Reliability Analysis of a Power System Based on the Multi-State System Theory Chunyang LI College of Mechatronics Engineering and Automation National University of Defense Technology Changsha, 410073, China E-mail: lichunyang.nudt@163.com Xun CHEN, Xiaoshan YI College of Mechatronics Engineering and Automation National University of Defense Technology Changsha, 410073, China Abstract—Reliability analysis of power systems using the traditional system reliability theory usually can not represent the real-life situation. The multi-state system theory is introduced to analyze the reliability of a power system. States and corresponding probabilities of the battery are defined. The reliability of the power system is estimated by the multi-state system theory. The results show that the system reliability estimated by the traditional system reliability theory is conservative, and the proposed method in this paper is better to analyze the reliability of power systems. Keywords- power system; multi-state system theory; reliability; universal ge nerating function I. I NTRODUCTION A power system composed of battery pack provides en ergy for other systems, and the reliability of this power system is very important. To analyze the reliability of the power system using the traditional system reliability theory, the reliability of the battery should be gained first, and then the system reliability can be computed according to the structure of the system [1]. This method is simple, but it can not be applied to power systems with required capacities. The power system will fail when the system capacity is less than the required capacity even though the batteries of the system are all working. The traditional system reliability theory defines the power system and the batteries are b inary, but they are all multi-state actually. The performance of the batteries can degrade, which results in performance degradation of the power system. So there can be several states of degradation. Compared with binary system, the multi-state system can perform its task with many different performance levels except failed and working [2, 3]. The research on multi-state systems began in the 1970s [4, 5], and gained a lot of researchers’ attention. Many papers have been devoted to estimating the reliability of the multi- state system [6-8] and optimizing the structure of the multi- state system [9-11]. The reliability of the power system will be analyzed by the multi-state system theory in this paper. The procedure of applying the multi-state system theory to the reliability analysis of the power system is studied. The relationship between the performance of the system and the performance of the batteries is analyzed. The results obtained by the traditional system reliability theory are compared with the results obtained by the multi-state system theory. II. P ROBLEM F ORMATION The power system is composed of eight identical batteries. A branch c onsists of two batteries connected in series, and the system consists of four branches connected in parallel as depicted in Fig. 1. The required capacity of the power system is not less than 22.8 Ah. To protect proprietary data, all parameters have been scaled. This does not in any way affect the validity of the method presented in this paper. Figure 1. Structure of the power system A test of 120 batteries shows that the capacities of the batteries follow the s-normal distribution with mean 6000, variance . In short, , where G is the capacity of the battery. 2 150 () 2 ~ 6000,150GN To analyze the reliability of the system by the traditional system reliability theory, we must gain the reliability of the battery first. The power system has to provide the required capacity, and the system will fail when the required capacity is not fulfilled. According to the structure and the required capacity of the system, the reliability of the battery is defined as the probability that the capacity of the battery is not less than 5700 mAh. Then the reliability of the battery is: { } Pr 5700 0.97725RG=≥ = . When the capacity of one battery is less than 5700 mAh, the required capacity of the power system may be not fulfilled. So the reliability of the power system estimated by the traditional system reliability theory is: . 8 0.83185 st RR== 978-1-4244-4905-7/09/$25.00©2009 IEEE 95 This method can solve the problem, but the result is conservative. Because actually the reliability of the power system is: . (1) { Pr 22.8 ss RG=≥ } Equation (1) indicates that when the capacity of the system is above 22.8 Ah, the system is reliable, though the capacity of a battery is lower than 5700 mAh. Suppose the capacity of the first branch is 5600 mAh and the capacity of other branches are all above 5800 mAh, the system is reliable because the required capacity is reached. But when we analyze the system reliability using the traditional system reliability theory, the system fails. So this problem will be solve by another method — the multi-state system theory. III. M ULTI - STATE S YSTEM T HEORY Assume that the component has M possible states, and the performance is { } 12 ,,, M g gg=g " , with the corresponding probability is { } 12 ,,, M qq q=q " , where { } Pr ll qG , is the performance of the component, . Then the universal generating function of the component is: g== G M1, ,l= " () 1 l M g jl l Uz qz = =⋅ ∑ . (2) To obtain the universal generating function of the system, the operat ors of the universal generating function are defined as follows [2, 3]: (3) () () () () , 12 11 , kl MM fg g kl kl UzUz qqz == Ω=⋅⋅ ∑∑ , () () ( )() () () () () () () 11 11 ,, , ,, ,, , ,, kk n kk n Uz U zU z U z Uz U zU z U z + + Ω =Ω , "" "" (4) . (5) () () () () ( () () () () () () 11 11 ,, , ,, ,, , ,, kk n kk n Uz U zU z U z Uz U z U z U z + + Ω ⎡ =Ω Ω Ω . ⎣ "" "" ) ⎤ ⎦ The ( ) , kl f gg is defined according to the structure of the multi-state system. When the performance of the system is eq ual to the sum of the performance of components, define the π operator: () () () 12 11 , kl MM g g kl kl UzU z qqz π + == =⋅⋅ ∑∑ . (6) When the performance of the system is equal to the minim um of the performance of components, define the σ operator: () () () min( , ) 12 11 , kl MM g g kl kl UzUz qqz σ == =⋅⋅ ∑∑ . (7) Of course, other operators can also be defined according to the situatio ns. The universal generating function for the system can be obtained usi ng simple algebraic operations over individual universal generating function of components: , (8) () () () () 1 1 ,, i m M G in m Uz Uz U z qz σ = == ∑ " m () () () () 1 1 ,, sys s M G N s Uz U z U z qz π = == ∑ " s , (9) where ( ) i Uz is the universal generating function of subsystem ; is the number of components in subsystem i ; i n i M is the number of possible states of subsystem i . is the performance of subsystem i ; is the corresponding probability; m G m q ( ) Uz is the universal generating function of the system; is the number of subsystems in the system; N sys M is the number of possible states of the system; s G is the performance of the system; s q is the corresponding probability. Define the following δ operator over ( ) Uz : , (10) () () ( 11 ,, sys sys ss MM GG ss ss Uz W qz W qz W δδ δ == ⎛⎞ == ⎜⎟ ⎜⎟ ⎝⎠ ∑∑ ) , , (11) () , , 0, , s ss G s s qGW qz W GW δ ≥ ⎧ = ⎨ < ⎩ where W is the required performance level of the system. Then the reliability of the system is: ( ) { }() () Pr , s sms GW RW G W UzW q δ ≥ =≥= = s ∑ . (12) IV. S TATES D EFINITION AND P ROBABILITIES E STIMATION Suppose that the capacity of the battery is divide d into 1 M + intervals: [ ) 1 0, w , …, [ ) 1 , MM ww − , [ ) , M w ∞ , 11 0 M M ww − w< << <" . The states of the battery can be defined as follows: state 0: 1 0 Gw≤ < state 1: 12 wGw≤ < … state 1 M − : 1M M wGw − ≤ < state M : M Gw≥ M can be determined by the analytical precision. In this paper, the capacity of the battery is divided into eight intervals, that is: [ ) 5200,5550 , [ ) 5550,5700 , , [ ) 5700,5850 [ ) 5850,6000 , [ ) 6000,6150 , [ ) 6150,6300 , [ , ) 6300,6450 [ ) 6450,6800 . 96 To obtain the lower bound of the system reliability, the performance of each state is defined as the minimum capacity of each interval, that is: 1 5200g = , , , , 2 5550g = 3 5700g = 4 5850g = 5 6000g = , , , . 6 6150g = 7 6300g = 8 6450g = State probability is defined as { } Pr l qGg== l , and then the corresponding state probabilities of the battery are: {} {} 1 8 Pr 5200 5550 Pr 6450 6800 0.00135 , qG qG =≤< == ≤< = {} {} 2 7 Pr 5550 5700 Pr 6300 6450 0.02140 , qG qG =≤< == ≤< = {} {} 3 6 Pr 5700 5850 Pr 6150 6300 0.13591, qG qG =≤< == ≤< = {} {} 4 5 Pr 5850 6000 Pr 6000 6150 0.34134 . qG qG =≤< == ≤< = Because { } { } 8 Pr 5200 Pr 6800 4.82130 10GG − <=≥= × , the intervals [ and ) 0,5200 [ ) 6800,∞ are not considered. Then we can get the state performance of the battery is , { 5200,5550,5700,5850,6000,6150,6300,6450=g } and the corresponding probability is {0.00135,0.02140,0.13591,0.34134, 0.34134,0.13591,0.02140,0.00135}. = q V. R ELIABILITY A NALYSIS OF THE P OWER S YSTEM The reliability of the power system is analyzed using the mult i-state system theory. According to (2), the universal generating fun ction of the battery is: ( ) 5200 5550 5700 5850 6000 6150 6300 6450 0.00135 0.02140 0.13591 0.34134 0.34134 0.13591 0.02140 0.00135 . j Uzzzzz zzz =+++ z + +++ Based on the operators defined in (7) and (8), the universal generating fun ction of the branches can be obtained, and then the universal generating function of the power system can be computed by (6) and (9). According to (12), the reliability of the power system is : { } Pr 22.8 0.98682 sm s RG=≥= . The results show that when the required capacity of the power sy stem is 22.8 Ah, the result gained by the multi-state system theory is larger than the result gained by the traditional system reliability theory. Fig. 2 is the reliability obtained by these two methods in different capacities. From Fig. 2 we know that the results obtained by the traditional syste m reliability theory are always conservative. For example, when the required capacity is 23.4 Ah, the reliability of the system obtained by the traditional system reliability theory is only 0.25107, but the reliability of the system obtained by the multi-state system theory is 0.55963. Figure 2. The results obtained by the two methods VI. C ONCLUSIONS The multi-state system theory is introduced to analyze the reliability of the power system in this paper, and is compared with the traditional system reliability theory. The results show that: 97 (1) The reliability of the power system obtained by the traditional system reliability theory is always conservative. (2) The power system is a multi-state system. The multi- state sy stem theory can define the relationship between component performance and system performance, and the reliability of the power system obtained by this method is much better. A CKNOWLEDGMENT The authors would like to thank the Graduate School of National Univer sity of Defense Technology for supporting this research work. R EFERENCES [1] X.F. Liu, J.Y. Zou, and L.W. Li, “Study on reliability of storage battery array with high capacity,” Journal of Jilin University (Engineering and Technology Edition), 2007, Vol.37, No.3, pp. 672–674. [2] G. Levitin, The Universal Generating Function in Reliability Anal ysis and Optimization, London: Springer, 2005. [3] A. Lisnianski, and G. Levitin, Multi-state System Reliability: Assessment, Optimization and Applications. 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Huang, “Reliability and performance assessment for fuzzy multi-state elements,” Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 2008, Vol. 222, No. 4, pp.675-686. 98 . Reliability Analysis of a Power System Based on the Multi-State System Theory Chunyang LI College of Mechatronics Engineering and Automation National. To analyze the reliability of the system by the traditional system reliability theory, we must gain the reliability of the battery first. The power system