Comparison of alternative methods for evaluating loss of load costs in generation and transmission systems

Electric Power Systems Research 50 (1999) 107 – 114

Comparison of alternative methods for evaluating loss of load costs in generation and transmission systems L.A.F. Manso a,b, A.M. Leite da Silva b,*, J.C.O. Mello c a

Electrical Engineering Department, Federal Uni6ersity, Sa˜o Joa˜o del-Rei, FUNREI, MG, Brazil b Institute of Electrical Engineering, Federal Uni6ersity, Itajuba´, EFEI, MG, Brazil c Electric Power Research Center-CEPEL, Rio de Janeiro, RJ, Brazil Received 18 February 1998; accepted 8 September 1998

Abstract This paper presents a comparison among different methods to evaluate the loss of load cost (LOLC) index for composite generation and transmission systems. A new methodology is proposed and used as a reference to evaluate LOLC indices. It considers all blocks of unsupplied energy per consumer class, per bus, and the respective duration, to characterize accurately the interruption process. The proposed methodology is implemented with two composite reliability evaluation algorithms based on Monte Carlo simulation: sequential and pseudo-sequential. Also two load curtailment policies, minimum load curtailment and minimum cost, are considered and their impact on the different LOLC evaluation methods assessed. Case studies with the IEEE-MRTS (modified reliability test system) are presented and discussed. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Composite reliability; Reliability worth; Generation and transmission reliability; Uncertainty modeling.

1. Introduction Traditionally, the minimum cost expansion planning of power systems are only based on the minimization of investment and production costs necessary to meet future load, for a given level of reliability. In this context, pre-established values for the LOLE (loss of load expectation) and EENS (expected energy not supplied) indices can be used as restrictions to the optimization problem [1]. These indices represent relative measures of adequacy of the power systems and, therefore, are not capable of taking into consideration economic impacts of interrupted energy for the consumers. The introduction of the loss of load cost (LOLC) in the minimum cost planning depends basically on the unit interruption cost (UC) of each consuming class, usually given in US$/kWh. The UCs are obtained through specific economic studies (consumers surveys) [2 – 4]. These studies describe different factors which may have some impact on the UCs, and the duration of * Corresponding author. Tel.: +55-35-6291249; fax: + 55-356291187. E-mail address: [email protected] (A.M. Leite da Silva)

the interruption is considered the most important one. Therefore, the accuracy level established to evaluate this duration interferes in the quality of the estimates of LOLC indices. For generation and transmission systems, the estimates of loss of load cost are obtained through composite reliability evaluation algorithms, which are based on two distinct representations: state space and chronological modeling. Usually, state space based algorithms follow three major steps [5]: 1. select a system state (i.e. load level, equipment availability, etc.); 2. analyze the performance of the selected states (i.e. check if available generating units and circuits are able to satisfy the associated load without violating any operating limits; if necessary, activate corrective measures such as generation redispatch, voltage correction, load curtailment, etc.); 3. estimate reliability indices (i.e. LOLP, loss of load probability; EPNS, expected power not supplied, etc.), if the accuracy of the estimates are acceptable, stop; otherwise go back to step (a). State enumeration and non-sequential Monte Carlo simulation methods are examples of state space based

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algorithms, where Markovian models are used for both equipment and load state transitions. Therefore, states are selected and evaluated without considering any chronological connection [6]. The necessary steps to evaluate reliability indices considering the chronological representation (sequential Monte-Carlo simulation) are conceptually the same described for the state space representation [7]. The basic difference is how system states are selected; i.e. step (a) of algorithm. In this case, the sequential approach moves chronologically through systems states, while the non-sequential approach selects system states in a random way. The sequential simulation can therefore perceive all chronological aspects and, hence, is able to correctly reproduce the whole cycle of interruptions. However, the chronological modeling implies that two consecutive state samples differ from each other from one state component. Therefore, this method requires a more substantial computational effort than the other approach.This paper presents a comparison among different methods to evaluate LOLC index for composite generation and transmission systems. A new methodology is proposed and used as a reference for comparison purposes. In order to characterize accurately the interruption process, the proposed methodology models all blocks of unsupplied energy per consumer class, per bus, and the respective duration, and is implemented into a sequential and pseudosequential Monte Carlo simulation algorithms. Also two load curtailment policies, minimum load curtailment and minimum cost, are considered and their impact on the different LOLC evaluation methods assessed. Case studies with the IEEE-MRTS (modified reliability test system) are presented and discussed.

