Analytical model for fast reliability evaluation of composite generation and transmission system based on sequential Monte Carlo simulation

Electrical Power and Energy Systems 109 (2019) 548–557

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Analytical model for fast reliability evaluation of composite generation and transmission system based on sequential Monte Carlo simulation

T

⁎

Lvbin Penga, Bo Hua, , Kaigui Xiea, Heng-Ming Taib, Kaveh Ashenayib a b

State Key Laboratory of Power Transmission Equipment & System Security and New Technology at Chongqing University, Chongqing 400044, PR China Department of Electrical and Computer Engineering, University of Tulsa, Tulsa, OK 74104, USA

A R T I C LE I N FO

A B S T R A C T

Keywords: Power system reliability evaluation Sequential Monte Carlo simulation Analytical model Sensitivity analysis Power system planning

This paper proposes an analytical reliability evaluation (RE) model for composite generation and transmission system (CGTS) based on the sequential Monte Carlo simulation (MCS) method. The main idea is to partition the chronological system state sequence produced by sequential MCS into a set of mutually exclusive events based on the law of total probability. An analytical model is developed by extracting the component reliability parameters (CRPs) from system reliability index calculation formula of each mutually exclusive event. Once the proposed model is established, system reliability indices with variable CRPs can be obtained directly, rather than repeatedly performing the time-consuming sequential MCS process. In addition, sensitivity coefficients of system reliability indices with respect to CRPs can be obtained using the developed analytical model. A 4-unit generation system, the CGTS of IEEE-RTS system and a 91-bus power system are used for case study to validate the correctness and effectiveness of the proposed analytical RE model.

1. Introduction Reliability evaluation (RE) of the composite generation and transmission system (CGTS) is very useful in the power system planning to ensure acceptable system reliability level. In the RE of CGTS, a large number of system state analyses are commonly needed, and each state analysis involves network analysis, power flow analysis, generation rescheduling, load curtailment, bus voltage correction, etc. [1–6]. Calculation complexity of the RE of CGTS is enormous. Fast and accurate RE of CGTS is a common concern of power system reliability researchers [7–16]. There are two fundamental approaches for RE of the CGTS: the analytical enumeration method and the Monte Carlo simulation (MCS) method. Analytical enumeration method has a clear physical concept and the system state enumeration process is relatively simple. However, when the system size is large, the “dimensional disaster” problem may arise [1,2]. On the other hand, the MCS method can be categorized as the non-sequential MCS method and sequential MCS method. Although the non-sequential MCS method avoids the “dimensional disaster” problem, it cannot be used by itself to calculate the actual frequency index [3]. Compared with the non-sequential MCS method and analytical enumeration method, the sequential MCS method has the following advantages [4–6]. First, it is not too difficult to consider the effect of temporal correlation of the load and the power output of ⁎

renewable energy units on system reliability. Second, sequential MCS method can be used to calculate the actual frequency index in a simple way. Third, the statistical probability distributions of reliability indices can be calculated in addition to the expected values. However, the sequential MCS method puts great demands on computation speed and memory capacity for its time-consuming simulation. To improve the computational efficiency of the sequential MCS method, various techniques have been proposed. They are the variance reduction technique [7,8], the pseudo-chronological simulation [9] and the quasi-sequential simulation [10], the cross-entropy method [11,12], accelerated state screening and evaluation [13–15], and parallel computing [16]. All of the above techniques reduce the computation time of RE of the CGTS to some degree. But the inherent characteristic of MCS limits its further reduction. If the component reliability parameters (CRPs) are changed, reevaluation of the CGTS is required to obtain the new reliability indices. For example, the sensitivity analysis of system reliability with respect to the component reliability requires multiple REs of CGTS with variable CRPs [17–19]. Reliability optimization or reliability design of the CGTS also needs to perform the RE of CGTS many times [20–22]. Although the time to perform one RE is reduced by the above techniques, applications requiring multiple REs still demand a large amount of computation time. An efficient solution would be avoiding the repeated REs using sequential MCS. To achieve this goal, a potential solution may be to

Corresponding author. E-mail addresses: [email protected] (B. Hu), [email protected] (H.-M. Tai), [email protected] (K. Ashenayi).

https://doi.org/10.1016/j.ijepes.2019.02.039 Received 24 June 2018; Received in revised form 15 February 2019; Accepted 21 February 2019 0142-0615/ © 2019 Published by Elsevier Ltd.

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Nomenclature RE CGTS CRP MCS LOLP LOLF EENS Al λl μl n m Ttotal td(s) Ψ

Ld(s) T P(Fj) P(E|Fj)

reliability evaluation composite generation and transmission system component reliability parameter Monte Carlo simulation loss of load probability loss of load frequency expected energy not supplied availability of the component l failure rate of the component l repair rate of the component l number of components in a CGTS number of components of interest in a CGTS total simulation time duration of state s set of all system states with loss of load

φju φjd Njd Kj Jj,k Jj,O Lj β

load curtailment of the system state s length of the given time interval (usually 8760 h) probability of the composite state Fj conditional probability of event E given the composite state Fj set of Components in the up state of the Fj set of Components in the down state of the Fj the number of components in the down state of the Fj coefficient of the analytical model of system LOLP coefficient of the analytical model of system LOLF corresponding to component k in φjd coefficient of the analytical model of system LOLF corresponding to components not in φjd coefficient of the analytical model of system EENS variance coefficient threshold for LOLP, LOLF, and EENS indices of multiple MCSs

fixed in the analytical model.

construct an analytical model for RE of the CGTS based on the sequential MCS method. Once the analytical model is constructed, system reliability indices with variable CRPs can be obtained directly, without resorting to the time-consuming sequential MCS process. To the best of our knowledge, there are no reports about the analytical model of RE based on the sequential MCS. The analytical model research is mainly based on the analytical enumeration method [23,24]. The sequential MCS method uses random numbers to simulate the sequential operating process of the CGTS, and obtains the system reliability indices through statistical analysis of the chronological system state sequence. Modeling based on the analytical enumeration method does not apply to the sequential MCS method, because the reliability index calculation formula of the sequential MCS does not contain any CRPs. In this paper, analytical RE model of the CGTS based on the sequential MCS is developed. The aim is to obtain the system reliability indices with variable CRPs in a direct and quick manner, instead of repeatedly performing the time-consuming sequential MCS. The proposed analytical model can be applied to calculate the sensitivity coefficients of system reliability indices with respect to CRPs. A 4-unit generation system, the CGTS of IEEE-RTS and a 91-bus power system are used in the case study to verify the accuracy and applications of the proposed analytical model.

