236.52 = 601.32 US$

USE = 2.92 MWh

EBE = 90. - (87.08 + 2.92) = 0.0 MWh

LOLP = Pr(L, > IC) = 0.272/4 = 0.0680.

4. NUMERICAL EVALUATION

This section applies the proposed approach explained in the preceding sections to a prototype generating system provided by the Reliability Test System Task Force of the IEEE Power Engineering Application of Probabilistic Methods Subcommittee [12]. This system is slightly modified with an installed capacity of 3400 MW. Table 3 gives the generation mix of the 32 units comprising the system their installed capacities and FORs. For this system the 13 winter weeks ( 48-52 and l-8 weeks ) hourly loads data of IEEE-RTS [12] is used as a load model of 2850 MW peak load 1102 MW base load and 2184 h time duration.

The evaluated expected energy generation and production cost of each type of generating unit based on a numerical integration with a step of 5 MW are given in Table 4. The loading order

350 A. K. Azad and R. B. Misra

@Segmentation method

0 1 5 10

Step Site I MW)

Fig. 7. Comparison of CPU time for the given system

as specified in this table is obtained from the average incremental cost (A) of each type of generating unit.

The same values of expected energy generation and production costs shown in the Table 4 are also observed from the existing accepted segmentation method with the similar type of generation data and step size. In the proposed method, the process of load sampling with a lower value of step size avoids the sampling error. The capacity of the generating unit is considered as a step size of that type of unit at the time of committing with demands. Finally, the step size of load sampling is considered as the basic step size of the proposed approach.

10.0

0 1 5 10

Step Size (HW)

Fig. 8. Comparison of CPU time for a system with 170 x 20 MW (0.1).

- z 3 Q Proposed method

r

r’ ‘F 2

2 v 1

0 1 S 10

Step Size I MW) Fig. 9. Comparison of CPU time for a system with 68 x 50 MW (0.1).

Reliability and production cost of a power generating system 351

The global expected energy generation production cost unserved energies and LOLP with

different step size of the system are given in Table 5. The CPU time for different methods with different step size obtained using CYBER mainframe 180/840A is presented in Fig. 7.

Though the evaluated results from both the methods are similar but the CPU time obtained is different as shown in Fig. 7. Table 5 and Fig. 7 prove the computational efficiency and accuracy of the proposed method. This approach may again be considered exact because of its zero energy balance. Note that the expected energy demand for this system is 4163.48031 GWh.

Figure 8 shows the CPU time comparison of different methods for 170 units of capacity 20 MW at FOR = 0.1 and Fig. 9 shows the same for 68 units of capacity 50 MW at FOR = 0.1. This approach, therefore, specially handles the identical generating unit efficiently. The time required for committing these type of generating units requires further less time if more machines of the particular type were included in the system. Similarly when multistate generating units are considered, the computational time requirement of proposed method is much less as compared to existing methods.

5. CONCLUSION

The paper has presented an effective approach to evaluate the expected energy generation, expected unserved energy, production costs and Loss of Load Probability for a power generating system. In this approach, no explicit approximation has been made apart from that arising from the discritization of continuous function which is necessary for any digital computer application. This approach can simulate multistate representations of generating units. The new simulation approach reduces the computations in the convolution of a group of identical generating units. This further reduces thousands of load impulses to a few discrete values making the convolution process numerically simpler. The approach is robust and not restricted to LDCs and unit outage density function of any shapes or size of the systems with a large number of generation units. Multiple generating units with same outage behavior can be committed with system demand efficiently. The method is efficient and easy to apply in comparison with accepted existing methods with the similar data. The accuracy of the method has been illustrated using an example. The proposed approach therefore provides an accurate and efficient technique for the evaluation of the expected energy generation, expected unserved energy, production costs and LOLP.

REFERENCES 1. 2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12. 13.

14.

Booth, R. R., Power system simulation model based on probability analysis. IEEE Transactions, 1972, PAS-91,62-69. Zahavi. J., Verdi, J. and Avi-Itzhak, B., Operating cost calculation of an electric power generating system under increm&tal loading procedure. IEEE Trans&rions,-1972, PAS-91, 2114-2124. Rau, N. S., Toy, P. and Schenk, K. F., Expected energy production costs by method of moments. IEEE Transactions, 1980, PAS-99, 1908-1917. Streiel, J. P., Jenkins, R. J., Babb, R. A. and Bayless, W. D., Production costing using the cumulant method of representing the equivalent load curve. IEEE Transactions, 1980, 99, 1947-1956. Stremel, J. P., Sensitivity study of the cumulant method in calculating generation system reliability. IEEE Transactions, 1981, PASlOO, 771-778. Levy, D. J. and Kahn, E. P., Accuracy of the Edgeworth approximation for LOLP calculations in small power systems. IEEE Trunsucrions, 1982, PAS-101, 986-996. Caramanis, M., Stremmel, J. V., Fleck, W. and Daniel, S., Probabilistic production costing: an investigation of alternative algorithms. Inr. J. Eiecrricul Power and Energy Systems, 1983, 5, 75-86. Schenk, K. F., Misra, R. B., Vassos, S. and Wen, W., A new method for the evaluation of expected energy generation and loss of load probability. IEEE Transactions, 1984, PAS-103, 297-303. Mazumdar, M. and Gaver, D. P., On the computation of power generating system reliability index. Technometrics, 1984, 26, 173-185. Mohan, P., Balasubramanian, R. and Prakasa Rao, K. S., A new fourier method for evaluating generation system reliability indices. IEEE Transactions on Power Systems, 1986, 1, 88-94. Chanan Singh and J&O. Kim, A new technique for generating capacity reliability evaluation. Electric Power System Research, 1992, 23, 181-187. IEEE Committee Report, IEEE reliability test system. IEEE Transactions, 1979, PAS-99, 2047-2054. Azad, A. K. and Misra, R. B., A new approach to evaluate the generating system reliability. Proceedings of the eighth Nurionul Power System Conference (NPSC-94), Vol.1, 1994, pp. 67-71. Allied Publishers Ltd, New Delhi. Azad, A. K., Misra, R. B. and Soman, K. P., Stochastic modelling of interconnected power generating system and its reliability evaluation by fast Fourier transform, Proc. of rhe Internurionul Conference on Srochustic Models, Optimizurion Techniques and Computer Applications (ZCSOC-94), 1994, pp. 64-76. Wiley Eastern Ltd, New Delhi.