2. Loss of load cost evaluation

2.1. Unit interruption cost From the economic point of view, the most relevant aspects to estimate the impact of an interruption are: the amount of unserved energy (kWh) and the unit interruption cost (US$/kWh). Surveys performed among consumers indicate that the unit interruption cost depends on several characteristics, such as duration, frequency, time of occurrence, warning time, depth of curtailment and geographical coverage. The most relevant is the duration of an interruption. Fig. 1 illustrates the curves of UC for the three consumer classes, residential, commercial and industrial, as a function of the energy shortage duration. Such curves were taken from a survey performed by Ontario Hydro [2].

2.2. E6aluation methods LOLC indices, can be evaluated by the following general expression:

 n

LOLC= E % Ki

(1)

i  I

where Ki, cost (US$) of a sampled interruption i; I, analyzed period; E[.], expected value operator. If a sequential Monte Carlo algorithm is used, the general expression (Eq. (1)) is estimated by: LOLC=



1 NI % NIn = 1

% Ki



(2)

i  In

where In is the nth of NI simulated periods. For the non-sequential and pseudo-sequential simulations, the adopted estimate of LOLC is: LOLC= % fi E[Ki ], or

!

i  I

LOLC=

"

1 NS Ki % NS i = 1 E[Di ]

(3) (4)

where NS, no. of non-sequential samples; fi, Ki and Di, estimates for frequency, cost and duration, associated with the sampled failure state i (non-sequential simulation) or the whole interruption I (pseudo-sequential simulation). The evaluation methods for LOLC index differ not only in the way expression (Eq. (1)) is estimated, but also in the way cost Ki is calculated. Taking into consideration that an energy shortage is, in most cases, composed by a sequence of system operation states, one can assert that the reliability algorithms based on chronological representation are more suitable to LOLC evaluations.

2.2.1. MR Method (method of reference) The proposed method for the evaluation of cost Ki, and taken as reference for the comparative analysis, was initially applied to generating systems [8]. It models accurately all blocks of unsupplied energy per consumer class, per bus, and the respective duration. Any inter-

Fig. 1. Unit interruption cost for Ontario Hydro.

L.A.F. Manso et al. / Electric Power Systems Research 50 (1999) 107–114

109

proposed by reference [9], the states which compose this interruption (and others) are separately identified by non-sequential Monte Carlo simulation. As an example, the cost associated with each occurrence of state 1 is: K1 = PS1 × SD1 × UC(SD1)

Fig. 2. Graphic representation of an interruption.

ruption i, can be described by a set Si of energy shortages related to successive failed states which compose this interruption. The associated cost Ki (US$) defined for a particular consumer class is given by: Ki = % ESj × UC(Dj )

(5)

j  Si

where ESj, energy shortage j Si ; Dj, duration of energy shortage j; UC(Dj ), unit interruption cost (US$/kWh). Observe that ESj is equal to the product: PSj × Dj, where PSj is the power shortage associated with energy shortage j.The previous concept is illustrated in Fig. 2. It shows an interruption between times t1 and t6, with three different blocks. For this interruption, Eq. (5) can be written as: K = ES1UC(D1)+ES2UC(D2) +ES3UC(D3)

(6)

where ES1 = PS1 × D1; PS1 =(P1 −0); and D1 =(t6 − t1). Other terms are similarly obtained.

