Multi-area reliability evaluation including frequency and duration indices with multiple time varying load curves

Electrical Power and Energy Systems 42 (2012) 276–284

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Multi-area reliability evaluation including frequency and duration indices with multiple time varying load curves Thatiana C. Justino a, Carmen L. Tancredo Borges b,⇑, Albert C.G. de Melo a a b

Electrical Power Research Center (CEPEL), P.O. Box 68007, 21944-970 Rio de Janeiro, Brazil Federal University of Rio de Janeiro (UFRJ), P.O. Box 68504, 21941-972 Rio de Janeiro, Brazil

a r t i c l e

i n f o

Article history: Received 27 December 2010 Received in revised form 23 March 2012 Accepted 9 April 2012

Keywords: Multi-area reliability Time varying load Monte Carlo simulation Frequency and duration indices

a b s t r a c t The reliability evaluation of multi-area systems is part of the expansion planning process of a power system. In order to accurately estimate the system and areas reliability indices, especially the frequency and duration (F&D) indices, the dynamic behavior of system components must be adequately represented with particular attention to the variation of each area load. Nowadays, the most efficient methods for multi-area reliability evaluation do not represent the chronological aspects of the different areas loads. The method that directly considers these aspects is the sequential Monte Carlo simulation, however requiring a high computational effort. This paper presents the application of some methods that represent the chronological aspects of the load to the reliability evaluation of multi-area systems and evaluates their performance and accuracy. The main purpose is to obtain a model for multi-area reliability evaluation that represents the different load curve of each area and accurately estimates the F&D indices. Results obtained by the studied methods are presented for two systems, one composed of two areas and the other based on a representation of the Brazilian interconnected power system with four areas. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The main function of a power system is to supply the load demand at the lowest possible cost and with an acceptable level of reliability. However, these goals are conflicting because to reach a high reliability level, it is necessary to perform a large amount of investment in the system. Therefore, the challenge of the power system expansion planning is to find a trade-off between its reliability and investment cost considering the uncertainties inherent to the problem, like variations of load and availability of components. For systems like the Brazilian power system, characterized by predominantly hydroelectric energy generation and large amounts of power exchanges between different regions of the country, the expansion planning of the generation capacity and the system areas interconnections has an important role in overall planning process. The generation expansion planning is performed in two steps: (1) Energy supply expansion planning. (2) Generation capacity expansion planning.

⇑ Corresponding author. Tel.: +55 21 25628027; fax: +55 21 25628080. E-mail addresses: [email protected] (T.C. Justino), (C.L. Tancredo Borges), [email protected] (A.C.G. de Melo).

[email protected]

0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.04.016

The first step solves the problem of attending the foreseen energy demand and defines an expansion scheduling of the power plants. The second step solves the problem of attending the foreseen load peak and defines the generation reserve margins at each stage of the planning horizon. Multi-area generation scheduling [1] and reserve constrained economic dispatch [2] are part of the planning problem. However, the reserve margins are not autonomous and fully reliable because, in fact, the generation system is composed by interconnected areas which may interchange power to attend their own loads. To compute the resulting realistic generation margins, a reliability evaluation of interconnected (or multiarea) power systems is performed at the end of the expansion planning process. This evaluation aims to introduce constraints into the power (and energy) exchange between areas (or subsystems) and assesses the overall impact of these constraints in the system reliability and, mainly, in the reliability of each area. Thus, it is important to develop efficient and reliable multi-area reliability evaluation methods. Most of the methods used for this evaluation use the state space representation to estimate the reliability indices. This representation disregards the chronological aspects of power system operation and analyzes its performance through snapshots of its operation cycle. One of the most important chronological aspects is the load variation that directly influences the reliability indices values. This influence is mainly sensed in the magnitude of load curtailment and in the frequency and duration of energy supply interruptions.

