WO2017041393A1 - 三级马尔科夫模型开关磁阻电机系统可靠性定量评估方法 - Google Patents

三级马尔科夫模型开关磁阻电机系统可靠性定量评估方法 Download PDF

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WO2017041393A1
WO2017041393A1 PCT/CN2015/099103 CN2015099103W WO2017041393A1 WO 2017041393 A1 WO2017041393 A1 WO 2017041393A1 CN 2015099103 W CN2015099103 W CN 2015099103W WO 2017041393 A1 WO2017041393 A1 WO 2017041393A1
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state
states
matrix
zero
failure
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PCT/CN2015/099103
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陈昊
徐帅
董金龙
王星
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中国矿业大学
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Priority to AU2015381975A priority patent/AU2015381975B2/en
Priority to CA2938533A priority patent/CA2938533A1/en
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/34—Testing dynamo-electric machines
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/40—Testing power supplies
    • G01R31/42—AC power supplies
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N7/00—Computing arrangements based on specific mathematical models
    • G06N7/01—Probabilistic graphical models, e.g. probabilistic networks

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  • the invention relates to a quantitative evaluation method, in particular to a three-level Markov model quantitative evaluation method suitable for reliability of various types and various phase numbers of switched reluctance motor systems.
  • the quantitative evaluation of reliability mainly includes the establishment of reliability model and the quantitative solution based on reliability model.
  • the traditional reliability modeling method can only represent the basic normal and failure states of the switched reluctance motor system, and it is impossible to characterize all operating states of the switched reluctance motor system during the entire operating cycle.
  • the dynamic fault tree and the Markov model can characterize all possible states of the system, the dynamic fault tree model establishment process requires complex theoretical analysis and is not conducive to subsequent quantitative solutions.
  • the commonly used Markov modeling methods are often used in the reliability evaluation of software and electronic equipment.
  • the established model does not exert Markov's excellent characteristics based on state transition. Generally, a fault is a Markov space state, which is increased.
  • the complexity of the solution at the same time, the operation of the system under multi-level faults is not analyzed, and the reliability and fault tolerance of the system cannot be fully evaluated.
  • the reliability model quantitative solution methods mainly include Boolean logic method, Bayesian method and Markov state space method. Boolean logic and Bayesian method can not meet the analysis requirements in the case of multi-component and multi-fault, while the conventional Markov state space method can solve the above problems, but the solution time is too long due to the influence of the number of spatial states. Can not meet the requirements of the rapidity of reliability modeling.
  • the object of the present invention is to overcome the deficiencies of the prior art, and provide a three-level Markov model with simple method, fast evaluation speed and wide application range to quantitatively analyze the reliability evaluation method of the switched reluctance motor system.
  • the three-level Markov model of the present invention quantitatively analyzes the reliability evaluation method of the switched reluctance motor system, and the steps thereof are: through the first-level fault, the second-level fault, and the switched reluctance motor drive system
  • the system operation situation under the third-level fault is analyzed.
  • a total of five first-level Markov states are obtained, including four valid states and one failure state.
  • the 18 secondary Markov states include 14 active states and four failure states.
  • the 57 three-level Markov states include 43 effective states and 14 failure states. Considering the initial normal state and the final failure state, there are 62 effective states and 20 failure states in the third-order Markov model.
  • the state transition diagram of the switched reluctance motor drive system under the three-level fault, the effective state transition matrix A under the three-level fault is obtained:
  • the state transition matrix A is a square matrix of 62 rows and 62 columns, the row of the state transition matrix A is in the effective state, the column of the state transition matrix A is the next state to be transferred, and the corresponding transition rate is corresponding in the state transition matrix A.
  • the element's transfer rate of its own state is the inverse of the transfer probability sum of the state to all states (including the failed state).
  • A1, A11, A12, A13, A2, A3, A4 is a non-zero matrix
  • O represents a zero matrix
  • submatrix A1 is a square matrix of 13 rows and 13 columns:
  • B1, B21, B31, B2, B3 are non-zero matrices
  • O represents a zero matrix
  • B21 and B31 have only one non-zero element, and the rest are 0 elements, five sub-elements.
  • the matrices are:
  • Submatrix A2 is a square matrix of 18 rows and 18 columns:
  • B5, B61, B71, B81, B6, B7, B8 are non-zero matrices
  • O represents a zero matrix
  • seven non-zero matrices are B61, B71 There is only one non-zero element in B81, and the rest are 0 elements.