2.2.2. Method M1 Using methods based on the state-space representation, it is possible to evaluate the LOLC index of each bus of the system (LOLCBUS) without calculating Ki. In this case, the LOLC of each bus is approximated by the following expression: LOLCBUS =EENSBUS ×UC(LOLDBUS)

where SD1 is the sampled duration from an exponential distribution with parameter equals to the summation of outages rates from state 1. For any of the three repetitions of state 1 (and possibly for the great majority of occurrences of this state), the sampled duration is significantly smaller than the total duration of the energy block (ES1) to which they belong to. Thus, the interruption cost evaluation without considering the possible chronological connection between states might endanger the results of LOLC indices. This will be verified during the tests.

2.2.4. Method M3 Based on a sequential Monte Carlo simulation, the method proposed by reference [10] takes into consideration the temporal connection between the failure states. However, this method evaluates the cost of a given interruption only by using the total duration of it. In the interruption case of Fig. 2, the cost is: K=(ES1 + ES2 + ES3)× UC(D1)

(9)

The total energy shortage (ES= ES1 + ES2 +ES3), and not only ES1, is multiplied by the unit cost obtained for the total failure duration (D1). This is equivalent to consider an interruption in which an average load would be shed: i.e. PAVERAGE = ES/D1. Method M3 was initially used along with the so called proportional load shedding policy, in which the interrupted energy in each bus is distributed proportionally among consumer classes. This policy simplifies the LOLC evaluation process. If other loading shedding policies are adopted (e.g. the policy shown in Fig. 3 where priorities are assumed for residential, commercial and industrial classes, in this order), it will be necessary

(7)

where EENSBUS and LOLDBUS are, respectively, the expected energy not supplied and loss of load duration indices at each bus of the system. Method M1 is implemented in a non-sequential simulation algorithm.

2.2.3. Method M2 In general, an interruption process is composed by many interrelated states. Taking Fig. 2. as an example, one can suppose that interruption i is composed by three operating states of the system. State 1 represents, in its three repetitions (intervals [t1 t2], [t3 t4] and [t5 t6]), power curtailment level P1. Other two states represent curtailment levels P2 and P3. Through the methodology

(8)

Fig. 3. Shortage of energy per consumer class.

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to individualize, per consumer class, the representation of each interruption. As shown Fig. 3, only the residential class would be affected by interruption i, because the other classes, which have higher priority over the residential one, would be above the mean value of the power to be curtailed.

3. Comparison of results In order to compare numerically the previous considerations, the three methods (M1, M2 and M3) and the proposed one (MR) were applied to the MRTS test system (modified IEEE reliability test system), which is a result of modifications performed in the system IEEERTS [11], with the objective of stressing the transmission network. Thus, the generating capacity and the load in each bus of the system are duplicated. The chronological curve adopted for the load corresponds to 52 repetitions of the peak week (week 51 of the original curve). The Ontario Hydro [2] unit interruption costs were used (Fig. 1). The participation of each consumer class per bus of the system is extracted from reference [12]. The CPU times shown are obtained in a PENTIUM-150 MHz microcomputer.

3.1. Case 1: minimum load shedding policy For each selected state, a system adequacy analysis is carried out by a DC linear power flow. If operating violations are detected, a remedial action model is used, based on an optimization, which performs active power redispatching and, in more severe cases, applies load shedding. For case 1, the minimization of load shedding is used as object function of the following optimization problem: n

z = Min % ri i=1

s.a. Bu + g+r = d g5 gmax g]gmin r 5d f

5fmax

Table 1 Performance of methods: case 1 LOLC (106 US$/year) MR B1

60.263

B3

26.456

B8

7.216

B14

72.919

B18

14.615

Syst

316.960

M1

M2

M3

59.176 (−1.8%) 21.404 (−19.1%) 7.421 (2.8%) 73.930 (1.4%) 14.012 (−4.1%) 307.310 (−3.0%)

58.375 (−3.1%) 29.506 (11.5%) 5.662 (−21.5%) 91.564 (25.6%) 19.963 (36.6%) 362.810 (14.5%)