T.C. Justino et al. / Electrical Power and Energy Systems 42 (2012) 276–284

For reliability evaluation of multi-area systems through state space representation, the load behavior has been approached by the use of one load level [3] or by a Markov load model aggregated in multi-states [4]. Although being significant and useful for identifying weakness and reinforcement necessities, reliability indices calculated at one load level do not incorporate the effects of load variation [5]. Some aspects of the load variation can be represented by a Markov load model aggregated in multi-states. This model is able to represent basic information about the load levels, such as their average durations, probability and frequency of occurrence and probability and frequency in which they transit for upper or lower levels. But the use of this model implies in considering that all areas loads of the system are fully correlated and, consequently, the different load patterns of the areas are not modeled [6]. Another approach is to use a multi-area load model which accommodates some load correlation between areas in the decomposition-simulation method [7], with no concerns related to frequency and duration (F&D) indices. The most direct way to represent the chronological aspects of the load in multi-area reliability evaluation is to use the chronological load curve in sequential simulations, where all system states are sampled consecutively throughout the analysis period, usually 1 year [8]. The chronological approach has been applied in [9] for assessing the load variation and equipment availability impact in the Available Transfer Capability (ATC). However, in general, the chronological assessment requires higher computational effort than those required by the methods that use the state space representation. This happens because most of the analyzed states do not contribute to the estimation of the reliability indices. So, the ideal approach would be to conciliate the efficiency of techniques based on the state space representation with the accuracy of considering the chronological evolution of power system operation. Such approach has been tested in some methods developed for composite reliability evaluation. Examples of these methods are the Pseudo-sequential [10], Pseudo-chronological [11] and non-sequential with one step forward state transition process [12] Monte Carlo simulations. The Pseudo-sequential simulation [10] uses the non-sequential simulation to select the failure states of the system and applies the sequential simulation to the fault subsequences formed by the failure states that are neighbor to the selected state. Such as in the sequential Monte Carlo simulation, the load behavior is represented by the chronological load curve. This approach has been applied for a restructured power system with a bilateral market in [13]. The Pseudo-chronological simulation [11] is a variation of the Pseudo-sequential method that combines the non-sequential simulation and a technique to estimate the Loss of Load Frequency (LOLF) index based on state transition sampling. Instead of using a chronological load curve, this technique uses a non-aggregated Markov load model to represent the behavior of the system load and its areas loads. This approach has been applied for expansion planning in [14]. The non-sequential simulation with one step forward state transition process [12] is similar to the traditional non-sequential simulation but instead uses a non-aggregated Markov load model to represent the system and areas loads and a different test function to calculate the LOLF index. Then, this paper presents the application and enhancement of some of these methods for reliability evaluation of multi-area systems and evaluates their performance and accuracy, since their original development was for composite reliability evaluation. Additionally, an approach to calculate the indices of a system whose loads are strongly correlated is presented, based on the non-sequential simulation that uses the conditional probability method [5] and the non-aggregated Markov load model. The main

277

purpose is to obtain a model for multi-area reliability evaluation that represents the different load curve of each area and accurately estimates the F&D indices. The methods were implemented into the program CONFINTÒ, developed by CEPEL (Brazilian Electric Power Research Center) for reliability evaluation of hydrothermal inter-connected power systems [15]. The results obtained by the studied methods are presented for two multi-area systems, one composed of two areas and the other based on a representation of the Brazilian interconnected power system with four areas. 2. Multi-area representation of a power system A multi-area power system is represented by a linear network flow, where the nodes represent the areas, and the arcs represent the interchanges among areas. Each interchange is represented by its maximum capacity, its failure and repair rates and a percentage that defines how much its capacity will be reduced if a failure happens. Therefore, the interchanges are modeled by constraints of the problem of networks flow in order to avoid deviations in the scheduled interchanges among system areas. The generation in each area is modeled as an arc coming into the node from a ‘‘source’’ node, S, also constrained by the maximum generation value. In turn, the area load is represented as an arc leaving the node to a ‘‘terminal’’ node, T. Fig. 1 illustrates a system composed of two areas [16]. The capacities associated with each arc are random variables, and can be obtained from the combination of the individual states of the equipments (generators and interconnections) and load levels. For example, the capacity of each generation arc is a random variable, corresponding to the sum of the available generation capacities in the corresponding area. The failure/success status of a given state can be assessed by calculating the maximum power flow from the source node S to terminal node T, taking into account the power balance at each node, and the arc capacities. If the max-flow is equal to the total demand, this means that all demand arcs arriving at T are at their limits. Therefore, all area loads are being supplied, i.e., there is no load curtailment. Conversely, if the max-flow is smaller than the total demand, this means that at least one of the area loads is not fully met. The amount of load curtailment is the difference between total demand and max-flow value. An alternative way of solving this problem is to solve its dual problem which corresponds to finding the minimum capacity cut1 between the source and terminal nodes. This is possible because, according to Maximum Flow–Minimum Cut Theorem [17], the max-flow value that crosses the system is equal to the min-cut. 3. Calculation of reliability indices In a multi-area system composed of generating units and interconnection lines modeled by a Markov process with two states (operation and failure), the system state can be represented by a vector x = (x1, x2, . . ., xk, . . ., xm), where xk represents the state of the kth component or a load level and m is the number of system components. The set of all possible system states is called the state space X. Since the component states are random variables, there is a probability associated to each system state P(x). The problem of calculating reliability indices is equivalent to the problem of calculating the expected value of a given test function, like expression (1)

EðFÞ ¼

X FðxÞ  PðxÞ

ð1Þ

x2X 1 A cut is a partition of the system areas into two disjoint subsets, one contained the source node and the other contained the terminal node.