The components of interest selected to construct the analytical model for RE can be any single component or multiple components. In practice, the selection of components of interest depends on the actual demand in power system planning or reinforcement. For example, assume the task is to improve the reliability of the whole system by decreasing the repair times of generators G1 and G2, then generators G1 and G2 are the components of interest. When performing RE using the proposed analytical model, only the variation of reliability parameters of G1 and G2 is considered. Reliability parameters of other components are assumed to be fixed. Assume that a CGTS contains n components, and m of them are components of interest. The availabilities of m components of interest are A1, A2, … , Am. A typical analytical model for LOLP, LOLF and EENS indices can be expressed as

LOLP = f1 (A1 , A2 , …, Am )

(1)

LOLF = f2 (A1 , A2 , …, Am )

(2)

EENS = f3 (A1 , A2 , …, Am )

(3)

where f1, f2, and f3 are nonlinear algebraic functions of A1, A2, … , Am. 3. Analytical RE model of CGTS based on sequential MCS

2. Analytical model for RE of CGTS

This section describes the process of constructing an analytical RE model based on the sequential MCS of CGTS. The frequently used sequential MCS method, which is also called the state duration sampling approach, is described in Section 3.1. The analytical RE model is developed in Section 3.2. The determination of the coefficients of the analytical model is given in Section 3.3.

Power system RE commonly involves finding the system (or bus) reliability indices based on the known CRPs. The system (or bus) reliability indices can be seen as functions of CRPs if the system operating parameter, electrical parameter, and network structure are fixed [1–3]. For RE of CGTS, widely used reliability indices are the system (or bus) loss of load probability (LOLP), loss of load frequency (LOLF), and expected energy not supplied (EENS) [1–3]. Other indices can be derived from these three indices. The commonly used CRPs include the availability A, failure rate λ and repair rate μ [1–3]. To simplify the description, this paper considers system LOLP, LOLF and EENS indices, and the component availability to introduce the analytical model for the RE of CGTS. The analytical model of the bus indices can be obtained in a similar way. To make the proposed analytical model more in line with the engineering practice, all components in the CGTS are grouped into two categories:

3.1. State duration sampling approach Suppose that each component of the CGTS has two states, i.e., the up state and the down state, and the component state duration is exponentially distributed. Procedure of the state duration sampling approach is described below [3]. (1) Suppose that the initial state of each component is up. (2) Sample a uniformly distributed random number between [0, 1] to determine the duration of each component residing in its present state. Suppose that ri is the sampled random number for the ith component. The state duration of the ith component can be calculated by

(1) Components of interest: reliability parameters appear as variables in the analytical model. (2) Components of no interest: reliability parameters are assumed to be 549

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tdi = −

1 ln ri λi

analytical model using the system LOLP index as an example. Let E be the event that the CGTS is in the state with loss of load, and F1, F2, … , FM be the M composite states of the m components of interest. Thus, P(E) = LOLP and {F1, F2, … , FM} is a partition of the sample space [25]. Based on the law of total probability and the definition of conditional probability [25], Eq. (8) can be rewritten as

(4)

where λi is the transition rate of the ith component. If the ith component is in the up sate, λi is failure rate; if the ith component is in the down state, λi is repair rate. (3) Repeat Step (2) until the total simulation time is equal to or greater than the given time span, and construct the chronological state transition process of each component. (4) Combine the component state transition processes of all components to construct the chronological system state transition process. The system state transition process of three components is shown in Fig. 1. In Fig. 1, numbers 1–8 correspond to 8 different system states. (5) For each system state, the circuit breaker in the branch with faulty components trips to isolate the faulty components. Then check if there is network splitting for the network with no faulty components. If no splitting, do the following analysis for the current network; otherwise, do the same analysis for each sub-network after system splitting. The analysis is to determine if there are problems such as the shortage of generation capacity, overloading of transmission lines or node voltage violation by conducting the generation rescheduling and AC power flow calculation [3]. If any of the above problems occurs, then perform the load curtailment minimization. If the curtailed load is not zero, record the amount of the curtailed load. Under this circumstance, the system state contributes to the unreliability indices. (6) Calculate the system reliability indices. The system LOLP, LOLF and EENS indices can be calculated by [3]:

LOLP =

LOLF =

EENS =

LOLP =

Kj = P (E |Fj ) =

(7)

j

Ttotal

1 P (Fj )

×

(12)

Ag

∏

(1 − Ah ). (13)

h ∈ φjd

+⋯+

+⋯+

∑s ∈ φ td (s ) j

Ttotal

∑s ∈ φ If (s )

∑s ∈ φ td (s ) × Ld (s ) × T 1

j

Ttotal

+⋯+

Ttotal ∑s ∈ φ td (s ) × Ld (s ) × T M

Ttotal

+⋯+

.

+⋯+

∏

(1 − Ah ). (14)

h ∈ φjd

M

∑ j=1

Jj ×

∏

Ag

g ∈ φju

∏

(1 − Ah ). (15)

h ∈ φjd

The analytical model of system EENS index can be expressed as Up Component 1 Up

Down Component 2

Up

Down

∑s ∈ φ td (s ) M

Ttotal

Ag

g ∈ φju

Similarly, the analytical model of system LOLF index can be expressed as

Assume each component of the system has two states, i.e. up state and down state, and component failures are independent of each other. For m components of interest, there are M = 2 m composite states [1–3]. Suppose the M states are denoted by 1st, 2nd, … , Mth of the m components of interest. Assume φj (j = 1, 2, … , M) denotes the set of the system states in set Ψ when the m components of interest are in their jth composite state, then the set Ψ in (5)–(7) can be partitioned into M subsets: φ1, φj, … , φM. Eqs. (5)–(7) can be rewritten as the sum of M terms

+

∑s ∈ φ td (s )