  • the seven sub-matrices are:
  • Submatrix A3 is a square matrix of 12 rows and 12 columns:
  • B10, B111, B121, B11, B12 are non-zero matrices
  • O represents a zero matrix
  • B111 and B121 have only one non-zero element, and the rest are 0 elements, five sub-mass
  • the matrices are:
  • Submatrix A4 is a square matrix of 19 rows and 19 columns:
  • B14, B151, B161, B171, B15, B16, B17 are non-zero matrices
  • O represents a zero matrix
  • B161 and B171 have only one non-zero element
  • the rest are For the 0 element
  • the seven sub-matrices are:
  • P A1 (t), P A2 (t), P A3 (t), and P A4 (t) in equation (31) represent the probabilities of the effective states in the A1 submodel, the A2 submodel, the A3 submodel, and the A4 submodel, respectively. , as shown in equations (32) through (35):
  • exp represents an exponential function
  • t represents time
  • A represents a state transition matrix
  • the three-level Markov model is used to quantitatively analyze the reliability of the switched reluctance motor system.
  • the reliability evaluation method of the switched reluctance motor system based on the quantitative analysis of the three-level Markov model not only effectively improves the reliability evaluation accuracy, but also if the switched reluctance motor system can tolerate three or more faults, the third level
  • the Markov model can characterize all possible operating states of a switched reluctance motor system under three-level faults. If the output of the system is within the allowable range, the state at this time can be reflected in the Markov model to maximize the representation.
  • the fault-tolerant performance of the switched reluctance motor system at the same time, the state transition based method in the Markov modeling process, the final impact of all possible faults on the switched reluctance motor system is state, greatly reducing the number of states, The rapidity of reliability quantitative evaluation is improved, and the reliability evaluation accuracy and speed can meet the requirements of high reliability switched reluctance motor system.
  • the third-order Markov model has the highest calculation accuracy and is suitable for environments with more equivalent failures and relatively loose failure determination conditions.
  • FIG. 1 is a Markov state transition diagram of a three-stage fault of a switched reluctance motor system of the present invention
  • Figure 2 is an A1 Markov submodel of the present invention
  • Figure 3 is an A2 Markov submodel of the present invention.
  • Figure 4 is an A3 Markov submodel of the present invention.
  • Figure 5 is an A4 Markov submodel of the present invention.
  • FIG. 6 is a schematic diagram of a switched reluctance motor system comprising a three-phase 12/8 structure switched reluctance motor and a three-phase double-switching power converter according to the present invention
  • 17 kinds of primary faults of the switched reluctance motor system are equivalent to 4 effective states and 1 failure state in the Markov space.
  • the four valid states are capacitor open circuit, turn-to-turn short circuit, phase loss, and lower tube short-circuit survival state, which are represented by A1, A2, A3, and A4 respectively.
  • the failure state is represented by A5, and the first-level fault enters five Markov state transitions.
  • the transfer rate is shown in Table 1.
  • the possible second-level faults Based on the first-level Markov state, consider the possible second-level faults, and summarize the possible second-level fault types as open-capacitance, turn-to-turn short circuit, upper tube short circuit, lower tube short circuit, phase loss fault, and Six cases of failure failure. There are five kinds of faults that may occur in the A1 state: the short circuit between the turn, the short circuit of the upper tube, the short circuit of the lower tube, the fault of the phase loss and the failure.
  • the state after the fault is summarized as four Markov states from B1 to B4.
  • the state transition rate of the Markov model under the second-level fault of the A1 state is shown in Table 2.
  • TTS Interturn short circuit
  • UMS Upper tube short circuit
  • DPH Phase loss
  • F Failure failure
  • Second level fault type B5 B6 B7 B8 B9 Open capacitor (CO) 1 0 0 0 0 2 Interturn short circuit (TTS) 0 0.1 0.9 0 0 3 Upper tube short circuit (UMS) 0 0 0.43 0 0.57 4 Down tube short circuit (DMS) 0 0 0.34 0.54 0.12 5 Phase loss (DPH) 0 0 0.88 0 0.12 6 Failure failure (F) 0 0 0 0 0 1
  • TTS Interturn short circuit
  • UMS Upper tube short circuit
  • DPH Phase loss
  • F Failure failure
  • TTS Interturn short circuit
  • UMS Upper tube short circuit
  • DPH Phase loss
  • F Failure failure
  • TTS Interturn short circuit
  • UMS Upper tube short circuit
  • DPH Phase loss
  • F Failure failure
  • TTS Interturn short circuit
  • UMS Upper tube short circuit
  • DPH Phase loss
  • F Failure failure
  • TTS Interturn short circuit
  • UMS Upper tube short circuit
  • DPH Phase loss
  • F Failure failure
  • phase loss fault includes five cases of open tube open circuit, open upper tube open circuit, upper diode short circuit, lower diode short circuit, turn-to-turn open circuit and position sensor open circuit, capacitor short circuit, upper diode open circuit, lower diode open circuit, inter-pole short circuit, and relative short circuit.