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1 G1

simulation, the chronological evolution of the load and the representation of different load patterns. These methods are called Pseudo-sequential (Pseq), Pseudo-chronological (Pchron) and nonsequential with one step forward state transition process (Nseq1St) Monte Carlo simulations and are described in Sections 4.2 and 4.3. Section 4.1 describes the non-sequential Monte Carlo simulation called traditional in this paper (NseqTrad) and Section 4.4 presents a new proposal, which combines the use of a new load model with the traditional simulation (NseqPrNag).

L1

S

T

G2 2

generation arcs

L2

4.1. Non-sequential Monte Carlo simulation with conditional probability method (NseqTrad)

load arcs

Fig. 1. Network flow model.

where E(F) is expected value of test function F. Its objective is to assess whether that specific combination of area generation capacities and interchange capacities is able to supply that specific set of area loads. Because the system states are random variables, the test result is also a random variable. The basic reliability indices, such as LOLP (Loss of Load Probability), EPNS (Expected Power Not Supplied) and the F&D indices as LOLF (Loss of Load Frequency) and LOLD (Loss of Load Duration), can be estimated by expression (1) through appropriate definition of the test function. For instance, the test functions for LOLP, EPNS and LOLF can be represented, respectively, by expressions (2)–(4).

 F LOLP ðxÞ ¼  F EPNS ðxÞ ¼

0; if x is a success state 1; if x is a failure state

ð2Þ

0; if x is a success state amount of load curtailment; if x is a failure state ð3Þ

8 0; if x is a success state > > > < sum of transition rates between x and all success F LOLF ðxÞ ¼ > states which can be reached from x in one transition; > > : if x is a failure state ð4Þ The definition of the appropriate test function for each index depends of the representation used in the reliability evaluation: by state space or chronological. The methods that use the state space representation are the analytical techniques and nonsequential Monte Carlo simulation. The estimation of indices by the chronological representation is performed using sequential Monte Carlo simulation. The uncertainty of the estimated indices is calculated by the coefficient of variation (b) [18], which is also used as the stop criterion of the Monte Carlo simulation. In multi-area reliability evaluations, the load behavior has been represented in analytical techniques and non-sequential simulation by a single load level and Markov load model aggregated in multi-states. In studies that require the knowledge of the chronological evolution of the system operation, it is used the sequential simulation that is able to represent the load by a chronological curve. However, it requires a high computational effort. Thus, it is important to develop, in the context of the multi-area reliability evaluation, methodologies that are able to represent the chronological aspects of each area load and calculate the reliability indices with more computational efficiency than the sequential simulation.

This method can produce any reliability index, including the frequency and duration indices whose calculation is more difficult because it requires the identification of all success states which can be reached from sampled failure state in one transition. In other words, if the system has m components, each with two states, we have in principle to carry out m + 1 assessments to update the frequency estimate. The number of evaluations increases further if there are components with multiple states. To avoid the high computational effort required for identifying the boundary between the failure and success states [5], proposed an efficient method for the evaluation of the F&D indices through the concepts of conditional probability and system coherence. It requires the same computational effort as for the estimation of LOLP and EPNS indices and allows the consideration of components with multiple states and unbalanced in frequency. This method developed a test function for the LOLF index presented in expression (5).

8 if x is a success state > < 0; m X F LOLF ðxÞ ¼ kin if x is a failure state > x ðjÞ; :

ð5Þ

j¼1

where kin x ðjÞ is the incremental transition rate of component j in the failure state x. This rate is calculated by expression (6), where mj is the number of states of component j; s is the state of component j in state vector x; kjsu is the transition rate of component j from state s to state u and P(xj = z) is the probability of state xj = z.

kin x ðjÞ ¼

mj X u¼sþ1

kjsu 

s1 X Pðxj ¼ v Þ j kvs v ¼1 Pðxj ¼ sÞ

ð6Þ

The limitation of this technique is that it is based on the assumption of system coherence2. Thus, for this assumption to be met, the system load and its areas (or subsystems) loads must have a high degree of correlation. To ensure that the coherence hypothesis would be met, it was applied in [5] a fully correlated load model, consisting of a single system load curve modeled by a Markov diagram, representing the load model aggregated in multi-states, and the areas (or buses) loads were obtained through participation factors. 4.2. Pseudo-sequential and Pseudo-chronological Monte Carlo simulations (Pseq and Pchron) The Pseudo-sequential simulation is a hybrid method in which the non-sequential simulation is used to select the failure states of the system and the sequential simulation is only applied to obtain the fault subsequences formed by neighbor failure states to the selected state [10]. This method defines any energy supply interrup-

4. Multi-area reliability evaluation with time varying loads The methods applied in this work to multi-area reliability evaluation use techniques for combining non-sequential Monte Carlo

2 Coherence Hypothesis: If a failure component is repaired, the system performance never gets worse; conversely, if a working component fails, the system performance never becomes better.