∑ Kj × ∏ j=1

3.2. Analytical model based on sequential MCS

EENS =

td (s )

M

LOLP =

Ttotal

Ttotal

M

Ttotal

Substituting (13) into (11) yields the analytical model of system LOLP index

∑s ∈ ψ td (s ) × Ld (s ) × T

1

∏ g ∈ φju

(6)

∑s ∈ φ If (s )

∑s ∈ φ

is the conditional probability of event E given the Fj. In fact, Kj denotes the loss of load probability of the system given that the composite state of the m components of interest is in the jth state. If the system operating parameter, electrical parameter, and network structure are fixed, P (E|Fj) only depends on the reliability level of n-m components of no interest. In this paper, the reliability parameters of the n-m components of no interest are assumed to be constants. So Kj, j = 1, 2, … , M should be constants. Since component failures are assumed to be independent of each other, P(Fj) can be obtained by

LOLF =

LOLF =

+⋯+

where

P (Fj ) =

Ttotal

1

j

Ttotal

(11)

∑s ∈ ψ If (s )

Ttotal

∑s ∈ φ td (s )

=K1 P (F1)+⋯+Kj P (Fj )+⋯+KM P (FM )

where If(s) is an indicator variable. If the preorder state of state s is a state without loss of load, If(s) = 1; otherwise If(s) = 0.

LOLP =

+⋯+

=P (E |F1) P (F1)+⋯+P (E |Fj ) P (Fj )+⋯+P (E |FM ) P (FM )

(5)

∑s ∈ φ td (s )

1

Ttotal

= P (E ∩ F1)+⋯+P (E ∩ Fj )+⋯+P (E ∩ FM )

∑s ∈ ψ td (s ) Ttotal

∑s ∈ φ td (s )

Component 3

(8)

Down

∑s ∈ φ If (s ) M

Ttotal

∑s ∈ φ td (s ) × Ld (s ) × T j

Ttotal

(9) System

+⋯

Time

(10)

Fig. 1. Chronological component and system state transition process.

The following describes the modeling process of the reliability index 550

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values of LOLP(i), LOLF(i), EENS(i), Ttotal, be zeros, and flag = 0. (3) Assume all the components are in the up state. Sample the uniformly distributed random number between [0, 1] to determine the duration of each component residing in the up state according to Eq. (4). (4) Suppose that the current system state is at state s. The duration of state s depends on the shortest duration of the present state for all components and the load. If the shortest duration occurs at the component l and its duration is ts, let Ttotal = Ttotal + ts, and flag = 0; If the shortest duration occurs at the load and its duration is ts, let Ttotal = Ttotal + ts, and flag = 1. (5) Determine whether the state s is a failure state with loss of load [3]. If yes, go to (6); otherwise, go to (8). (6) Check whether there is a component whose state changed from the preorder system state to the current system state. If yes, let ltr be the index of this component; otherwise, ltr = 0. (7) Let td(s) = ts and calculate the indicator variable If(s) and loss of load Ld(s) of the state s. According to the composite state of m components of interest in state s, determine the subset φj, to which the state s belongs, for j = 1, 2, … , M. Let Kj(i) = Kj(i) + td(s), Jj(i) = Jj(i) + If(s), and Lj(i) = Lj(i) + td(s)Ld(s)T. If ltr ∈ φjd and If(s) = 1, let k be the index of component ltr in φjd and Jj,k(i) = Jj,k(i) + td(s). (8) Update LOLP(i), LOLF(i) and EENS(i). Calculate the variance coefficient of EENS(i). If the variance coefficient is less than the pre-specified value, let

M

EENS =

∑ Lj × ∏ j=1

g ∈ φju

∏

Ag

(1 − Ah ). (16)

h ∈ φjd

The Jj in (15) denotes, Njd

J j = J j, O +

∑ (Jj,k × μk ),

(17a)

k=1

where Jj,O and Jj,k are coefficients of (15) whereas μk is a variable. Jj,k and Jj,O are obtained as,

J j, k =

J j, O =

∑s ∈ φ td, k (s ) j

×

Ttotal ∑s ∈ φ If (s ) j

Ttotal

×

1 , P (Fj ) 1 − P (Fj )

k = 1, 2, …, Njd

(17b)

Njd

∑ (Jj,k × μk ) k=1

(17c)

and Njd is the number of components in the down state of the composite state Fj. The td,k(s) is duration of current system state s on condition that the state of kth component in φjd changes from the preorder system state to the state s and If(s) = 1. The coefficients in (16) can be obtained by

Lj =

∑s ∈ φ td (s ) × Ld (s ) × T j

Ttotal

×

1 . P (Fj )

(18)

It can be seen from (14), (15), and (16) that system reliability indices are explicit functions of the reliability parameters of components of interest. If reliability indices need to be reevaluated due to the change of reliability parameters of components of interest, we can obtain these reliability indices directly by substituting reliability parameters of components of interest into (14), (15), and (16), rather than performing the time-consuming sequential MC simulation.

Njd

J j(,iO) = J j(i) −

∑ (J j(,ik) × μk ), k=1

and go to (10); otherwise, go to (9). (9) If flag = 0, change the present state of the component l, then sample the uniformly distributed random number between [0, 1] to determine the duration. Go to 4). If flag = 1, go to (4) directly. (10) Let N = N + 1 and

3.3. Coefficients of the analytical model This section focuses on obtaining the coefficients Kj, Jj, and Lj of the proposed analytical model for the system LOLP, LOLF, and EENS indices. At least one sequential MCS has to be performed to obtain these coefficients, according to (12), (17) and (18). For multiple repetitive MCSs, the reliability results of CGTS are not fixed but fluctuate randomly because the durations of the component or the system state are randomly sampled in the sequential MCS [3]. Fig. 2 shows the histogram of annual system EENS index of the IEEERTS [26] from 100 sequential MCS runs. Data of the IEEE-RTS and load curtailment strategy are described in Section 5. It can be seen from Fig. 2 that the distribution of EENS index approximates the normal distribution. If the coefficient Lj (j = 1, 2, … , M) is determined by only one MCS, Lj vary randomly, which conflicts with the assumption of the proposed analytical model that Lj should remain the same. Averaging over results from multiple MCSs alleviates the effect of randomness. Then the average of reliability indices would fluctuate within a smaller range, which is governed by the variance coefficient of reliability indices obtained from multiple MCSs. A variance coefficient threshold is set to ensure that the average varies within an acceptable scope. If the variance coefficient is less than the pre-specified threshold, the average can be used to replace the distribution so that the proposed analytical model holds. Let β be the variance coefficient threshold for LOLP, LOLF, and EENS indices of the repetitive MCSs. A chronological load model is used in the sequential MCS. Procedure for obtaining the coefficients Kj, Jj, and Lj, j = 1, 2, … , M, is given below.