60.284 (0.0%) 21.850 (−17.4%) 7.815 (8.3%) 69.192 (−5.1%) 13.055 (−10.7%) 301.820 (−4.8%)

The solution algorithm adopted is basically the DualSimplex, modified to explore the characteristics of electrical network sparsity equations. Aiming at reducing the computational storage requirements and computational effort, the algorithm uses the reduced base method [13]. The obtained results for some bus and systems are shown in Table 1, where percentage errors made by M1, M2 and M3 in relation to the reference method (MR) are included. These errors appear in parentheses right under the corresponding value of LOLC. The buses which appear in Table 1 correspond to the buses that show the greatest values of LOLC, for the current case (B1, B3 and B14) and for case 2 (B8, B14 and B18), which will be discussed in Section 3.2. The convergence achieved by the system LOLC corresponds to a relative uncertainty (b) of 5% [6,10]. For methods MR and M3, which use sequential Monte Carlo simulation, 140 synthetic annual series are necessary. The CPU time is : 42 min. In the evaluation of the LOLC indices by methods M1 and M2, implemented in a non-sequential Monte Carlo simulation algorithm, 39 423 samples are used, which result in a CPU time equals : 1.4 min. From the obtained results two performance indices are calculated: AAE, absolute average error; and PAE, percentage average error, whose expressions are given as follows: n AEi PEi and PAE= % i=1 n i=1 n n

(10)

where ri, load shedding at bus i; n, total number of system buses; B, susceptance bus matrix of the system, equivalent to the admittance bus matrix for a system with no losses; u, bus angle vector; g, bus generating vector; d, bus load demand vector; r, load shedding vector; gmax, bus maximum capacity generating vector; gmin, bus minimum capacity generating vector; f, circuit flow vector; fmax, circuit maximum flow vector.

AAE= %

(11)

Table 2 Error performance of methods: case 1

AAE (106 US$/year) PAE (%)

MI

M2

M3

1.484 6.013

5.737 20.845

1.867 8.969

L.A.F. Manso et al. / Electric Power Systems Research 50 (1999) 107–114

where AEi and PEi are, respectively, the absolute and percentage errors for each one of the n load buses of the system. Table 2 presents the AAE and PAE indices obtained from methods M1, M2 and M3 in relation to the reference method MR. As expected, the results presented in Tables 1 and 2 show that method M2 had a worse performance than M3, for both the system and for the majority of buses. However, the good performance of M1, similar to M3, was not expected for the system nor for the majority of buses. M1 method is based on two simplifying hypotheses: “ E[UC(DBUS)]: UC(E[DBUS]) =UC(LOLDBUS), where UC is a non-linear function; “ Consider that the energy not supplied in each bus and the unit interruption cost UC(DBUS) are independent random variables, and then calculate the expected value of its product, as the product of its expected values; Eq. (7). In general, the unit cost curves present distinct behaviors for different consumer classes; see Fig. 1. These behaviors may produce a compensation among the absolute LOLC errors of the consumer classes in certain system buses. Compensation may also occur among the absolute LOLC errors of system buses. Although an approximate method may produce very high errors for the LOLC index associated with a specific consumer class of a certain bus, it may fortunately yield accurate LOLC index for the system or even at the bus level. It will depend on the characteristics of the system, consumer class composition per bus and the shape of the unit cost curves.

3.2. Case 2: minimum cost policy The objective function of the optimization problem (Eq. (10)) yields the minimization of energy indices (e.g. EENS). However the same does not occur in relation to the reliability worth (LOLC). The interruption cost is not only a function of the amount of energy being curtailed, but also of the unit interruption cost (UC), which may vary significantly among consumer classes. Therefore, the minimization of the loss of load cost is more attractive from the economical point of view as compared with the minimization of the load curtailment. Bearing in mind that the UC depends on the failure duration, in order to implement a minimum cost policy, it would be necessary to predict the duration of each interruption and so reject those loads associated with the lowest values of UC. Clearly, in practice, it is not possible to perform such forecasting. In principle, it is possible to implement sophisticated load curtailment policies into any reliability evaluation algorithm. However, the most important question is how close these policies will be to those adopted in practice by the concessionaires of energy.