T.C. Justino et al. / Electrical Power and Energy Systems 42 (2012) 276–284

tion through a forward/backward simulation around the selected failure state. The duration of this interruption DI is given by the sum of the durations of failure states that form it. On the other hand, the Pseudo-chronological simulation is a variation of the Pseudo-sequential method combining the System State Transition Sampling method [19] and non-sequential simulation. Instead of using a chronological load curve, this technique uses a non-aggregated Markov load model to represent the behavior of the system load and its areas loads [11]. This model is composed by a set of T multiple levels connected in the same order they appear in the load historical to keep the chronological information about the load, as shown in Fig. 2. The model uses a constant transition rate kL = 1/DT, where DT represents the time unit used to discrete the period T. For each one of the m considered areas, it is given the load level by time interval. In these methods, the LOLP and EPNS indices are calculated as in the non-sequential simulation, i.e., through expressions (2) and (3), respectively. And for the LOLF index, a new test functions is used. For example, the Pseudo-chronological simulation uses expression (7).

F LOLF ðxÞ ¼



0;

if x is a success state;

1=EðDI Þ; if x is a failure state:

ð7Þ

4.3. Non-sequential Monte Carlo simulation with one step forward state transition process (Nseq1St) This method combines the non-sequential simulation and nonaggregated Markov load model and dispenses the hypothesis of system coherence [12]. The LOLF index is calculated by the one step forward state transition process, where, for each sampled failure state, an additional analysis of a new state reached through a single transition is performed. In this method, the LOLP and EPNS indices are estimated as in the non-sequential simulation, i.e., through expressions (2) and (3), respectively. The LOLF index is estimated by a new test function, expression (8).

279

4.4. Non-sequential Monte Carlo simulation combining conditional probability method and non-aggregated Markov load model (NseqPrNag) As described in Section 4.1 [5], considered a single load curve to the system, modeled by one Markov diagram, and used participation factors to dissociate the system load by the buses. Therefore, it used a fully correlated load model. However, non-aggregated Markov load model has the advantage of representing more accurately the buses (or areas) load curves of the system through a single Markov diagram. By this representation, the correlations between the buses (or areas) loads are preserved. Thus, a proposal presented in this paper is to calculate the indices, including F&D, of a system whose loads are strongly correlated, by non-sequential simulation with the conditional probability method and non-aggregated Markov load model. This can be performed because when a high degree of correlation between areas loads exists, the coherence hypothesis tends to be respected. Conversely, in situations where the correlation is weak, the system coherence is not guaranteed, which may invalidate the use of the conditional probability method to estimate the F&D indices. The contribution of the load for calculating the LOLF index in expression (5), i.e., the incremental transition rate of the load, is modified due to the nature of the non-aggregated Markov load model. Thus, this contribution is given by expression (9), where Lh is load level evaluated at hour h.

8 kL ; if the system load in level Lh is higher than in the > > > > > levels Lhþ1 andLh1 ; > > > > > 0; if the system load in level Lh is lower than in level > > > load in level Lh is higher than in level Lh1 and > > > > > lower than in level Lhþ1 ; > > > > > kL ; if the system load in level Lh is lower than in the > > : levels Lhþ1 and Lh1 ð9Þ

4.5. Conceptual algorithm

8 if xi is a success state; > < 0; F LOLF ðxi Þ ¼ kout ; if xi is a failure state and xk is a success state; i > : 0; if xi is a failure state andxk is a failure state: ð8Þ where kout is the sum of the transition rates from sampled failure i state xi for all states directly linked to it, and xk is any state reached from xi through a single transition.

Fig. 2. Non-aggregated Markov load model.

The conceptual algorithm for the multi-area reliability evaluation based on the non-sequential simulation methods presented in this paper is given below: (a) Sample a system state xi, i.e., sample a load level, the generators states, the hydrologic time series and the interchanges among areas states. (b) Evaluate the system state xi by the Maximum Flow–Minimum Cut Theorem and compute the test functions of the reliability indices, depending on the method used. (c) Estimate the expected values of the test functions. (d) Compute the coefficient of variation of the system LOLF index (bLOLF). (e) If the accuracy of estimated LOLF index is acceptable, stop the process; otherwise, return to step (a). Step (b) of the previous algorithm varies depending on the simulation method used. For the NseqTrad, the test functions are calculated by applying Eqs. (2), (3), (5), and (6). For the NseqPrNag, the test functions are calculated by the same equations plus Eq. (9). For the Pchron, if xi is a failure state, a synthetic fault subsequence around it is obtained and then the test functions are calculated by Eqs. (2), (3), and (7). And for the Nseq1St, if xi is a failure state, a