Kj − sum = Kj − sum +

K j(i) Ttotal P (Fj )

,

j = 1, 2, …, M

(19)

(i)

J j, O ⎧J ⎪ j, O − sum = Jj, O − sum + Ttotal P (Fj) , j = 1, 2, …, M ⎨ J (j,ik) ⎪ Jj, k − sum = Jj, k − sum + Ttotal P (Fj) , k = 1, 2, …, Njd ⎩

(1) Let N = 0, i = 1, and all the parameters ltr, Kj-sum, Jj,k-sum, Jj,O-sum, and Lj-sum, for j = 1, 2, … , M, are zeros. (2) Let Kj(i) = 0, Jj(i) = 0, and Lj(i) = 0, for j = 1, 2, … , M. Let the initial

(20)

Fig. 2. Histogram of annual system EENS index of the IEEE-RTS from 100 sequential MCS runs. 551

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Lj − sum = Lj − sum +

Lj(i) Ttotal P (Fj )

,

j = 1, 2, …, M ,

various reliability test systems. Once the analytical model for each reliability index is established, RE is performed by substituting CRPs into the analytical model. It can be seen from (14), (15), and (16) that the evaluation is very efficient. For example, if 100 sets of CRPs of components of interest are required for RE, then 100 sequential MCSs must be executed when using the conventional MCS method. However, the proposed method only performs N sequential MCSs to establish the analytical model. Thus, the computation time can be reduced by nearly (100-N) times of the time required for a single sequential MCS.

(21)

(11) Calculate variance coefficients of LOLP(i), LOLF(i) and EENS(i) indices for the repetitive MCSs. If all of them are less than β, let Kj = Kj-sum/N, Jj,k = Jj,k-sum/N, Jj,O = Jj,O-sum/N and Lj = Lj-sum/N, j = 1, 2, … , M, and terminate; otherwise, i = i + 1, go to (2). The computational efficiency of the above procedure can be improved by applying the RE efficiency acceleration techniques in [7–16]. After coefficients of the analytical model is determined, RE is performed straightforward without any MCS. Reliability evaluation using the conventional MCS method is required to undertake sequential MCS for each set of CRPs. In contrast, the proposed method only undertakes N sequential MCSs upfront to construct the analytical model. N denotes the required number of sequential MCS runs to determine the coefficients of the analytical model. We find that N is no more than six after a number of tests for different combinations of components of interest on

4. Application to sensitivity analysis The proposed analytical model can be applied to facilitate the sensitivity analysis of system reliability with respect to CRPs. The sensitivity coefficient provides the implication that the larger the sensitivity, the greater the impact of component failure is on system reliability. This feature can be used to identify the weak link of the system [1–3,23,24]. The sensitivity coefficient in this paper is defined by taking the

Bus 17 L30

Bus 18 Bus 21 L33

Bus 22

L38

L32 L31

Bus 23

L28

L29

L36

L34 L35

Bus 16

L37

Bus 19

230 kV

Bus 20

L22

L24 L25

L23 L26

Bus 15

Bus 14

Bus 13

L21

L18 L27

L19

L20

Bus 24

Bus 12

Bus 11

L16 L7

L14

L17

L15

Bus 3

Bus 10

Bus 9

L6

L10

L8 Bus 5

138 kV

L2

Bus 6

Cable

Bus 4

L13

L9

Bus 8

L12 L4

L3

L5

L11

Cable Bus 1

Bus 2

L1

Fig. 3. Single line diagram of the IEEE-RTS. 552

Bus 7

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partial derivative of system reliability index with respect to the CRP. It can be derived easily because the proposed analytical models of system reliability indices, as shown in (14), (15), (16), are expressed as the polynomial sum of the reliability parameters of m components of interest. Analytical expressions of sensitivity coefficients for system LOLP, LOLF, and EENS indices with respect to the availability of the component l are given in (22), (23), and (24), respectively.

∂EENS = ∂μl

∑ Kj ×

∂LOLF = ∂Al

∑ Jj×

∂EENS = ∂Al

∑ Lj ×

j=1

∏

∏

Ag

g ∈ φju and g ≠ l

∏

∏

Ag

g ∈ φju and g ≠ l

(22)

(1 − Ah ) × I (l) (23)

h ∈ φjd and h ≠ l

M j=1

∏

∏

Ag

g ∈ φju and g ≠ l

(24)

h ∈ φjd and h ≠ l

⎧ I (l) = 1 l ∈ φju ⎨ I (l) = −1 l ∈ φjd . ⎩

(25)

μl . λl + μl

(26)

Substituting (26) into (14), (15), and (16) yields the analytical expressions of system LOLP, LOLF and EENS indices in terms of the component failure rate and repair rate. Analytical expressions of sensitivity coefficients for system LOLP, LOLF, and EENS indices with respect to the failure rate of the component l can be derived as

5.2. A 4-unit generation system

M

∂LOLP = − ∑ Kj × ∂λl j=1

∏

Ag

g ∈ φju and g ≠ l

∏ h ∈ φjd and h ≠ l

μl (1 − Ah ) × (λl + μl )2

A simple 4-unit generation system is investigated first to show how the proposed method works. The single line diagram of a 4-unit generation system is shown in Fig. 4. Capacities and reliability parameters of the units are given in Table 1. For simplicity, the system load is set as 80 MW and unchanged in the evaluation. Take the Eq. (14) which is the analytical model of LOLP index as an example to show the finding of model coefficients and validation. Assume that units G1 and G4 are the components of interest. Suppose that each component has two states (i.e., up and down states). The up state is denoted by 0 and the down state by 1. For two components of interest, there are 4 composite states. Table 2 lists such composite states. In the following, we describe the procedure to determine the value of coefficients Kj in (14) by the sequential MCS. The coefficients K1 can be obtained by substituting (13) into (12). In finding the coefficients, the availabilities of G1 and G4 are set as A1 = 0.95 and A4 = 0.924, whereas the initial availabilities are A1 = 0.9747 and A4 = 0.9604 as shown in Table 1. At least one sequential MCS has to be performed to obtain the coefficients K1. The variance coefficient threshold β for multiple MCSs is 0.01. For the ith MCS, let tj = 0, j = 1, 2, 3, 4. For any system state s with loss of load generated by the sequential MCS, determine the subset φj,