  • the phase-to-phase short circuit constitutes a failure fault.
  • first-level Markov states including four active states and one failure state are obtained.
  • the 18 secondary Markov states include 14 active states and 4 failure states.
  • the 57 tertiary Markov states include 43 active states and 14 failure states, taking into account the initial normal state and the final failure state. There are 62 effective states and 20 failure states in the three-level Markov model.
  • the switched reluctance motor system has a fault of the fourth or higher level, it is generally considered that the switched reluctance motor system fails.
  • the state transition diagram of the three-level Markov model of the switched reluctance motor system is obtained, as shown in Fig. 1.
  • the Markov space state is represented by a circle.
  • 00 is the first valid state
  • A1 corresponds to the second valid state
  • B1 corresponds to the third valid state
  • C1 to C3 corresponds to the valid state 4 to 6
  • B2 is the first 7 valid states
  • C5 to C6 correspond to valid states 8 to 9
  • B3 is the 10th valid state
  • C8 to C10 correspond to valid states 11 to 13
  • A2 corresponds to the 14th valid state
  • B5 corresponds to the 15th valid state
  • C12 To C14 corresponding to the valid state 16 to 18, B6 is the 19th active state
  • C16 to C18 correspond to the active state 20 to 22
  • B7 is the 23rd active state
  • C20 to C22 correspond to the valid state 24 to 26
  • B8 is the 27th Valid state
  • C24 to C27 correspond to valid state 28 to 31
  • the remaining states in the state transition diagram are A5, B4, B9, B13, B18, C4, C7, C11, C15, C19, C23, C28, C31, C35, C39, C43, C48, C52, C57 and F are all failed states. .
  • Table 22 shows the state transition rate symbols and calculation formulas for the primary and secondary faults in Figure 1.
  • the three-level fault state transition rate is shown in Table 23.
  • the transfer rate of the three-level effective state to the final state is as shown in Table 24.
  • ⁇ DP ⁇ UMO + ⁇ DMO + ⁇ UDS + ⁇ DDS + ⁇ TTO + ⁇ PSO
  • ⁇ DP1 0.88 ⁇ DP +0.34( ⁇ DMS + ⁇ PSS )+0.43 ⁇ UMS +0.9 ⁇ TTS
  • ⁇ SP ⁇ CS +3( ⁇ UDO + ⁇ DDO + ⁇ POS + ⁇ PGS + ⁇ PHS )
  • ⁇ SP1 ⁇ CS +2( ⁇ UDO + ⁇ DDO + ⁇ POS + ⁇ PGS + ⁇ PHS )
  • the three-level Markov model consists of four sub-models: A1, A2, A3, and A4, as shown in Figure 2, Figure 3, Figure 4, and Figure 5, respectively.
  • the reliability is the probability of being in an effective state
  • a quantitative evaluation of the reliability can be achieved by simply finding the probability sum of the effective states.
  • first-level Markov states including four active states and one failure state are obtained.
  • the 18 secondary Markov states include 14 active states and 4 failure states.
  • the 57 tertiary Markov states include 43 active states and 14 failure states, taking into account the initial normal state and the final failure state.
  • the state transition diagram of the switched reluctance motor drive system under the three-level fault is established, and the effective state transition matrix A under the three-level fault is obtained:
  • the state transition matrix A is a square matrix of 62 rows and 62 columns, the row of the state transition matrix A is in the effective state, the column of the state transition matrix A is the next state to be transferred, and the corresponding transition rate is corresponding in the state transition matrix A.
  • the element's transfer rate of its own state is the inverse of the transfer probability sum of the state to all states (including the failed state).
  • A1, A11, A12, A13, A2, A3, A4 are non-zero matrices
  • O represents a zero matrix
  • submatrix A1 is a square matrix of 13 rows and 13 columns:
  • B1, B21, B31, B2, B3 are non-zero matrices
  • O represents a zero matrix
  • B21 and B31 have only one non-zero element, and the rest are 0 elements, five sub-elements.