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system state transition is sampled, the new state is evaluated and then the test functions are calculated by Eqs. (2), (3), and (8). 5. Results This section presents the results of reliability evaluation of two test systems by the studied methods. The Pchron, Nseq1St and NseqPrNag simulations were implemented in CONFINTÒ program where, currently, the calculation of reliability indices is performed by an analytical technique or by NseqTrad simulation. The sequential simulation was also implemented and served as reference for comparison between the methods. The Pchron, Nseq1St and NseqPrNag simulations used the nonaggregated Markov load model with 8760 levels to represent the system load and its areas loads. The NseqTrad simulation used the Markov load model aggregated in multi-states and the sequential simulation used hourly load curves. For all analysis, it was used a Pentium IV 3.0 GHz computer. 5.1. Power system with two areas This system is used as the test case of the CONFINT program [15] to analytically compute the reliability indices. Its installed capacity, peak load and failure statistic data of the generators are given in Table 1 for each area. The interconnection capacity between areas is equal to 20 MW and its failure probability is 5.848  103. The areas load curves were generated by 52 repetitions of the weekly peak load of the southeast and south areas of Brazilian system. These curves, which represent, respectively, the loads of the areas 1 and 2, are illustrated in Fig. 3. For this system, two study cases were analyzed. The convergence criterion for the analyzed cases was the coefficient of variation of the system LOLF index (bLOLF) less or equal to 1%.

5.1.1. Case A.1 In this case, the behavior of the system and its areas loads was represented by a single load curve, the system equivalent curve shown in Fig. 3. The NseqTrad simulation used the Markov load model aggregated in 166 levels to represent this curve. Table 2 presents the system reliability indices and their coefficients of variation for this case. The LOLP is shown in%, EPNS in MW, LOLF in occ./year and LOLD in 102 h/year. The coefficient of variation of the LOLD index is not presented because its calculation is not implemented in the CONFINTÒ program. The speedup is in relation to the sequential Monte Carlo simulation. The areas reliability indices are shown in Table 3. The reliability indices estimated by the NseqTrad, Pchron, Nseq1St and NseqPrNag simulations are similar to those obtained by the sequential simulation. Excluding Nseq1St, these simulations spent less processing time than the sequential simulation. It may cause some curiosity the small difference between the processing times of the sequential and non-sequential simulation. However, due to the expressions used to estimate the LOLF index, the convergence of the LOLF estimator used in methods based on sequential simulation is faster than those based on non-sequential simulation. Thus, by construction, the convergence of LOLF in sequential methods demands less computational time than in non-sequential techniques. The time difference also varies with the numerical value of the index and since this test system has relatively high frequency of failure, this behavior is accentuated. The extra simulation required by the Nseq1St method combined with this characteristic made the processing time become even higher than sequential simulation. It is worth mentioning the behavior of the NseqPrNag simulation which obtained good results in terms of accuracy of the indices, especially in relation to the LOLF index. But this was expected because, in this case, the system load and its areas loads are fully correlated, which preserves the assumption of system coherence used in developing expressions (5) and (9).

Table 1 Generation and load data – two areas system. Area

Generator number

Installed cap. (MW)

Failure rate (occ./h)

Repair rate (occ./h)

Peak load (MW)

1

1 2 3

30 20 10

0.010 0.015 0.028

0.490 0.285 0.372

20 20

1 0.9 0.8 0.7

Load (p.u.)

2

0.6 0.5 0.4 0.3

Southeast System

0.2 0.1

South

0 0

12

24

36

48

60

72

84

96

108

120

Time (h) Fig. 3. First week of the areas loads used in the cases A.1 and A.2.

132

144

156

168

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T.C. Justino et al. / Electrical Power and Energy Systems 42 (2012) 276–284 Table 2 System indices – case A.1. Indices

Sequential

NseqTrad

Pchron

Nseq1St

NseqPrNag

LOLP (bLOLP) EPNS (bEPNS) LOLF (bLOLF) LOLD Sampled years or states (103) Analyzed states (103) Processing time (s) Speedup