(27)

× I (l) M

∂LOLF = − ∑ Jj× ∂λl j=1

∏

Ag

g ∈ φju and g ≠ l

∏

(1 − Ah ) ×

h ∈ φjd and h ≠ l

μl (λl + μl )2 (28)

× I (l) M

∂EENS = − ∑ Lj × ∂λl j=1

∏

Ag

g ∈ φju and g ≠ l

∏

(1 − Ah ) ×

h ∈ φjd and h ≠ l

μl (λl + μl )2 (29)

× I (l)

Analytical expressions of sensitivity coefficients for system LOLP, LOLF, and EENS indices with respect to the repair rate of the component l can be derived as M

∑ Kj × j=1

∏

Ag

g ∈ φju and g ≠ l

∏

(1 − Ah ) ×

h ∈ φjd and h ≠ l

λl (λl + μl )2 (30)

× I (l) ∂LOLF

(32)

A 4-unit generation system, the IEEE-RTS and a 91-bus practical system are selected to conduct the case study. The IEEE-RTS consists of 32 generating units, 33 transmission lines and 5 transformers with a peak load of 2850 MW. Refer to [26] for detailed system data. The single line diagram of the IEEE-RTS is shown in Fig. 3. The proposed method is also applied to a 91-bus system derived from the power system at the Chuanyu region of China. The 91-bus Chuanyu system (CS) consists of 64 generating units and 173 transmission lines (500 kV or 220 kV) with a peak load of 9732 MW [24]. The installed capacity is 10,684 MW. Suppose that the operating and repair state durations of each component are exponentially distributed. Load models of both test systems are based on the annual load curve of IEEE-RTS [26]. The DC flow-based optimal load shedding approach [2,3] is used to determine the load curtailments for the IEEE-RTS and the AC flow-based optimal load shedding approach for the Chuanyu system.

Let λl and μl be the failure rate and repair rate of the component l, respectively. We have [1–3]

∂LOLP = ∂μl

h ∈ φjd and h ≠ l

λl (λl + μl )2

5.1. Test system and data

(1 − Ah ) × I (l)

where Al is the availability of the component l and I(l) is an indicator variable

Al =

g ∈ φju and g ≠ l

(1 − Ah ) ×

In this section, case studies are presented to demonstrate the correctness and merit of the proposed analytical model. RE indices obtained from the analytical model are compared with that of the sequential MCS method. Parameters that affect the accuracy of the analytical model are investigated. Applications to the sensitivity analysis are also evaluated. Simulations are performed in MATLAB (R2016a).

M j=1

∏

Ag

5. Case study

(1 − Ah ) × I (l)

h ∈ φjd and h ≠ l

j=1

∏

× I (l)

M

∂LOLP = ∂Al

M

∑ Lj ×

M

⎧ ∂μ = ∑ j = 1 Jj × ∏g ∈ φju ⎪ l ⎪ I (l) = 1

and g ≠ l

Ag ∏h ∈ φ

jd and h ≠ l

(1 − Ah ) ×

λl , (λl + μl )2

⎨ ∂LOLF = ∑M Jj × ∏ A ∏h ∈ φ and h ≠ l (1 − Ah ) × j=1 g ∈ φju and g ≠ l g jd ⎪ ∂μl ⎪ + Jj, k × ∏g ∈ φ Ag ∏h ∈ φ (1 − Ah ), I (l) = −1 ju jd ⎩

G1

G2

G3

G4

−λl (λl + μl )2

Load Fig. 4. Single line diagram of a 4-unit generation system.

(31) 553

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bus 16), G18 (400 MW unit in bus 18), L13, L18 and L21. In the modeling process, the availabilities of G13, G16 and G18 are set as 0.93 and the availabilities of L13, L18 and L21 are set as 0.9994. The variance coefficient β is 0.01. Analytical models in the case study and respective components of interest are listed below:

Table 1 Capacities and reliability parameters of the 4-unit generation system. Units

Capacity (MW)

λ (Occ./yr)

μ (Occ./yr)

A

G1 G2 G3 G4

20 30 30 40

4 6 6 8

154 195 195 194

0.9747 0.9701 0.9701 0.9604

Model-1: Model-2: Model-3: Model-4: Model-5: Model-6: Model-7: Model-8: Model-9:

Table 2 Composite states of components of interest of a 4-unit generation system. Composite states

G1 state

G4 state

F1 F2 F3 F4

1 0 1 0

0 1 1 0

After the above nine analytical models are established, the reliability of IEEE-RTS can be evaluated by substituting the availabilities of components of interest into the corresponding models. Accuracy of the proposed analytical model can be examined by comparing the RE results with that of the sequential MCS method. In the test, the values of the availability of G13, G16, G18, L13, L18, and L21 are 0.95, 0.96, 0.88, 0.9995, 0.9995, and 0.9993, respectively. These are the initial availability values given in the IEEE-RTS. For the CS, the components of interest are G1 (550 MW unit in bus 1, the largest capacity in CS), G23 (100 MW unit in bus 23), G33 (175 MW unit in bus 33), G52 (45 MW unit in bus 52), L1, L4, L10, and L13. In the modeling process, the availabilities of G1, G23, G33, and G52 are set as 0.998 and the availabilities of L1, L4, L10, and L13 are set as 0.999. Analytical models in this case study and the respective components of interest are listed below:

to which the state s belongs, and let tj = tj + td(s). For example, assume states of G1-G4 in the system state s are 1, 0, 1 and 0, respectively. Since G1 is in the down state and G4 in the up state, the composite state of G1 and G4 is F1 from Table 2. The state s belongs to subset φ1, and let t1 = t1 + td(s). When the convergence criterion of MCS is met, K1 obtained by ith MCS can be rewritten as

