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电源系统是由配电设备、储能设备、调波设备、开关设备、充电控制设备以及相关电线电路组成的总体[1],为各种电机提供高/低频、交/直流电源,其可靠性直接影响整个电机系统的平稳运行[2]。然而,电源系统结构复杂、设备繁多,如何理清其系统结构,识别主要脆弱点和关键薄弱点,影响并制约着供电任务的平稳实施。传统的静态可靠性分析方法只能应用于固定时刻的系统可靠性分析,在所有电源设备集成为一个动态系统时,其复杂结构、高危高压、动态失效的运行特点,使得该系统可靠性问题需采用动态的分析方式。
动态系统可靠性分析方法是近年的研究热点及难点,DBN从这些方法中脱颖而出。机器学习[3]、数据挖掘[4]等技术的兴起也为DBN的发展和应用提供了更为广阔的空间。目前,DBN的理论成果得到了进一步的发展完善,包括因果推断[5]、不确定性知识表达[6]、模式识别[7]和聚类分析[8]等。DBN也因其在双向推理及故障诊断方面的优势,被广泛应用于复杂系统的动态可靠性分析。文献[9]通过引入
$\,\beta $ 因子,结合模糊DBN,提出了一种针对数据缺失的多态系统可靠性分析方法。文献[10]基于贝叶斯网络对海上浮式风机进行了可靠性分析,大幅提升了故障诊断的准确率。文献[11]利用隐马尔科夫模型度量证据节点的先验概率,结合DBN,实现了化工设备的任务可靠性预计与评估。现有的DBN算法主要分为:离散时间贝叶斯网络算法(discrete-time Bayesian network, DTBN)[12]与连续时间贝叶斯网络算法(continues-time Bayesian network, CTBN)[13]。其中,DTBN以离散任务时间的方式,通过定义节点的条件概率表(conditional probability table, CPT)及边缘概率表(marginal probability table, MPT),求解出系统在不同任务时间片的后验概率。其中,文献[14]基于DTBN与改进的GO-FLOW方法,研究分析了多阶段多状态系统的共因失效问题。文献[15]通过引入区间分析理论,对无人机中表决系统的不确定性问题进行了量化处理,并基于DTBN算法对该系统进行了可靠性建模分析。然而,随着系统节点数与时间片数的增加,CPT&MPT的维度呈指数增加[16],导致求取系统可靠度所需的计算量也指数增长。
CTBN可在连续任务时间条件下,建立各个节点的概率密度函数(probability density function, PDF)的解析解,并借此得出任意时刻节点的后验概率,极大地减少了计算时间。文献[17]基于CTBN构建车辆系统的性能函数,并将其应用于该系统的可靠性优化设计。文献[18]通过模糊函数量化了失效数据的参数不确定性,并针对DBN中的动态逻辑门,构建了基于CTBN算法的模糊函数解析解模型。然而,现有的CTBN算法,需要针对不同的分析对象建立特定的分析模型,缺少通用的建模方法[19]。
因此,本文提出一种改进的DBN概率表建模方法,在无需离散任务时间的情况下,实现电源系统的动态可靠性分析与故障诊断。
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在DTBN算法中,任务时间被离散为多个时间片段,且节点的条件概率与边缘概率以CPT和MPT的形式给出。以2时间片段的DTBN为例,“与”门的CPT&MPT如图1所示。对于节点A和B,
${P_1}$ 和${P_2}$ 分别代表节点在对应时间片中的失效概率,而状态3的概率$1 - {P_1} - {P_2}$ 表示该节点在任务时间内未发生失效的概率。而在“与”门的CPT中,$P\left( {T{\rm{ = }}1\left| {A = 1,B = 1} \right.} \right) = 1$ 表示节点A和B在第一个时间片中失效时,该“与”门在此时间片内失效的条件概率为1,其他的概率也类似定义。基于图1的CPT,可通过式(1)~式(3)求解出节点T在各个状态的边缘概率。$$\begin{split} P\left( {T = 1} \right) &= \sum\limits_{i = 1}^3 {\sum\limits_{j = 1}^3 {P\left( {\left. {T = 1} \right|A = i,B = j} \right)} } \times \\ & P\left( {A = i} \right)P\left( {B = j} \right) = {P_1}^2 \\ \end{split} $$ (1) $$\begin{split} P&\left( {T = 2} \right) = \sum\limits_{i = 1}^3 {\sum\limits_{j = 1}^3 {P\left( {\left. {T = 2} \right|A = i,B = j} \right)} } \times \\ &P\left( {A = i} \right)P\left( {B = j} \right) = {P_1}{P_2}{\rm{ + }}{P_2}{P_1}{\rm{ + }}{P_2}^2 \end{split} $$ (2) $$\begin{split} P \left( {T = 