1.238 (1.64%) 0.073 (1.81%) 56.110 (1.0%) 0.022 218 2104 3.14 –

1.218 (0.64%) 0.072 (0.86%) 55.500 (1.0%) 0.022 1960 1960 2.92 1.073

1.225 (0.73%) 0.072 (0.98%) 56.213 (1.0%) 0.022 1510 1647 2.46 1.277

1.221 (0.57%) 0.072 (0.76%) 55.275 (1.0%) 0.022 2503 2534 3.78 0.830

1.219 (0.64%) 0.071 (0.86%) 55.857 (1.0%) 0.022 1959 1959 2.92 1.074

Table 3 Areas indices – case A.1. Indices

Sequential

Table 5 Areas indices – case A.2. NseqTrad

Pchron

Area 1 LOLP EPNS LOLF LOLD

0.718 0.024 46.528 0.015

0.710 0.024 46.141 0.015

0.715 0.024 46.530 0.015

Area 2 LOLP EPNS LOLF LOLD

1.237 0.049 56.101 0.022

1.218 0.048 55.491 0.022

1.225 0.048 56.213 0.022

Nseq1St

NseqPrNag

Indices

Sequential

NseqTrad

Pchron

Nseq1St

NseqPrNag

0.710 0.024 45.975 0.015

0.708 0.024 46.212 0.015

Area 1 LOLP EPNS LOLF LOLD

0.763 0.044 54.519 0.014

0.710 0.024 46.141 0.015

0.754 0.043 54.077 0.014

0.754 0.042 53.595 0.014

0.756 0.043 54.20 0.014

1.221 0.048 55.268 0.022

1.219 0.048 55.858 0.022

Area 2 LOLP EPNS LOLF LOLD

1.244 0.066 65.623 0.019

1.218 0.048 55.491 0.022

1.211 0.064 64.925 0.019

1.219 0.064 64.219 0.019

1.226 0.064 64.574 0.019

5.1.2. Case A.2 In this case, the behavior of the system areas loads was represented by a different curve for each area, as shown in Fig. 3. The results are presented in Tables 4 and 5 for the system and its areas, respectively. The Pchron, Nseq1St and NseqPrNag simulations obtained good performances in terms of accuracy of the indices, i.e., the indices estimated by them, mainly the LOLF index, are similar to those computed by the sequential simulation. However, the NseqTrad simulation did not obtain reliability indices with the same accuracy because it is not able to represent two different load patterns for the system areas. Concerning processing time, Pchron and NseqPrNag obtained little speedups in relation to the sequential simulation (1.42 and 1.11, respectively), but the Nseq1St required higher time than the other simulations, as in case A.1. It is worth highlighting that, in this case, the NseqPrNag simulation also obtained a good result for the LOLF indices of system and its areas, since they are similar to those computed by the sequential simulation. However, it is important to notice that the areas load curves have high correlation with the system load curve, preserving the coherence hypothesis used to calculate the LOLF index. 5.2. Power system with four areas This system has four areas and four interconnections, as shown in Fig. 4. Its generation system is composed of 266 generating units

Fig. 4. Power system with four areas.

Table 6 Generation and load data – four areas system. Area number

Plants number

Installed cap. (MW)

Peak load (MW)

1 2 3 4 Total

51 14 12 1 78

20645.6 6111.6 9934 24.6 36715.8

20000 6300 9900 15 36215

Table 4 System indices – case A.2. Indices

Sequential

NseqTrad

Pchron

Nseq1St

NseqPrNag

LOLP (bLOLP) EPNS (bEPNS) LOLF (bLOLF) LOLD Sampled years or states (103) Analyzed states (103) Processing time (s) Speedup

1.245 (1.58%) 0.109 (1.61%) 65.623 (1.0%) 0.019 212 2046 3.28 –

1.218 (0.64%) 0.072 (0.86%) 55.500 (1.0%) 0.022 1960 1960 2.93 1.12

1.211 (0.76%) 0.106 (1.41%) 64.927 (1.0%) 0.019 1410 1516 2.31 1.42

1.219 (0.62%) 0.106 (1.15%) 64.234 (1.0%) 0.019 2108 2134 4.32 0.76

1.226 (0.70%) 0.107 (1.31%) 64.597 (1.0%) 0.019 1628 1628 2.96 1.11

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1 0.9 0.8

Load (p.u.)

0.7 0.6 0.5 0.4 0.3 0.2

Southeast

South

Northeast

0.1

North

System

0 0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

Time (h) Fig. 5. First week of the areas loads used in the cases B.1, B.2 and B.3.

Table 7 System indices – case B.1.

Table 8 Area indices – case B.1.

Indices

Sequential

NseqTrad

Pchron

Nseq1St

NseqPrNag

Indices

Sequential

NseqTrad

Pchron

Nseq1St

NseqPrNag

LOLP (bLOLP)

2.166 (2.52%) 12.287 (3.07%) 175.0 (1.88%) 0.012 9

2.176 (1.79%) 12.147 (2.28%) 179.007 (1.88%) 0.012 140,876

2.149 (1.66%) 11.907 (2.11%) 174.44 (1.88%) 0.012 165,898

2.158 (1.44%) 11.894 (1.83%) 176.519 (1.88%) 0.012 219,154

2.105 (1.78%) 11.777 (2.27%) 172.547 (1.88%) 0.012 147,015

Area 1 LOLP EPNS LOLF LOLD

2.107 6.719 172.778 0.012

2.132 6.649 177.611 0.012

2.112 6.536 172.121 0.012

2.110 6.508 173.244 0.012

2.061 6.441 171.194 0.012

121,382

140,876

178,155

223,883

147,015

Area 2 LOLP EPNS LOLF LOLD

2.107 2.116 172.778 0.012

2.132 2.095 177.611 0.012

2.112 2.059 172.121 0.012

2.110 2.050 173.244 0.012

2.061 2.029 171.194 0.012

14.43

17.96

22.54

28.69

18.80

Area 3 LOLP EPNS LOLF LOLD

2.163 3.447 174.667 0.012

2.174 3.397 178.664 0.012

2.146 3.308 174.093 0.012

2.155 3.331 176.256 0.012

2.101 3.302 172.165 0.012

Area 4 LOLP EPNS LOLF LOLD

2.108 0.005 172.778 0.012

2.132 0.005 177.309 0.012

2.110 0.005 171.747 0.012

2.113 0.005 173.490 0.012

2.064 0.005 171.337 0.012

EPNS (bEPNS) LOLF (bLOLF) LOLD Sampled years or states Analyzed states Processing time (s)