K1(i) =

t1 1 × . Ttotal (1 − A1 ) A 4

(33)

After performing five sequential MCSs, the variance coefficient of LOLP is 0.00678. Similarly, the variance coefficients of LOLF and EENS indices are 0.00562 and 0.00953, respectively. Since variance coefficients of all three reliability indices are less than β, the convergence criterion of multiple MCSs is met. Table 3 lists the values of K1(i), K2(i), K3(i), K4(i) obtained from five sequential MCSs. By averaging over Kj(i) from five MCS runs, it can be seen from Table 3 that the coefficients K1, K2, K3, and K4 are 0.0581, 0.0577, 1.003, and 0.0009, respectively. Then the analytical model for the system LOLP index can be expressed as

Model-10: G1, G23, G33, and G52 Model-11: G1, G23, G33, G52, L1, L4, L10, and L13 In the test, the values of availability of G1, G23, G33, G52, L1, L4, L10, and L13 are 0.9980, 0.9936, 0.9977, 0.9985, 0.9973, 0.9991, 0.9991, and 0.9991, respectively. These are the initial availability values given in the CS. Table 5 shows the RE results of the annual system LOLP, LOLF, and EENS indices of IEEE-RTS obtained from nine different analytical models. Values of LOLP, LOLF, and EENS in the row of SMCS represent the mean of reliability indices obtained from 100 MCS runs. It is observed from Table 5 that the proposed analytical model is capable of finding annual system reliability indices with high accuracy (less than 2% error) no matter what the components of interest are generating units, transmission lines, or both. Table 6 shows the annual system LOLP, LOLF, and EENS indices of the CS obtained from two different analytical models. It is observed from Table 6 that the proposed method is able to find the annual system reliability indices with high accuracy. The errors of reliability indices obtained by the proposed method are within 2% from that by the sequential MCS method. Once the analytical model is established, the system reliability

LOLP = 0.0581(1 − A1 ) A 4 + 0.0577A1 (1 − A 4 ) + 1.0031(1 − A1 )(1 − A 4 ) + 0.0009A1 A 4 .

(34)

For accuracy validation, Table 4 shows the RE results of the analytical model and that of the sequential MCS method. Substituting the availabilities of G1 and G4 from Table 1 (A1 = 0.9747, A4 = 0.9604) into (34) yields the system LOLP index of 0.00551. The RE results of the sequential MCS method are included for comparison. Values of LOLP, LOLF, and EENS in the row of “SMCS” represent the average of reliability indices obtained from 100 MCS runs. They can be considered as the true values of system reliability indices. The term ERLOLP% is defined as

ERLOLP \% =

|LOLPAM − LOLPSMCS | × 100\% LOLPSMCS

G13 G16 G18 G13, G16 and G18 L13 L18 L21 L13, L18 and L21 G13, G16, G18, L13, L18 and L21

(35)

where LOLPAM and LOLPSMCS are system LOLP indices calculated by the proposed analytical model and the sequential MCS method, respectively. ERLOLF% and EREENS% are defined similarly. It can be seen from Table 4 that the reliability indices obtained by the proposed method are in close agreement with that by the sequential MCS method.

Table 3 Coefficients Kj(i) and system LOLP index obtained by five sequential MCS for the 4-unit generation system.

5.3. Results and analysis of the CGTS Applicability and accuracy of the proposed RE analytical model is investigated by evaluating various analytical models constructed based on different groups of components of interest. The considered components of interest are G13 (197 MW unit in bus 13), G16 (155 MW unit in 554

Kj(i)

K1

K2

K3

K4

LOLP

1st 2nd 3rd 4th 5th Average

0.05719 0.05913 0.05580 0.06185 0.05663 0.05812

0.06033 0.05833 0.05803 0.05358 0.05842 0.05774

1.05033 0.99420 0.99749 1.01280 0.96080 1.00312

0.00086 0.00094 0.00088 0.00097 0.00095 0.00092

0.01168 0.01149 0.01128 0.01137 0.01127 0.01142

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when availability of the transmission lines of interest varies. This indicates that variations of availability of components of interest in the model building do not affect the RE results by the analytical model. Results for various availabilities of components of interest used in the construction of Model-11 in the CS are listed in Table 9. The availability of the generating units of interest varies from 0.993 to 0.999 and that of transmission lines varies from 0.997 to 0.9995. These availabilities almost cover all possible availabilities of generators and transmission lines in the CS. It can be seen from Table 9 that all the LOLP, LOLF and EENS indices are very close to that obtained by the MCS method. The relative errors are less than 3.0%. This demonstrates that the proposed method is also effective for large systems such as the 91-bus system. Next, we examine the accuracy of the proposed analytical model for different values of the variance coefficient threshold β. The Model-4 is used for test. The RE results of Model-4 for various values of β are shown in Table 10. It can be seen from Table 10 that β has great impact on the error of the analytical model. The smaller the value of β is, the higher the accuracy of the analytical model. However, smaller β also means larger number of computations are needed in the modeling process.

Table 4 Comparison of RE results by analytical model and the sequential MCS (SMCS) method for a 4-unit generation system. Method

LOLP(×10−3)

LOLF (Occ./yr)

EENS (MW h/yr)

Proposed method SMCS ER (%)

5.51 5.54 0.49

2.0036 2.0122 0.43

1067.06 1074.60 0.70

Table 5 Comparison of RE results by analytical models and the sequential MCS (SMCS) method for IEEE-RTS. Model

LOLP (×10−3)

LOLF (Occ./yr)

EENS (MW h/yr)