111

In general, loads are classified according to their importance in two categories: curtailable (C) and firm (F). C-loads involve residential and commercial sectors, while F-loads involve, for instance, industrial sectors, large commercial consumers, hospitals, etc. Usually, C-loads are shed from the less to the most important ones. If more than one bus is equally qualified, the amount of power to be shed is divided among these buses according to some criterion. Also, whenever it is possible, residential loads are the first to be shed. F-loads are curtailed after shedding all C-loads in the system or area. Usually, the amount of load to be curtailed from buses containing F-loads are shed among these buses according to some criterion: buses with large industrial loads will be more affected than buses with small industrial consumers. Priorities can also be assumed if some sort of agreement or contract has been established a priori between industrial consumers and the utility. In fact, any philosophy or policy for curtailing bus loads can be considered. The main goal is always to minimize system interruption costs. An alternative adopted in this work establishes three groups with different priority le6els (GL1, GL2 and GL3) to shed load for each bus composed by different consumer classes. This grouping aims at controlling the energy curtailment, starting by group level 1 (GL1), composed by consumer classes with smaller damage costs (i.e. UC), until group level 3 (GL3), with consumer classes with higher damage costs. The model is flexible enough to allow network restrictions interfere in the grouping process. The load shedding policy is implemented according to two distinctive actions: 1. the objective function of the load shedding optimization model has to be slightly changed in order to include penalties associated with the fictitious generating buses (load shedding-vector r): n

z= Min % ai ri

(12)

i=1

2. once established the amount and the bus of load shedding ri (previous step), it has to be applied to the different consumer groups at bus i, according to the criterion previously established. In terms of optimization model, it would be correct to use the energy shedding cost in each bus, and not the penalties above. However, the unit interruption cost is a function of the load shedding duration, and this is not known at the beginning of the interruption. Even if possible, the utilization of costs in the objective function would depict an unrealistic situation of the real world. Thus, the penalties ai try to restrict the load shedding only to buses with high priority groups (GL2 and GL3), and therefore reproducing the general practice of a concessionaire of energy which establish a kind of merit order for system buses, according to certain social-economic standards. To simplify and also to

L.A.F. Manso et al. / Electric Power Systems Research 50 (1999) 107–114

112 Table 3 Performance of methods: case 2 LOLC (106 US$/year) MR B1

0.379

B3

0.714

B8

54.187

B14

31.523

B18

4.067

Syst

109.910

M1

M2

M3

0.325 (−14.3%) 0.604 (−15.3%) 7.205 (−86.7%) 4.136 (−86.9%) 1.01 (−75.1%) 20.196 (−81.6%)

0.494 (30.3%) 0.938 (31.5%) 39.097 (−27.9%) 32.799 (4.1%) 4.306 (5.9%) 96.149 (−12.5)

0.348 (−8.2%) 0.660 (−7.5%) 55.665 (2.7%) 32.054 (1.7%) 4.212 (3.6%) 111.870 (1.8%)

better compare with the results obtained in case 1, it will be assumed that groups GL1, GL2 and GL3 are composed by classes residential, commercial and industrial, respectively. In order to implement the previously proposed minimum cost policy, the merit order among buses must be known to define penalties ai. As this information is not available for the IEEE-MRTS, the adopted criterion is to perform the load curtailment according to the amount of residential class (GL1) at a particular bus. Eq. (13) describes this criterion:



n



ai = 1− LGL1i / % LGL1i 100

(13)