3} \right) = \sum\limits_{i = 1}^3 {\sum\limits_{j = 1}^3 {P\left( {\left. {T = 3} \right|A = i,B = j} \right)} } \times \quad \\ P\left( {A = i} \right)P\left( {B = j} \right) = \left( {1 - {P_1} - {P_2}} \right)\left( {1 + {P_1} + {P_2}} \right) = \\ 1 - {\left( {{P_1} + {P_2}} \right)^2} = 1 - P\left( {T = 1} \right) - P\left( {T = 2} \right) \quad \end{split}$$ (3) -
与DTBN不同的是,在CTBN中,引入了单位阶跃函数和脉冲函数[13],所有节点的概率分布以PDF的形式给出。以图1中的“与”门为例,T的
${f_{\left. T \right|A,B}}\left( {\left. T \right|A,B} \right)$ 为:$${f_{\left. T \right|A,B}}\left( {\left. T \right|A,B} \right) = u\left( {b - a} \right)\delta \left( {t - b} \right) + u\left( {a - b} \right)\delta \left( {t - a} \right)$$ (4) 式中,
$u\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{0\quad\;\;\;\; {\rm{if}}\;x < 0}\\{1/2\quad {\rm{if}}\;x = 0}\\{1\quad\;\;\;\; {\rm{if}}\;x > 0}\end{array}} \right.$ 当
$x \ne {\rm{0}}$ ,$\delta \left( x \right) \!=\! 0$ ,且$\displaystyle\int\nolimits_0^\infty {\delta \left( t \right){\rm{d}}t} \!=\! 1$ ,$\displaystyle\int\nolimits_0^\infty {f\left( t \right)\delta \left( {t - \tau } \right)} $ $ {{\rm{d}}x} \!=\! f\left( \tau \right)$ 。则求解出节点T的PDF为:
$$\begin{split} {f_T}\left( t \right) = \int_0^\infty {\int_0^\infty {{f_{\left. T \right|A,B}}\left( {\left. T \right|A,B} \right){f_B}\left( b \right){f_A}\left( a \right){\rm{d}}a{\rm{d}}b} } = \\ \int_0^\infty {\int_0^\infty {\left[ {u\left( {b - a} \right)\delta \left( {t - b} \right) + u\left( {a - b} \right)\delta \left( {t - a} \right)} \right]}}\times\;\;\\ {{{f_B}\left( b \right){f_A}\left( a \right){\rm{d}}a{\rm{d}}b} } = {f_B}\left( t \right){F_A}\left( t \right) + {f_A}\left( t \right){F_B}\left( t \right) =\;\;\;\\ [{F_A}\left( t \right){F_B}\left( t \right)]' \qquad\qquad\qquad \end{split} $$ (5) 但是,正如引言中介绍的,现有的CTBN算法,需要针对不同的分析对象建立特定的解析解模型。并且随着动态逻辑门和节点数的增加,该方法的建模难度也会随之增加。因此,本文提出一种改进的CPT&MPT建模方法。
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以“与”门的CPT&MPT为例,如图2所示,
$P\left( {A = 1} \right) = {F_A}\left( t \right)$ 表示节点A在该时间点的失效概率,而$P\left( {A = 2} \right) = {R_A}\left( t \right) = 1 - {F_A}\left( t \right)$ 表示节点A在该时间点的可靠度。通过图2中节点T的CPT,则可由式(6)得出该节点的边缘概率分布。$$\begin{split} {F_T}\left( t \right) = &P\left( {T = 1} \right) = \sum\limits_{i = 1}^2 {\sum\limits_{j = 1}^2 } {{P\left( {\left. {T = 1} \right|A = i,B = j} \right)} } \times \\ &P\left( {A = i} \right)P\left( {B = j} \right) = {F_A}\left( t \right){F_B}\left( t \right) \\ \end{split}$$ (6) 可以看出,该CPT与传统CTBN算法中的单位阶跃函数和脉冲函数有相同的功能。且通过算法1,可以计算出任意数量节点的“与”门CPT,且该“与”门的父节点可以服从任意的失效分布类型。