ERLOLP %

ERLOLF %

EREENS %

Model-1 Model-2 Model-3 Model-4 Model-5 Model-6 Model-7 Model-8 Model-9 SMCS

1.22 1.21 1.21 1.22 1.22 1.22 1.22 1.21 1.21 1.22

2.1938 2.1896 2.1584 2.1854 2.2026 2.2031 2.2018 2.2052 2.2090 2.1960

1312.76 1315.36 1314.95 1296.13 1306.14 1327.75 1335.78 1311.69 1311.32 1315.29

0 0.82 0.82 0 0 0 0 0.82 0.82 –

0.10 0.29 1.71 0.48 0.30 0.32 0.26 0.42 0.59 –

0.19 0.01 0.03 1.46 0.70 0.95 1.56 0.27 0.30 –

5.5. Computation time In this section, we show that the computation time by the proposed method is significantly reduced from that by the sequential MCS method. The RE is conducted on the IEEE-RTS and the 91-bus Chuanyu system, respectively. For the CGTS of IEEE-RTS, suppose that we want to improve the system reliability to a pre-specified level by reducing the repair time of G13 and G16. To solve this problem, a reliability optimization model [20–22] needs to be established. Continuous changes on the repair time of G13 and G16 and repeated RE are needed in the process of finding the solution. Assume that 50 sets of the repair time of G13 and G16 are available for RE. These sets are obtained by multiplying the original repair time of G13 and G16 by 50 pre-specified multipliers. For sequential MCS, 50 MCS runs are performed. For the proposed method, several MCSs are performed to determine coefficients of the analytical model. Then RE is conducted by substituting each set of repair time of G13 and G16 into the proposed analytical model. Computation time of the proposed method and that of the MCS method for RE with 50 repair time sets for the CGTS of IEEE-RTS are shown in Table 11. It is observed from Table 11 that it takes about 42,097.5 min (almost 30 days) for the MCS method to obtain the evaluation results, whereas only 4,249.35 min (about 3 days) for the proposed method. This is due to the fact that the MCS method needs to run one sequential MCS for each set of repair time, but the proposed method only requires five sequential MCSs to establish the analytical model. Once the model is constructed, RE only takes less than one

Table 6 Comparison of RE results by analytical models and the sequential MCS (SMCS) method for the 91-bus system. Model

LOLP (×10−3)

LOLF (Occ./yr)

EENS (MW h/yr)

ERLOLP %

ERLOLF %

EREENS %

Model-10 Model-11 SMCS

2.713 2.686 2.669

3.9973 3.8848 3.9490

3023.58 3009.81 2974.50

1.65 0.64 –

1.22 1.63 –

1.65 1.19 –

indices with variable CRPs can be obtained directly, rather than repeatedly performing the time-consuming sequential MCS. This feature enables us to easily construct the visual relationship of the system reliability indices against various CRPs. And the visual relationship would be very useful in power system planning, as illustrated below. Assume that an analytical model for the IEEE-RTS considering G13 and G16 as the components of interest is constructed. Fig. 5 is a threedimensional plot of the system EENS index versus the availabilities of G13 and G16. It can be seen from Fig. 5 that EENS increases along with the directions indicated by the arrows. If we plan to modify or replace components G13 and G16 to achieve system reliability at a pre-specified level, then the question is: what would be the change and the new availabilities of G13 and G16 such that EENS = EENSobj? To solve this problem, we can superimpose a new plane of value EENSobj in parallel to the (G13, G16)-plane on Fig. 5. Intersection of these two planes gives the solutions, i.e., the availabilities of G13 and G16.

5.4. Parameter variation in modeling Now we investigate the accuracy of analytical model against the variation of availability of the components of interest in the model building. The RE results for various availabilities of G13, G16 and G18 used in Model-4 construction are shown in Table 7. The RE results for various availabilities of L13, L18 and L21 used in the construction of Model-8 are shown in Table 8. It can be seen from Tables 7 and 8 that, although the availability of the generating units of interest used in the construction of Model-4 varies from 0.84 to 0.99, the relative errors of annual system LOLP, LOLF and EENS indices are no more than 2.0%. Similar results occur

Fig. 5. Annual system EENS index versus the availabilities of G13 and G16. 555

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Table 7 RE results for various availabilities of G13, G16 and G18 used in Model-4 construction. A

LOLP (×10−3)

LOLF (Occ./yr)

EENS (MW h/yr)

ERLOLP %

ERLOLF %

EREENS %

0.84 0.87 0.90 0.93 0.96 0.99

1.23 1.22 1.22 1.22 1.22 1.22

2.1987 2.1974 2.2207 2.1854 2.2021 2.2032

1330.03 1314.70 1330.40 1296.13 1321.14 1308.39

0.82 0 0 0 0 0

0.12 0.06 1.12 0.48 0.28 0.33

1.12 0.04 1.15 1.46 0.44 0.52

Table 12 Sensitivity coefficients of LOLP with respect to various CRPs.

LOLP (×10−3)

LOLF (Occ./yr)

EENS (MW h/yr)

ERLOLP %

ERLOLF %

EREENS %

0.9494 0.9594 0.9694 0.9794 0.9894 0.9994

1.22 1.20 1.21 1.21 1.21 1.21

2.2308 2.1936 2.1994 2.1765 2.1755 2.2052

1319.80 1298.13 1304.94 1315.76 1323.71 1311.69

0 1.64 0.82 0.82 0.82 0.82

1.58 0.11 0.15 0.89 0.93 0.42

0.34 1.30 0.79 0.04 0.64 0.27

Table 9 RE results for various availabilities used in Model-11 construction. AG

AL

LOLP (×10−3)

LOLF (Occ./yr)

EENS (MW h/yr)

ERLOLP %

ERLOLF %

EREENS %

0.993 0.994 0.996 0.998 0.999

0.997 0.9975 0.9980 0.9990 0.9995

2.683 2.708 2.729 2.685 2.690

3.8926 4.0044 3.9622 3.8848 3.9610

3012.64 3045.16 3040.33 3009.81 3031.53

0.52 1.48 2.25 0.62 0.78

1.43 1.40 0.34 1.63 0.30

1.28 2.38 2.21 1.19 1.92

Table 10 RE results of Model-4 with various β. β

LOLP (×10−3)

LOLF (Occ./yr)

EENS (MW h/yr)

ERLOLP %

ERLOLF %

EREENS %

0.005 0.01 0.015 0.02

1.22 1.22 1.23 1.25

2.1941 2.1854 2.2587 2.2665

1313.82 1296.13 1342.68 1354.51

0 0 0.82 2.46

0.09 0.48 2.86 3.21

0.11 1.46 2.08 2.98

Table 11 Computation time of the proposed method and the MCS method for various repair time of the CGTS of IEEE-RTS. Multiplier of repair time