i=1

where LGL1i is the total amount of residential load at bus i. Therefore, higher is the amount of residential load at a certain bus of the system, higher will be the chances of shedding load during a emergency condition at this bus. Table 3 presents the results for the LOLC of the system and of the same buses considered in Table 1. To achieve the same accuracy of case 1, the methods using sequential Monte Carlo simulation (i.e. MR and M3) has converged after simulating 200 years (43% more than in case 1), and the methods using non-sequential Monte Carlo simulation (i.e. M1 and M2) has converged after 81 247 samples (106% more than in case 1). The reduction in the system LOLC is the main reason for the observed increase in the number of simulations and consequently in the computing time. It can be concluded from the results presented in Table 3 that the LOLC indices, not only for the system but for most buses, are considerably smaller than those shown in Table 1. The system LOLC evaluated by the proposed MR method (Table 1) is o6erestimated by :190%. Therefore, the minimization of load curtailment, described by Eq. (10), should not be used in minimum cost expansion planning of power systems.

The performance indices AAE and PAE are shown in Table 4. Undoubtedly the method M3 is superior to methods M1 and M2. The low performance of method M1 is explained by its inability to capture the power curtailment deepness for different bus interruptions. In this method, the distribution of the interrupted energy among the different consumer classes or group levels (GL1–GL3) is carried out according to an average curtailed power obtained by the ratio EENSBUS/LOLEBUS. For most system buses this average curtailment is even below the total amount of the residential load. This fact explains the low values of costs obtained with the method M1 for this particular case. Therefore, although this method has presented a good performance for the minimum load shedding policy, it has shown to be inadequate with the minimum cost policy. The performance of method M2 for the LOLC of the system is − 12.5%, and the error performance among the buses is still lower (19.6%). Another test is carried out to confirm this performance: the composition of consumer classes at bus 8 is changed from 50% commercial and 50% residential to 50% industrial and 50% residential. The results for the system LOLC indices are: MR“ 87.746×106 US$/year and M2 “ 100.97× 106 US$/year. Therefore, the percentage error goes from − 12.5% to +16.4%. Thus, as an important conclusion, the LOLC evaluation without considering the possible chronological connection between states might endanger the results of LOLC indices. In relation to the method M3, it can be observed that there is a slight improvement in its performance when considering the minimum cost policy: performance index PAE goes from 8.9 to 6.1%. However, the additional computational effort to evaluate the costs by considering each block of unsupplied energy per class, per bus and its respective duration instead of the total duration of the interruption, as used by M3, is negligible. In fact, the success of M3 depends on durations of the deeper energy interruptions, which, in turn, are usually shorter.

4. Pseudo-sequential simulation In the previous sections, it was demonstrated that methods based on state–space representation are not able to provide reliable estimates for the loss of load Table 4 Error performance of methods: case 2

AAE (106 US$/year) PAE (%)

M1

M2

M3

9.971 48.700

1.874 19.620

0.292 6.195

L.A.F. Manso et al. / Electric Power Systems Research 50 (1999) 107–114 Table 5 Comparison between PSM and sequential simulations LOLC (106 US$/year) Case 1 SEQ B1

60.263

B3

26.456

B8

7.216

B14

72.919

B18

14.615

Syst

316.960

Case 2 PSM 57.155 (−5.2%) 24.679 (−6.7%) 7.216 (−0.0%) 74.964 (2.8%) 13.949 (−4.6%) 309.350 (−2.4%)

SEQ 0.379 0.714 54.187 31.523 4.067 109.910

PSM 0.351 (−7.5%) 0.729 (2.2%) 52.424 (−3.2%) 33.409 (6.0%) 4.269 (5.0%) 108.520 (−1.3%)

costs. Two reasons can be pointed out: the Markovian representation of the load and the failure representation. To demonstrate that the impact of the latter assumption does compromise the accuracy of the results, the pseudo-sequential simulation with Markovian representation of loads (PSM) [14] will be used in this analysis. The PSM algorithm can be interpreted as a non-sequential Monte Carlo simulation considering the chronological simulation only to model the failure sequences. Therefore, the concepts brought by method MR are also implemented into a PSM algorithm. The load model is assumed as Markovian but no clustering among load states are considered. Tables 5 and 6 show the results for both algorithms performance: the sequential and the PSM simulations. Also two policies are considered: minimum load shedding (case 1) and minimum cost (case 2). As can be observed in both tables, the performance of the PSM is very good in terms of accuracy. However the CPU time obtained with the sequential simulation was 42 min (case 1) while with PSM was just 1.2 min, i.e. approximately the same achieved with the non-sequential simulation.