which are distributed in 78 plants. The installed capacity and peak load of each area are presented in Table 6. This test system is derived from a configuration of the Brazilian Power System with some changes made in order to evaluate the performance of the implemented methods, mainly regarding the estimation of the LOLF index. Further details about this system can be found in [20]. The areas load curves were generated by 52 repetitions of the weekly peak load of the southeast, south, northeast and north areas of Brazilian system. These curves, which represent, respectively, the loads of the areas 1–4, are illustrated in Fig. 5. For this system, three study cases were analyzed. The convergence criterion for the analyzed cases was the coefficient of variation of the system LOLF index (bLOLF) less or equal to 2%.

5.2.1. Case B.1 In this case, the behavior of the system and its areas loads were represented by a single load curve, the system equivalent curve, shown in Fig. 5. The NseqTrad simulation used the Markov load model aggregated in 164 levels to represent this curve. Tables 7 and 8 present the results for the system and the areas, respectively. The reliability indices estimated by the NseqTrad, Pchron, Nseq1St and NseqPrNag simulations are similar to those obtained

Table 9 System indices – case B.2. Indices

Sequential

Pchron

Nseq1St

NseqPrNag

LOLP (bLOLP) EPNS (bEPNS) LOLF (bLOLF) LOLD Sampled years or states Analyzed states Processing time (min) Speedup

1.154 (2.60%) 30.404 (1.57%) 93.742 (2.0%) 0.012 31 419,509 1.52 –

1.134 (1.73%) 30.063 (2.57%) 91.734 (2.0%) 0.012 290,692 302,976 0.78 1.95

1.149 (1.51%) 30.821 (2.24%) 93.997 (2.0%) 0.012 376,931 381,261 0.87 1.75

1.129 (1.76%) 29.621 (2.62%) 86.924 (2.0%) 0.013 282,904 282,904 0.50 3.04

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T.C. Justino et al. / Electrical Power and Energy Systems 42 (2012) 276–284 Table 10 Area indices – case B.2.

Table 13 Area indices – case B.3.

Indices

Sequential

Pchron

Nseq1St

NseqPrNag

Indices

Sequential

Pchron

Nseq1St

NseqPrNag

Area 1 LOLP EPNS LOLF LOLD

0.896 16.726 81.774 0.011

0.891 16.557 77.808 0.011

0.898 16.992 76.498 0.012

0.890 16.311 81.039 0.011

Area 1 LOLP EPNS LOLF LOLD

0.898 118.229 80.933 0.011

0.905 117.530 79.895 0.11

0.912 118.097 79.903 0.011

0.887 113.310 78.762 0.011

Area 2 LOLP EPNS LOLF LOLD

0.896 4.726 81.774 0.011

0.891 4.681 77.808 0.011

0.898 4.802 76.498 0.012

0.890 4.612 81.039 0.011

Area 2 LOLP EPNS LOLF LOLD

0.898 38.502 80.933 0.011

0.905 38.275 79.895 0.11

0.912 38.459 79.903 0.011

0.887 36.90 78.762 0.011

Area 3 LOLP EPNS LOLF LOLD

1.153 8.918 93.645 0.012

1.134 8.812 91.734 0.012

1.148 9.015 93.997 0.012

1.128 8.686 86.853 0.013

Area 3 LOLP EPNS LOLF LOLD

0.984 60.457 85.40 0.012

0.984 59.974 86.371 0.11

0.992 60.328 86.780 0.011

0.950 57.925 80.659 0.012

Area 4 LOLP EPNS LOLF LOLD

0.906 0.033 82.774 0.011

0.903 0.012 78.740 0.011

0.916 0.013 78.502 0.012

0.901 0.012 81.993 0.011

Area 4 LOLP EPNS LOLF LOLD

0.897 0.089 80.933 0.011

0.906 0.089 79.926 0.11

0.912 0.089 79.857 0.011

0.886 0.086 78.553 0.011

Table 11 Confidence intervals of system indices – sequential simulation – case B.2. Indices LOLP EPNS LOLF

1.154 30.404 93.742

95% Confidence level

99% Confidence level

1.095 29.468 90.086

1.064 28.972 88.146

1.213 31.340 97.398

Table 14 Confidence intervals of system indices – sequential simulation – case B.3. Indices