  • the matrices are:
  • Submatrix A2 is a square matrix of 18 rows and 18 columns:
  • B5, B61, B71, B81, B6, B7, B8 are non-zero matrices
  • O represents a zero matrix
  • B71 and B81 have only one non-zero element
  • the rest are For the 0 element
  • the seven sub-matrices are:
  • Submatrix A3 is a square matrix of 12 rows and 12 columns:
  • B10, B111, B121, B11, B12 are non-zero matrices
  • O represents a zero matrix
  • B111 and B121 have only one non-zero element, and the rest are 0 elements, five sub-mass
  • the matrices are:
  • Submatrix A4 is a square matrix of 19 rows and 19 columns:
  • B14, B151, B161, B171, B15, B16, B17 are non-zero matrices
  • O represents a zero matrix
  • B161 and B171 have only one non-zero element
  • the rest are For the 0 element
  • the seven sub-matrices are:
  • P A1 (t), P A2 (t), P A3 (t), and P A4 (t) in equation (31) represent the probabilities of the effective states in the A1 submodel, the A2 submodel, the A3 submodel, and the A4 submodel, respectively. , as shown in equations (32) through (35):
  • the three-level Markov model is used to quantitatively analyze the reliability of the switched reluctance motor system.
  • a switched reluctance motor system consisting of a three-phase 12/8 structure switched reluctance motor and a three-phase two-switch power converter, as shown in Figure 6, passes through the switched reluctance motor system shown in Figure 1.
  • the Markov state transition diagram under the fault of the level, establishes the state transition matrix A under the three-level fault, solves the probability matrix P(t) of the switched reluctance motor system in the effective state, and calculates the effective state probability matrix P(t).
  • the sum of the elements obtains the reliability function R(t) of the switched reluctance motor system.
  • the integral of the reliability function curve R(t) in the time domain 0 to infinity can be calculated.
  • the MTBF of the switched reluctance motor system is 3637112 hours, which realizes the quantitative evaluation of the reliability of the three-stage Markov model of the three-phase switched reluctance motor system.
  • the MTBF of the mean time between failures reflects the area of the reliability function curve R(t) and the abscissa axis and the ordinate axis in the first quadrant. The larger the area, the more reliable the system.

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Abstract

三级马尔科夫模型定量分析开关磁阻电机系统可靠性评估方法，通过对开关磁阻电机驱动系统在第一级故障、第二级故障和第三级故障下系统运行情况进行分析，得到一级故障下有4个有效状态和1个失效状态，二级故障下有14个有效状态和4个失效状态，三级故障下有43个有效状态和14个失效状态，并考虑初始正常状态和最终失效状态，则三级马尔科夫模型中共有62个有效状态和20个失效状态，建立开关磁阻电机系统三级故障下的状态转移图，得到状态转移矩阵，解得系统处于有效状态的概率矩阵P(t)，计算有效状态概率矩阵P(t)各元素之和，由可靠度函数R(t)计算出平均无故障时间，从而实现了三级马尔科夫模型定量分析开关磁阻电机系统可靠性评估。具有良好的工程应用价值。

Description

三级马尔科夫模型开关磁阻电机系统可靠性定量评估方法 技术领域
本发明涉及一种定量评估方法，尤其是适用于各种类型、各种相数的开关磁阻电机系统可靠性的三级马尔科夫模型定量评估方法。
背景技术
