Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 3, 238-245, 216 doi:1.21311/1.39.3.31 Reliability Assessment Algorithm of Rural Power Network Based on Bayesian Network Yunfang Xie College of Mechanical and Electrical Engineering, Agricultural University of Hebei, Baoding 711, Hebei, China Wenhui Hou* Department of Mechanical Engineering, Tangshan Polytechnic College, Tangshan 6311, China *Corresponding author(e-mail: 4757198@qq.com) Limin Shao* College of Mechanical and Electrical Engineering, Agricultural University of Hebei, Baoding 711, Hebei, China *Corresponding author(e-mail: shaolm@126.com) Abstract This paper puts forward a new algorithm for the reliability assessment of the rural power network which is based on the Bayesian network. This algorithm is on the basis of the benchmarking convergence method, changes the distribution of the system node s time for power supply and power failure using the node model of causality, combined with the characteristics of the Bayesian timing simulation method s diagnostic reasoning and causal reasoning, and then we can obtain the rural power network s reliability index simply and usefully and find the weak link which clamps the rural power network. Experiments show that this algorithm can make the benchmarking convergence method speed up the convergence rate and meanwhile it can improve the convergence rate of the system index well. The effectiveness of this method is verified by the actual operation results. Key words: Bayesian Network, Reliability Assessment, Rural Power Network, Timing Simulation. 1. INTRODUCTION Without the reliability assessment of the rural power network, the operation and the optimal planning of the rural power network are impossible. Generally speaking, the reliability assessment of the rural power network has two kinds, which are analytical method and Monte Carlo simulation method. The analytical method is a method which calculates the reliability index by analytical method. The analytic method mainly includes failure mode effect analysis, network-equivalent approach, interval algorithm, etc. Monte Carlo simulation method selects the condition by sampling, and gets the reliability index by statistics. Both this methods can calculate the rural power network s reliability index, but they cannot show the main factors which affect the system reliability, and some element s or elements position in the whole system reliability cannot be given quantitatively. When some elements state in the system is known, existing methods are hard to calculate the elements conditional probability which affects the whole system or part of the system. However, the conditional probability is very helpful in improving and increasing reliability of the power system. The Bayesian Network can describe the interrelation of the random variable, expressing the random variable in probability, and get the result of the uncertain problem by reasoning. Therefore, the Bayesian Network is widely used in the reliability assessment of the power network. The literature (Billinton and Wang, 1999) applies the Bayesian Network to the reliability assessment of the interconnected systems; the literature (Billintion and Wang,1998;Zhang and Guo, 22; Zhang and Wang, 24) applies the Bayesian Network to the reliability assessment of the distribution system, and all the literatures use the exact inference algorithm of the Bayesian Network. But when the system scale is big, the exact inference algorithm is hard to realize. Cooper proved that all of the exact inference algorithms of haphazard texture in the Bayesian Network are np-hardness, so people devote to the research of the Bayesian Network s approximate reasoning algorithm (Cooper,199). However, the exact inference algorithm and the approximate reasoning algorithm of the Bayesian Network cannot calculate the problems related to time when they are used in the reliability assessment of the power system, for example, the problem of time varying load, of the maintenance scheduling of deterministic, of the operating mode of the kinds of seasonal variations, etc. And both of the two algorithms cannot calculate the failure frequency index and the fault duration index, and they cannot accurately reflect the actual value of the 238
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 3, 238-245, 216 element error rate and the component repair rate. The literature (David, Thanh and Peter, 1999) proposes a sequential simulation reasoning algorithm of Bayesian Network used in the reliability assessment of the transformer substation. But this net and algorithm are only suitable for the reliability assessment of the transformer substation, not suitable for the reliability assessment of the rural power network, because this algorithm cannot deal with the causal node whose conditional probability is not 1 or when it is used in the reliability assessment of the rural power network, and its convergence rate is slow. For this reason, this paper has improved the algorithm in the literature (David, Thanh and Peter, 1999), binding the Bayesian Network and the timing simulation, and has done the research on the new algorithm suitable for the reliability assessment of the large-scale rural power network. 