1.244 31.836 99.338

LOLP EPNS LOLF

by the sequential simulation, especially for the LOLF index. However, their performance concerning processing time is not so good because they required higher time than the sequential simulation. This is due to the same reason described in Case A.1, i.e., the convergence of the LOLF index is slower in non-sequential methods and faster in the sequential simulation. It can be noticed that the LOLF index for this case is extremely high, what accentuates even more this behavior. Again the LOLF indices estimated by the NseqPrNag simulation are similar to those estimated by sequential simulation. This was expected because the assumption of system coherence is being respected, since the use of one load curve to represent the system load and its areas loads implies in considering them fully correlated. 5.2.2. Case B.2 In this case, different load curves were used to represent the areas loads of system, as shown in Fig. 5. The results of this case are presented in Tables 9 and 10 for the system and the areas, respectively. The indices calculated by the NseqTrad simulation are not presented because, as shown in case A.2, it is not able to represent different load patterns of the system. In terms of accuracy of the LOLP and EPNS indices, all the simulations obtained similar accuracy as the sequential simulation. In

1.007 217.278 87.60

95% Confidence lEVEL

99% Confidence level

0.962 216.596 84.235

0.938 216.235 82.449

1.052 217.959 90.965

1.076 218.321 92.751

terms of processing time, they spent less time than the sequential simulation, the NseqPrNag being the fastest simulation (speed gain higher than 3 times). However, the LOLF index estimated by NseqPrNag is not statistically similar to the LOLF calculated by the other simulations. This fact can be confirmed analyzing the confidence intervals of the indices estimated by the sequential simulation, shown in Table 11. The LOLF index calculated by NseqPrNag is not within any of the confidence intervals. The explanation for this bad performance of the NseqPrNag simulation is that the load curve of area 3 (northeast), which causes most of the load curtailments of the system (see Table 10), is not highly correlated with the system load curve. Since the transition rates for obtaining the load related part of the LOLF index in this simulation is based on the system load curve, the adoption of test function (9) leads to wrong estimates of the LOLF index. This problem is even intensified because the load curve of area 3 is heavier than the system curve, as shown in Fig. 5. 5.2.3. Case B.3 In order to confirm the conclusion about the restriction of applying NseqPrNag to evaluate the LOLF index, a test with the ex-

Table 12 System indices – case B.3. Indices

Sequential

Pchron

Nseq1St

NseqPrNag

LOLP (bLOLP) EPNS (bEPNS) LOLF (bLOLF) LOLD Sampled years or states Analyzed states Processing time (min)

1.007 (2.27%) 217.278 (0.16%) 87.60 (1.96%) 0.011 15 202,640 0.44

1.006 (1.75%) 215.868 (2.28%) 88.559 (1.96%) 0.011 320,422 330,645 0.77

1.015 (1.55%) 216.973 (2.02%) 89.432 (1.96%) 0.011 405,113 409,223 1.01

0.974 (1.88%) 208.221 (2.45%) 82.747 (1.96%) 0.012 287,963 287,963 0.60

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change of the areas load curves was performed in this case. The objective is to analyze the performance of the NseqPrNag simulation if the load curve of the less reliable area is more correlated with the system curve, for example, the south area curve. Thus, in this case, the curves of the south and northeast areas represent the loads of areas 3 and 2, respectively. Tables 12 and 13 present the results for the system and the areas, respectively, while Table 14 shows the confidence intervals of the indices estimated by the sequential simulation. In this case, the indices values obtained by the Pchron, Nseq1St and NseqPrNag simulations are equivalent to those estimated by the sequential simulation, including the LOLF index obtained by the NseqPrNag simulation, which belongs to the 99% confidence interval of the LOLF estimated by the sequential simulation. As expected, this good result happened because the load curve of the south area, which represents the load of the less reliable area 3, is more correlated with the system load curve than the northeast area curve. 6. Conclusion This paper presented multi-area reliability evaluation methods that represent the proper load curve of each area and calculates the system and areas reliability indices, including the F&D indices. The Pchron and Nseq1St obtain accurate reliability indices with no restriction in relation to the system areas load curves. The NseqPrNag combines the use of the non-aggregated Markov load model and the conditional probability method to estimate the LOLF index. Based on published papers, it was suggested that representing different load patterns for the system areas with the conditional probability method would not obtain good estimates of the LOLF index. However, the studies performed in this paper showed some exceptions. It was observed that it is not necessary that the system loads are fully correlated to determine the LOLF index using the conditional probability method. Instead, it is necessary that the load curve of the area that contributes most to the load curtailments have a high correlation with the system load curve. When this is the case, the indices obtained are accurate and the simulation is more computationally efficient than the other methods. Therefore, this paper attended its proposal of presenting a practical solution for multi-area reliability evaluation with different load curve for each area, what produces more accurate indices when planning the expansion of generation system capacity and interconnections.

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