5. Conclusions A new methodology to evaluate the loss of load costs (LOLC) in generating and transmission systems is preTable 6 Error performance of cases: PSM

AAE (106 US$/year) PAE (%)

Case 1

Case 2

1.416 5.343

0.416 8.364

113

sented in this paper. This methodology is based on a sequential or pseudo-sequential Monte Carlo simulation and evaluates accurately the cost of each block of unsupplied energy per consumer class and per bus, considering the associated duration. It is used as a reference method to evaluate different approximations based on state-space and chronological sampling of system states. Although very efficient from the computational point of view, state-space Markovian methods are not able to provide accurate results for LOLC indices. Eventually, these estimates may be considered as acceptable, in terms of accuracy, but this can not be always ensured for any analyzed case, since it depends on many factors including damage function characteristics, remedial actions (redispatching and load curtailment policies), composition of consumer classes. Conversely, chronological representation of system states does allow for an accurate evaluation of LOLC indices, although it is very time consuming. In fact, the chronological representation has only to be ensured for the sequence of failure states; this is the principle of the pseudo-sequential simulation. The important aspect is the evaluation of the loss of load costs according to the proposed methodology. Also, the Markovian representation of the load does not impact the accuracy of the LOLC indices but its temporal disconnection does. It is also possible to conclude that the minimization of the load shedding, in general, lead to high results for the LOLC estimates. Thus, it is prohibitive its application in cost/benefit analysis, such as the minimum cost planning. Through minimization cost policies it is possible to reproduce more closely the concessionaire practices. References [1] A.P. Sanghvi, N.J. Balu, M.G. Lauby, Power system reliability planning practices in North America, IEEE Trans. Power Syst. 6 (1991) 1485 – 1492. [2] EPRI, Customer demand for service reliability, Report RP-2810 (1989). [3] G. Wacker, R. Billinton, Customer cost of electric service interruptions, Proc. IEEE 77 (6) (1989) 919 – 930. [4] A.G. Massaud, M. Th. Schilling, J.P. Hernandez, Electricity restriction costs, IEE Proc. C 141 (1994) 229 – 304. [5] EPRI, Transmission system reliability models, Report EL-2526 (1982). [6] M.V.F. Pereira, N.J. Balu, Composite generation/transmission reliability evaluation, Proc. IEEE 80 (4) (1992) 470 – 491. [7] L. Salvaderi, Monte Carlo simulation techniques in reliability assessment of composite generation transmission systems, IEEE Tutorial Course 90EH0311-1-PWR, (1990). [8] A.M. Leite da Silva, A.G. Perez, J.W. Marangon Lima, J.C.O. Mello, Loss of load costs in generation capacity reliability evaluation, Elec. Power Sys. Res. 41 (1997) 109 – 116. [9] Li Wenyuan, R. Billinton, A minimum cost assessment method for composite generation and transmission system expansion planning, IEEE Trans. Power Syst. 8 (1993) 628 – 635.

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allocation approach for generation and transmission facilities, IEEE Trans. Power Syst. 5 (1990) 842 – 851. [13] B. Stott, J.L. Marinho, Linear programming for power system network security applications, IEEE Trans. Power Appar. Syst. 98 (1979) 837 – 848. [14] J.C.O. Mello, A.M. Leite da Silva, M.V.F. Pereira, Efficient loss of load cost evaluation by combined pseudo-sequential and state transition simulation, IEE Proc. C 144 (1997) 147 – 154.