2. THE TIMINGSIMULATIONINFERENCE ALGORITHMBASED ON BAYESIAN NETWORK 2.1. The simulation of the element state based on the benchmarking convergence method Take the example of two states of the element; the element s operation process on the simulation time of sample is a process of alternative conversion between the normal state and the fault state, as shown in figure 1. 1 State Ti Ti1 Tik Figure 1. Component working state In the figure1, T ik shows the duration time of the element i on the state at that time, which can be determined by the sampling of the stochastic process artificially generated by the computer simulation technology. The definition of the benchmarking convergence method is: when the number of simulations is n, the reliability of the simulation value element itself can easily get whose error to the exact value is small. Based on the simulation value A s, we can get a reliability rate A m by simulating to the times of m. And we compare A m and A s, if A m <A s, we will improve A m by producing the working time on the time of (m+1) according to the formula 1; if A m >A s, we will decrease A m by producing the fault time on the time of (n+1) according to the formula 1. To the benchmarking convergence method, the state of element is not an alternative conversation between the normal state and the fault state. It is determined by the simulation value A m and A s. Take the example of the element L1, as shown in figure 2, when the number of simulation is n, both A m and A s are extremely close to the exact value ( ), among which the convergence rate of the component reliability simulation is faster after using the benchmarking convergence method. Error( ).9.8 L1 L1(Convergence).7.6.5.4.3.2.1 5 1 15 2 Figure 2. The benchmarking convergence method When we analyze the reliability of engineering system, generally we assume that the occurrence time of the event (fault, repairing) obeys the exponential distribution, so using the inverse transformation method we get the result as follows: TMTBF ln r x ( Ai As ), Tik (1) TMTTR ln r x 1 ( Ai As ), In the formula, r is the random number distributing uniformly between (, 1). T MTBF and T MTTR are respectively the element s mean time to failure and its mean repair time. When both the distribution functions of the continuous operating time and the distribution function of the continuous outage time obey the exponential distribution, T MTBF and the fault rate are reciprocal, and T MTTR and the repair rate are reciprocal. 2.2. The Bayesian Network model for the reliability of the rural power network 1, The Bayesian node model The Bayesian Network model based on the reliability assessment of the rural power network is consist of the logical relationship of the node model of Noisy-And and the logical relationship of the node model of Noisy-Or as shown in figure 3. The node model of Causal is the generalization of the node model Noisy- 239
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 3, 238-245, 216 And and the node model Noisy-Or, and the node model Noisy-And and the node model Noisy-Or are the special cases of the node model of Causal. In figure 3, the capital letter A, B, C show the node variable, which is normal when the value is 1, and when the value is, it show there is breakdown in somewhere. d1, d2, d3 and d4 are respectively the specific value of the conditional probability P(C=1 A, B) which is confirmed according to the specific circumstance. A C B A B P(C=1 A,B) 1 1 1 1 1 (a) Noisy-And node A C B A C B A B P(C=1 A,B) 1 1 1 1 1 1 1 (b) Noisy-Or node A B P(C=1 A,B) d1 1 d2 1 d3 1 1 d4 (c) Causal node Figure 3. Models of nodes We suppose that A and B in figure 3 respectively shows two load points in the rural power network, and the number of their users are respectively n A and n B, and combination point C expresses the rural power network combined by the load points A and B, in which it shows the user in the load point get power when the variable value is 1, and when the variable value is, it shows the user in the load point cannot get power. If the above assumption is established, then in the conditional probability table, when the value of d1, d2, d3 and d4 is respectively n B /(n A +n B ) n A /(n A +n B ) and 1, the four values represent the time distribution of the users time of power supply and blackout respectively. 