算法 1 由
$i$ 个节点组成的“与”门的$\rm{CPT}$ 计算Input:
$i$ ← the number of parent nodes$\rm{CPT}$ ← zeros($\left[ {2,{2^i}} \right]$ )for
${P_1}$ ← 1 to 2for
${P_2}$ ←$\left( {\left( {{P_1} - 1} \right) \times 2 + 1} \right):{P_1} \times 2$ …
for
${P_i}$ ←$\left( {\left( {{P_{i - 1}} - 1} \right) \times 2 + 1} \right):{P_{i - 1}} \times 2$ $h = \max \left[ {{P_1},{P_2} - \left( {{P_1} - 1} \right) \times 2,\cdots,{P_i} - \left( {{P_{i - 1}} - 1} \right) \times 2} \right]$ ${\rm{CPT}}\left( {\left[ h,{P_i} \right]} \right) = 1$ end for
…
end for
end for
Output
$\rm{CPT}$ 而针对动态贝叶斯网络中的动态逻辑门,以温备份门为例(warm spare, WSP),如图3所示,节点B为节点A的温备份节点。当节点A正常运行时,节点B处于温备份状态,其失效率
${\lambda _B}$ 下降为$\alpha {\lambda _B}$ ,$\alpha $ 为备份因子,且$0 \leqslant \alpha \leqslant 1$ 。当$\alpha = 0$ 时,节点B转变为冷备份节点;$\alpha = 1$ 时,节点B转变为热备份节点。当A失效时,B转变为工作状态,且失效率转变为${\lambda _B}$ 。则节点B的条件概率可由式(7)~式(10)得出。$$P\left( {B = 1\left| {A = 1} \right.} \right) = \frac{{\displaystyle\int_0^t {{f_A}\left( {{t_A}} \right)\int_0^{{t_A}} {{f_{\alpha B}}\left( {{t_B}} \right){\rm{d}}{t_B}{\rm{d}}{t_A} + \int_0^t {{f_A}\left( {{t_A}} \right)\left[ {\left( {1 - \int_0^{{t_A}} {{f_{\alpha B}}\left( {{t_B}} \right){\rm{d}}{t_B}} } \right)\int_{{t_A}}^t {{f_B}\left( {{t_B} - {t_A}} \right)} {\rm{d}}{t_B}} \right]{\rm{d}}{t_A}} } } }}{{\displaystyle\int_0^t {{f_A}\left( {{t_A}} \right){\rm{d}}{t_A}} }}$$ (7) $$P\left( {B = 2\left| {A = 1} \right.} \right) = \frac{{\displaystyle\int_0^t {{f_A}\left( {{t_A}} \right)\left[ {\left( {1 - \int_0^{{t_A}} {{f_{\alpha B}}\left( {{t_B}} \right){\rm{d}}{t_B}} } \right)\int_t^{ + \infty } {{f_B}\left( {{t_B} - {t_A}} \right)} {\rm{d}}{t_B}} \right]{\rm{d}}{t_A}} }}{{\displaystyle\int_0^t {{f_A}\left( {{t_A}} \right){\rm{d}}{t_A}} }}$$ (8) $$P\left( {B = 1\left| {A = 2} \right.} \right) = \displaystyle\int_0^t {{f_{\alpha B}}\left( {{t_B}} \right){\rm{d}}{t_B}} $$ (9) $$P\left( {B = 2\left| {A = 2} \right.} \right) = \displaystyle\int_t^\infty {{f_{\alpha B}}\left( {{t_B}} \right){\rm{d}}{t_B}} $$ (10) 式中,