– 0.65 0.67 …… 1.48 1.50 Total time

MCS method

4,249.3 3e−4 3.2e−4 …… 3e−4 3e−4 4,249.4

– 929.2 926.1 …… 743.3 738.9 42,097.5

∂LOLP ∂λl

∂LOLP ∂μl

G18 G23 G13 G16 G7 G22

−0.004699 −0.003787 −0.002932 −0.001590 −0.000743 −0.000317

0.00006231 0.00003659 0.00001510 0.00000669 0.00000391 0.00000071

−0.00000850 −0.00000318 −0.00000079 −0.00000028 −0.00000016 −0.00000001

6. Conclusions This paper has presented an analytical RE model for CGTS based on the sequential MCS method. The proposed analytical model is constructed by partitioning the chronological system state sequence produced by sequential MCS into a set of mutually exclusive events and extracting the CRPs from system reliability index calculation formula. Once the proposed model is established, the system reliability index

Computation time (min) Proposed method

∂LOLP ∂Al

sensitivity analysis of CRPs and subsequent identification of the weak component should focus on the generating unit rather than on the transmission line. Generating units with different capacities, such as G7 (100 MW unit in bus 7), G13, G16, G18, G22 (50 MW unit in bus 22), and G23 (350 MW unit in bus 23), are selected to test the correctness of the analytical expressions of sensitivity coefficients developed in Section 4. Suppose that a RE analytical model is constructed considering G7, G13, G16, G18, G22, and G23 as components of interest. Based on (22)–(32), sensitivity coefficients of annual system LOLP with respect to the availability, failure rate, and repair rate of each of these six generating units can be calculated and shown in Table 12. Similarly, sensitivity coefficients of LOLF and EENS are shown in Tables 13 and 14, respectively. The sensitivity coefficients in Tables 12–14 are listed in descending order according to their magnitudes. The sensitivity coefficient indicates the impact of component failure on system reliability. It is observed from Table 12 to Table 14 that the sensitivity coefficient of G18 is the largest and that of G22 is the smallest among six generating units. This implies that the failure of G18 has the highest impact on system reliability compared with other five generating units. Therefore, G18 is considered as the weak link of the components of interest. To verify the above observation, we calculate the annual system EENS index by the sequential MCS method using various component failure rates. Results are plotted in Fig. 6. The failure rates are set from 0.3 times to 2.1 times of their initial values. One can see from Fig. 6 that the reliability indices increase along with the failure rate. The increase of G18 is the most significant among all generating units. The rates of change of the reliability index from the largest to smallest are in sequence of G18, G23, G13, G16, G7, and G22. This result is in consistent with the results shown from Table 14 obtained by the analytical model. Similar results are also observed for annual system LOLP index and LOLF index versus the component failure rate using the sequential MCS method.

Table 8 RE results for various availabilities of L13, L18 and L21 used in Model-8 construction. A

Component

Table 13 Sensitivity coefficients of LOLF with respect to various CRPs.

second. 5.6. Application to the sensitivity analysis The CGTS of IEEE-RTS is a system with sufficient transmission network against relatively inadequate generating capacity [27]. Thus 556

Component

∂LOLF ∂Al

∂LOLF ∂λl

∂LOLF ∂μl

G18 G23 G13 G16 G7 G22

−6.72 −5.16 −4.52 −4.41 −2.06 −1.69

0.0891 0.0499 0.0233 0.0185 0.0108 0.0038

−0.01215 −0.00434 −0.00123 −0.00077 −0.00045 −0.00004

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Table 14 Sensitivity coefficients of EENS with respect to various CRPs. Component

∂EENS ∂Al

∂EENS ∂λl

∂EENS ∂μl

G18 G23 G13 G16 G7 G22

−6415.78 −6108.18 −2692.46 −1660.03 −989.21 −480.73

85.08 59.02 13.87 6.99 5.20 1.08

−11.60 −5.13 −0.73 −0.29 −0.22 −0.01

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Fig. 6. Annual system EENS index versus the multiplier factor of component failure rate using the sequential MCS method.

with variable CRPs can be obtained directly, rather than repeatedly performing the time-consuming sequential MCS process. The proposed analytical model has been applied to calculate the sensitivity coefficients of system reliability indices with respect to CRPs. Case study using a 4-unit generation system, the IEEE-RTS and a 91-bus power system has been presented to verify the correctness and effectiveness of the proposed model. Simulation results have demonstrated the following findings: (1) The proposed analytical model can obtain system reliability indices with high accuracy (less than 2% difference from that by the sequential MCS method). (2) The analytical model enables easy construction of the visual relationship of the system reliability index against CRPs. This visual relationship is useful in power system planning. (3) The proposed analytical model facilitates the sensitivity analysis. The results can be used to identify the weak link of the components of interest.

Lvbin Peng is a Ph.D. candidate in the School of Electrical Engineering, Chongqing University, China. Currently, His research areas include power system reliability and planning. Bo Hu was born in 1983 in Henan, P.R. China. He received the Ph.D. degree in electrical engineering from Chongqing University, Chongqing, China, in 2010. Currently, he is an associate professor in the School of Electrical Engineering at Chongqing University, China. His research interests include power system reliability, parallel computing techniques in power systems.

Since the proposed analytical model is based on the sequential MCS, it can be easily applied to the reliability analysis of power systems incorporating renewable energy.

Kaigui Xie is a full professor in the School of Electrical Engineering, Chongqing University, China. His main research interests focus on areas of power system reliability, planning and analysis. He is an editor of IEEE Transactions on Power Systems.

Acknowledgement This work was supported in part by the National Natural Science Foundation of China (No. 51677011).

Heng-Ming Tai is professor in the Department of Electrical and Computer Engineering at the University of Tulsa, OK, USA. His research interests are in the areas of power system reliability and industrial electronics.

References

Kaveh Ashenayi is professor and chairman in the Department of Electrical and Computer Engineering at the University of Tulsa, OK, USA. His research areas are in power system, renewable energy, and robotics

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