2, The sequential Bayesian node model of Causal The sequential Bayesian nod model of Causal distributes the time of the system node s power supply and cut, as shown in figure 4. Load point A Load point B T A2 T A4 T T A1 T A5 A3 T B 2 T B 4 T B 1 T B 3 T B 5 Node C t t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 1 t 11 t 12 Figure 4.Simulation for Cause-Result We can suppose that the number of users in the load point A is n A, and the number of users in the load point B is n B. The state of the combinatorial node C (S) is still determined by the logical relation of Noisy-And and Noisy-Or through the state of the load point A and the load point B. Take the time distribution of the power supply and cut of the node C s previous 12 segments for example, it is determined by the rural power network s logical relation of Causal, which is: t1 ( S) TB 1 *1; t12( S 1) TA 1 TB 1* d2; t23( S ) TA 1 TB 1* d3; t34( S 1) TB 2 TB 1 TA 1*1; t45( S ) TA1 TA2 TB 1 TB 2 * d2; t T T T T * d ; 56( S 1) A1 A2 B1 B2 3 3 2 t ( T T )*1; 67( S ) Bi Ai i1 i1 t T * d ; 78( S1) B4 2 t 89( S ) T * d ; B4 3 3 4 t ( T T )*1; 91( S ) Ai Bi i1 i1 t T * d ; 111( S1) A4 3 t T * d 1112( S) A4 2 When A and B s power is cut respectively, the time of the combinatorial point C is divided into the time of power cut influenced by the load point of power cut and the time of power supply influenced by the load point of power supply. This paper will deal with the time distribution of power supply and cut in the combinatorial point C according to the random number. When A and B cut their power supply at the same time, the t 24
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 3, 238-245, 216 combinatorial point C s time of power cut is A and B s time of power cut together. The combinatorial point C s time of power supply is A and B s time of power supply together. 2.3. The reasoning process of the timing simulation This paper handles the conversed Bayesian Network with multi layers. The time and state of component layer both belong to the logical relation layer of Noisy-And and Noisy-Or. The state of the load point is still the logical relation of Noisy-And and Noisy-Or when it comes down to the user level of the load point. The time coefficient of the power supply and blackout is decided by the logical relation of Causal. We can suppose that the number of the components in reliability system is m (including electric transmission line, transformer, breaker, and buses) and the number of load points is n. Each state of each component can be simulated by the method in formula 1. The state of each load point and each system is combined by the state of every component, and the time of power supply and blackout in system is decided by the causal relationships among every load point. If the system state is x k when in the time of t k, then x k ={x 1k, x 2k, x 3k x mk }. The system state changes if and only if the state of arbitrary element in the system changes. The running state of the system is shown as figure 5. Element 1 Element 2 Element 3 t t 1 t 2 t 3 t k tk+ 1 Figure 5. State changes of system (1)The simulation process begins from the system is normal, when we suppose t =, all the components are in normal running state. At this time, the system state can be expressed as x ={x 1, x 2 x m }. We can take sample of every component s state and of their duration in this state according to formula 1, then we can get the result T i, (i=1, 2 m). If we suppose the duration x of the system is T, then T =min{t i }. (2)We suppose the simulation clock has pushed on to t k, when the load node is in the state x k ={x 1k, x 2k x mk }, among which x ik is the ith component s state at the time of t k, then the duration of the system state x k can be expressed as T k = min{t ik }. (3) Next time the simulation clock has pushed on to is t k+1 =t k +T k, we can get that the system s state is x k+1 at this time. Caculate T i,k+1 : T T ( T T ) (2) T ik, 1 ik k ik k SampleT i, k1 according (1) ( Tik Tk ) (4) Judging the component load point when it is on the minimum time T k, we can obtain the proportional relation that the number of users of the load occupies the N number of load points according to formula 3 and formula 4, thus getting this load point s gain and loss tie in system nodes. f x1, x2, x3, x n ( ) t n T n n i i1 Tk * p ( e m) t( s d) Tk * l ( e m) Formula 3 is the power outage ratio coefficient of the load point; n T is the total number of users in the outage load point, and n i is the number of the users in every load point. In formula 4, d is the system state, when d is 1 and the system is in fault condition, p= f(x 1, x 2, x 3 x n ), among which e is the number of the fault components. When d is and the system is in running state, p=1-f(x 1, x 2, x 3... x n ), among which e is the number of the normal components. (5) We can add up t got by formula 3 and formula 4 and the state t is in to the corresponding system node memory, and at the same time every component and every load point also records their t and the state they are in. (6) Continuing to simulate, repeating step 2-step 4, we can record the system parameter until the simulating time or the simulating times to the specified value: the normal running time of the system T N ; the total simulating time span T A ; the system fault time F N ; and we can calculate system s reliability index and reason the conditional probability of the causal reasoning. Adding t to the total system fault time, we add t to every system component s corresponding accumulator according to the component s state in the system and on the time of t k. The task of Bayesian network reasoning is calculating the probability of every system component in fault or normal condition supposing the system is in fault condition. The conditional probability of reasoning is the cumulative time of every system component s any time divided by the total time of the system failure. (3) (4) 241