${f_A}\left( t \right)$ 、${f_B}\left( t \right)$ 分别为节点A和B工作时失效概率的PDF;${f_{\alpha B}}\left( t \right)$ 为节点B在温备份状态下的PDF;${t_A}$ 、${t_B}$ 分别为节点A和B的失效时间。针对电子系统,当节点A和B的失效类型服从指数分布时,即${f_A}\left( t \right) = {f_B}\left( t \right) = \lambda {{\rm{e}}^{ - \lambda t}}$ ,${f_{\alpha B}}\left( t \right) = \alpha \lambda {{\rm{e}}^{ - \alpha \lambda t}}$ ,则节点B的CPT如表1所示。表 1 温备份节点B的CPT
A 1 2 B 1 $1 - \dfrac{{{{\rm{e}}^{ - \lambda t}}\left( {1 - {{\rm{e}}^{ - \alpha \lambda t}}} \right)}}{{\alpha \left( {1 - {{\rm{e}}^{ - \lambda t}}} \right)}}$ $1 - {{\rm{e}}^{ - \alpha \lambda t}}$ 2 $\dfrac{{{{\rm{e}}^{ - \lambda t}}\left( {1 - {{\rm{e}}^{ - \alpha \lambda t}}} \right)}}{{\alpha \left( {1 - {{\rm{e}}^{ - \lambda t}}} \right)}}$ ${{\rm{e}}^{\left( { - \alpha \lambda t} \right)}}$ 为验证所提方法的准确性,以图3中的温备份门(节点T)为例,将各节点的CPT&MPT代入算法2中,当
${\lambda _A} = {\lambda _B} = 0.000\;01$ ,$\alpha = 0.5$ 时,可计算出T节点的可靠度曲线,分别与蒙特卡罗仿真(Monte Carlo simulation, MCS)[18]以及DTBN的计算结果进行了对比,如图4所示。算法 2:DBN后验概率求解算法
Input:
$N$ ← number of nodesdag(
$N$ ,$N$ ) ← adjacent matrix for DBNstate number of each node ←
$2\times {\rm{ones}}\left( {1,N} \right)$ for
$i$ ← 1 to 50 do$t$ ←$i \times 10\;000$ bnet.CPD{
$N$ } ←CPT&MPT;define inference engine;
define evidence;
end for
Output posterior probabilities of each node
由图4结果可知,所提方法可在未离散任务时间的情况下,获取与DTBN相同的系统可靠度计算结果。而且,本方法B节点的CPT维度仅为
$\left( {2 \times 2} \right)$ ,即便是与最简单的2时间片段DTBN相比,其建模难度也远小于DTBN的CPT[16]。与MCS的计算结果的对比分析结果也验证了所提方法的准确性。由表1的CPT可求解出B节点
${F_B}\left( t \right)$ 的解析解,如式(11)所示,其结果与CTBN[19]一致。并且,通过将构建的CPT&MPT与算法2相结合,即可计算得到观测节点的后验概率,而无需针对不同分析对象建立特定的解析解模型。$$\begin{split} {F_B}\left( t \right) = &P\left( {B = 1} \right) = \sum\limits_{i = 1}^2 {P\left( {\left. {B = 1} \right|A = i} \right)} P\left( {A = i} \right) = \\ &1 - \frac{1}{\alpha }{{\rm{e}}^{ - \lambda t}} + \frac{{1 - \alpha }}{\alpha }{{\rm{e}}^{ - \lambda \left( {1 + \alpha } \right)t}} \end{split} $$ (11) -
本文以某电源系统为研究对象,该系统组成结构如图5所示。
该系统通过将民用电转换成高压直流电,给规定容量的主预储能电容器充电。在规定的时间范围内达到指定值,在规定的时序触发下,主、预储能电容通过晶闸管固态开关、平波电感给负载放电,为负载提供合适的能量脉冲,使它们的形状、幅度和时序保持一致。根据该电源系统的结构框图,可建立其DBN如图6所示,事件编号及失效率如表2所示。
表 2 事件节点编号及失效率
节点
编号名称 节点
编号名称 失效率(${\lambda _i}$)/(10−6/h) TM 电源系统失效 AM 配电单元 2.67900 BM1 充电失效 EM 调波单元 0.01120 BM2 充电模块失效 B1 低压供电模块 0.29400 CM1 储能失效 B2 主充电模块 2.16040 CM2 储能电容失效 B3 预充电模块 1.08020 DM 开关失效 B4 控制采集模块 7.62380 C1 主储能电容模块 0.00580 C2 预储能电容模块 0.00058 C3 泄放单元模块 0.65470 C4 充电保护单元模块 0.00780 D1 主开关单元模块 7.53300 D2 预开关单元模块 5.27310