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 3, 238-245, 216 The specific process which achieves causal reasoning can be described as follows: in step 5 of the simulation of system state, firstly we distinguish the state of the component which is set as evidence whether confirms to the evidence or not. If it confirms to, the evidence element accumulates its own simulated time, and the system accumulates the final t which is obtained by causal relationship. If it doesn t confirm to, then we will go to step 6. The conditional probability of causal reasoning is the cumulative time of system failure divided by the evidence element s cumulative time in evidence state. 3. APPLICATION EXAMPLE In order to prove this algorithm s adaptability and validity to rural power network, this paper has done reliability analysis to the experimental system of RBTS bus 2 feeder F1 distribution systems by using this algorithm. The system ringing of Feeder F1 is shown as in figure 6 and the reliability parameters are all cited from the original documents (Allan, Billinton, Sjarief, Goel and So, 1991). LP1 LP3 LP5 LP7 T1 T3 T5 T7 F1 F2 L2 L5 F1 F3 F5 L1 L4 L7 L1 D1 D2 D3 F2 F4 F6 L12 T2 L3 LP2 L13 F3 LP8 D4 T4 L6 LP4 L14 L15 F3 Figure 6. Connection diagram of the distribution system for RBTS bus 2 This system has given 6 kinds of mode of connection: 1, the main frame has disconnecting switch, branch protection, back-up power, and no standby transformer; 2, the main frame has not disconnecting switch, branch protection, back-up power and no standby transformer; 3, the main frame has not disconnecting switch, has branch protection, has not back-up power and has standby transformer; 4, the main frame has disconnecting switch, has not branch protection, back-up power and has no standby transformer; 5, the main frame has disconnecting switch, branch protection, back-up power and standby transformer; 6, the main frame has disconnecting switch, branch protection, back-up power and no standby transformer. According to the Bayesian network method; this paper only draws the Bayesian network corresponding to connection mode5 for F1. The several other connection mode s corresponding Bayesian network is similar. L1 D1 L2 L3 T1 T2 L4 D2 L5 L6 T3 T4 L7 D3 L8 L9 T5 T6 L1 L1 1 T7 LP9 T6 L8 L9 LP6 L11 F7 A M1 M2 M3 M4 M5 M6 M7 M8 M9 M1 M11 LP1 LP2 LP3 LP4 LP5 LP6 LP7 F1 Figure 7. The Bayesian networks for mode 5 In figure 7, the first layer node expresses the circuit element L1~L11 and transformer component T1~T7. Because they have not father node, their conditional probability is their prior probability. The second layer node is the intermediate node led in for simply drawing. The type from the third node to the system node is the causal relationship. This paper has got the random number by the linear congruence and has the simulating calculation to the Bayesian network s reliability index in the distribution system as shown in figure 7, and has the diagnostic reasoning and the cause and effect simulation to the distribution network. 3.1. The reliability index of the power distribution system The reliability index of the power distribution system is the measure used to quantitatively evaluate the reliability of the power distribution system. The data relating to the last system node obtained in this paper is easily got to the reliability index of the power distribution system. ASAI T N / T A ; ASUI 1ASAI ; 242
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 3, 238-245, 216 SAIDI ASAI *876 ; SAIFI C * F / N * T ; A N T N CDIDI SAIDI / SAIFI ; ENS ASUI *876* L ; a AENS ENS / N T ; Among them, L a is the total of the average load of very load point, N T is the total users of the system, and C A is the average number of users of the load points. Using the algorithm mentioned in this paper, we have calculated the distribution network index of F1 in IEEE-RBTS. Having simulated this system 3*1 4 times, the result has shown as the following graph. From the graph we can know that the deviation of every distribution index value after converging with the value in the original books is within the permitted range. In addition, the convergence rate of every index value is in high speed after using benchmarking convergence method. The results are shown in figure 8 to figure 12..99998.99997.99996.99995.99994.99993.99992.99991.9999.99989.99988.9.8.7.6.5.4.3.2.1.4.35.3.25.2.15.1.5 ASAI.1.5 1 5 1 15 2 25 Figure 8. ASAI data comparison chart SAIDI.1.5 1 5 1 15 2 25 Times(*1) Figure 9.SAIDI data comparison chart SAIFI.1.5 1 5 1 15 2 25 ASAI(convergence) SAIDI(convergence) SAIFI(convergence) Figure 1. SAIFI data comparison chart Times(*1) Times(*1) 243