Reliability Analysis and Fault Diagnosis for Power System via Dynamic Bayesian Network
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摘要: 动态系统的可靠性分析与故障诊断一直是可靠性领域的热点及难点问题,作为该领域热门的分析工具之一,动态贝叶斯网络(DBN)得到了充分的应用与开发。但是,现有的DBN算法受限于系统的失效分布类型,且建模难度也随着系统复杂度的增加而呈指数增长。针对以上问题,该文提出一种改进的动态贝叶斯网络概率表建模方法,在连续任务时间的条件下,实现动态系统的可靠性分析。然后,结合DBN双向推理算法,求解系统失效时部件失效的后验概率,并将计算结果应用于系统故障诊断及薄弱部件定位。最后,结合某电源系统的可靠性分析与故障诊断,验证了该方法的实用性。Abstract: Reliability analysis and fault diagnosis for dynamic systems have always been hot topics in this field. As one of the popular reliability analysis methods, dynamic bayesian network (DBN) has been fully studied. However, the existing DBN algorithm has no general inference engines, and the modeling difficulty increases exponentially with the system complexity. This paper proposes a general probability table modeling method, which can also be applied on the dynamic reliability analysis of the system under the continuous mission time. Additionally, via the Bayesian inference algorithm, the posterior probability of component failure can be obtained, which can also be applied on system fault diagnosis. Finally, the validation of proposed method is verified by the reliability analysis and fault diagnosis of the power system.
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Key words:
- dynamic Bayesian network /
- fault diagnosis /
- power system /
- reliability analysis
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表 1 温备份节点B的CPT
A 1 2 B 1 $1 - \dfrac{{{{\rm{e}}^{ - \lambda t}}\left( {1 - {{\rm{e}}^{ - \alpha \lambda t}}} \right)}}{{\alpha \left( {1 - {{\rm{e}}^{ - \lambda t}}} \right)}}$ $1 - {{\rm{e}}^{ - \alpha \lambda t}}$ 2 $\dfrac{{{{\rm{e}}^{ - \lambda t}}\left( {1 - {{\rm{e}}^{ - \alpha \lambda t}}} \right)}}{{\alpha \left( {1 - {{\rm{e}}^{ - \lambda t}}} \right)}}$ ${{\rm{e}}^{\left( { - \alpha \lambda t} \right)}}$ 表 2 事件节点编号及失效率
节点
编号名称 节点
编号名称 失效率( ${\lambda _i}$ )/(10−6/h)TM 电源系统失效 AM 配电单元 2.67900 BM1 充电失效 EM 调波单元 0.01120 BM2 充电模块失效 B1 低压供电模块 0.29400 CM1 储能失效 B2 主充电模块 2.16040 CM2 储能电容失效 B3 预充电模块 1.08020 DM 开关失效 B4 控制采集模块 7.62380 C1 主储能电容模块 0.00580 C2 预储能电容模块 0.00058 C3 泄放单元模块 0.65470 C4 充电保护单元模块 0.00780 D1 主开关单元模块 7.53300 D2 预开关单元模块 5.27310 -
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