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 3, 238-245, 216 4. 3.5 3. 2.5 2. 1.5 1..5 CAIDI.1.5 1 5 1 15 2 25 CAIDI(convergence) Figure 11. CAIDI data comparison chart Times(*1) 3 25 2 15 1 ENS 5.1.5 1 5 1 15 2 25 ENS(convergence) Times(*1) Figure 12. ENS data comparison chart 3.2. Diagnostic reasoning and cause and effect simulation The concept of diagnostic reasoning is judging every element s probability value in normal or in failure operation when the system is in normal or in failure operation probability. This paper has taken P (E= LP4=1 )(E stands for every element, LP4 stands for the load points; 1 represents failure, represents normal) to check every element s normal operation probability when the system is in failure. Figure 13 is the comparison between the result got by the simulation algorithm mentioned in this paper and the result got by the exact algorithm. 1.2 1..8.6.4.2 Simulation value Exact value L1 L2 L3 L4 L5 L6 L7 L8 L9 L1 L11 T1 T2 T3 T4 T5 T6 T7 Elements Figure 13. The comparison between the results of the two diagnostic reasoning algorithms From the figure 13 we can know that when LP4 is in failure, the failure rate of element L4 L6 T4 is higher, and the second higher of the failure rate is L1, L7, L1. Raising the reliability rate of these elements can enhance the reliability rate of load point LP4. The concept of cause and effect simulation is judging the system s probability value in normal or in failure operation when it is in normal or in failure operation probability. This paper has taken P(S= E=1)(E stands for every element, S stands for the system; 1 represents failure, represents normal) to check the system s normal operation probability when the element is in failure. Figure 14 is the comparison between the result got by the simulation algorithm mentioned in this paper and the result got by the exact algorithm mentioned in document (Huo, Zhu and Su, 23). 244
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 3, 238-245, 216 1.2 1..8.6.4.2 Simulation value Exact value L1 L2 L3 L4 L5 L6 L7 L8 L9 L1 L11 T1 T2 T3 T4 T5 T6 T7 Elements Figure 14. The comparison between the results of the two causal inference algorithms From the figure 14 we know that element L1, L4 has caused great hidden danger to the system s normal operation, followed by L2, L3, L5, T1, T2, T3, and L7, L1 have influence on the system s normal operation to some extent. Raising the above-mentioned elements reliability and shortening the repair time can both advance the system s reliability appropriately. 4. CONCLUSION This paper has made the combination element and the related individual element do associated action on the basis of sequential Bayesian reasoning algorithm. In this paper we have adopted the logical relationship of Causal to deal with the time of the load node on the basis of using the benchmarking convergence method and have redistributed the power supply and black out of the load point. After the system testing of the Feeder 2 in IEEE RBTS2, its simulation result has shown that the improved algorithmic model is reasonable and valid and is approximate to the data result in the original document. What s more, the calculated result of the posterior probability using this algorithm is very close to the calculated value obtained by the exact algorithm reasoning. The sequencing which doesn t influence the posterior probability can effectively recognize the bottleneck muzzling the reliability of the power system, and can effectively guide the planning department and the operation department to take effective measures to improve the level of reliability of the system. Acknowledgements This work was supported by Baoding Science and Technology Research and Development Project (11ZG29, 14ZG4). REFERENCES Allan R N, Billinton R, Sjarief I, Goel L, So S. K. (1991) A Reliability Test System for Education Purposes: Basic Ditribution_system Datand Results,IEEE Trans on Power system, 6(2), pp.813~82. Billinton R, Wang P. (1999) Teaching distribution system reliability evaluation using monte carlo simulation,teee Transactions on Power Systems, 14 (2), pp.397-43. Billintion R, Wang P(1998) Reliability-nrtwork-equivalent approach to distribution-system-reliability evaluation,iee Proc, 145 (2), pp.149~153. Cooper G F. (199) The computational complexity of probability inference using Bayesian belief networks,artificial Intelligence, 42(3), pp.393-45. David C.Yu, Thanh C.Nguyen, Peter Haddaway (1999) Bayesian Network Model for Reliability Assessment of Power Systems,IEEE Transactions on Power Systems, 14 (2), pp.426-432. Huo Limin, Zhu Yongli, Su Haifeng (23) Reliability Analysis of Distribution Networks Based On Bayesian Networks, Journal of North China Electric Power University, 3(6), pp.6~1. Zhang Peng, Guo Yongji (22) Large scale distribution system reliability evaluation based on failure mode and effect analysis, Journal of Tsinghua University, 42 (3), pp.353~357. Zhang Peng, Wang Shouxiang (24) A novel interval method f orreliability evaluation of large scale distribution system,proceedings of the CSEE, 24 (3), pp.77~83. 245