可靠性定量评估主要包含可靠性模型的建立和基于可靠性模型的定量求解两部分。传统的可靠性建模方法只能表示开关磁阻电机系统基本正常和失效两种状态，无法实现对开关磁阻电机系统整个运行周期所有运行状态的表征。动态故障树和马尔科夫模型虽然能够表征系统所有可能出现的状态，但是动态故障树模型建立过程需要复杂的理论分析，同时不利于后续的定量求解。现在常用的马尔科夫建模方法多用在软件和电子设备可靠性评估，建立的模型没有发挥马尔科夫基于状态转移的优良特性，一般是一个故障即为一种马尔科夫空间状态，增加了求解的复杂度；同时没有对多级故障下系统的运行情况进行分析，不能完整评估系统的可靠性和容错能力。可靠性模型定量求解方法主要有布尔逻辑法、贝叶斯法和马尔科夫状态空间法。布尔逻辑法和贝叶斯法无法满足多部件和多故障情况下的分析要求，而常规的马尔科夫状态空间法采用虽能解决上述问题，但由于受空间状态数目的影响使求解时间过长，不能满足可靠性建模快速性的要求。因此，急需对开关磁阻电机系统实现马尔科夫模型分级可靠性定量评估，顾及一种故障可以进入不同的马尔科夫状态，能表示处于正常和失效状态之间的开关磁阻电机系统带故障有效运行状态，并减少马尔科夫空间状态数，快速实现开关磁阻电机系统可靠性的定量评估。
发明内容
本发明的目的是克服已有技术的不足之处，提供一种方法简单，评估速度快，使用范围广的三级马尔科夫模型定量分析开关磁阻电机系统可靠性评估方法。
为达到上述技术目的，本发明的三级马尔科夫模型定量分析开关磁阻电机系统可靠性评估方法，其步骤为：通过对开关磁阻电机驱动系统在第一级故障、第二级故障和第三级故障下系统运行情况进行分析，共得到5个一级马尔科夫状态包括4个有效状态和1个失效状态，18个二级马尔科夫状态包括14个有效状态和4个失效状态，57个三级马尔科夫状态包括43个有效状态和14个失效状态，同时考虑初始正常状态和最终失效状态，则三级马尔科夫模型中共有62个有效状态和20个失效状态，建立开关磁阻电机驱动系统在三级故障下的状态转移图，得到三级故障下的有效状态转移矩阵A:
Figure PCTCN2015099103-appb-000001
状态转移矩阵A是62行62列的方阵，状态转移矩阵A的行是所处有效状态，状态转移矩阵A的列是要转移的下一状态，对应的转移率为状态转移矩阵A中对应的元素，自身状态的转移率是该状态向所有状态(包含失效状态)转移的转移概率和的相反数。式(1)中，A1，A11，A12，A13，A2，A3， A4为非零矩阵，O表示零矩阵，子矩阵A1是13行13列的方阵：
Figure PCTCN2015099103-appb-000002
式(2)中，B1，B21，B31，B2，B3为非零矩阵，O表示零矩阵，五个非零矩阵中B21和B31仅有1个非零元素，其余均为0元素，五个子矩阵分别是：
Figure PCTCN2015099103-appb-000003
Figure PCTCN2015099103-appb-000004
Figure PCTCN2015099103-appb-000005
Figure PCTCN2015099103-appb-000006
Figure PCTCN2015099103-appb-000007
子矩阵A2是18行18列的方阵：
Figure PCTCN2015099103-appb-000008
式(8)中，B5，B61，B71，B81，B6，B7，B8为非零矩阵，O表示零矩阵，七个非零矩阵中B61，B71 和B81仅有1个非零元素，其余均为0元素，七个子矩阵分别是：
Figure PCTCN2015099103-appb-000009
Figure PCTCN2015099103-appb-000010
Figure PCTCN2015099103-appb-000011
Figure PCTCN2015099103-appb-000012
Figure PCTCN2015099103-appb-000013
Figure PCTCN2015099103-appb-000014
Figure PCTCN2015099103-appb-000015
子矩阵A3是12行12列的方阵：
Figure PCTCN2015099103-appb-000016
式(16)中，B10，B111，B121，B11，B12为非零矩阵，O表示零矩阵，五个非零矩阵中B111和B121仅有1个非零元素，其余均为0元素，五个子矩阵分别是：
Figure PCTCN2015099103-appb-000017
Figure PCTCN2015099103-appb-000018
Figure PCTCN2015099103-appb-000019
Figure PCTCN2015099103-appb-000020
Figure PCTCN2015099103-appb-000021
子矩阵A4是19行19列的方阵：
Figure PCTCN2015099103-appb-000022
式(22)中，B14，B151，B161，B171，B15，B16，B17为非零矩阵，O表示零矩阵，七个非零矩阵中B151，B161和B171仅有1个非零元素，其余均为0元素，七个子矩阵分别是：
Figure PCTCN2015099103-appb-000023
Figure PCTCN2015099103-appb-000024
Figure PCTCN2015099103-appb-000025
Figure PCTCN2015099103-appb-000026
Figure PCTCN2015099103-appb-000027
Figure PCTCN2015099103-appb-000028
Figure PCTCN2015099103-appb-000029
式中，λA1、λA2、λA3、λA4、λA5、λB1、λB2、λB3、λB4、λB5、λB6、λB7、λB8、λB9、λB10、λB11、λB12、λB13、λB14、λB15、λB16、λB17、λB18、λC1、λC2、λC3、λC4、λC5、λC6、λC7、λC8、λC9、λC10、λC11、λC12、λC13、λC14、λC15、λC16、λC17、λC18、λC19、λC20、λC21、λC22、λC23、λC24、λC25、λC26、λC27、λC28、λC29、λC30、λC31、λC32、λC33、λC34、λC35、λC36、λC37、λC38、λC39、λC40、λC41、λC42、λC43、λC44、λC45、λC46、λC47、λC48、λC49、λC50、λC51、λC52、λC53、λC54、λC55、λC56、λC57、λF1、λF2、λF3、λF4、λF5、λF6、λF7、λF8、λF9、λF10、λF11、λF12、λF13、λF14、λF15、λF16、λF17、λF18、λF19、λF20、λF21、λF22、λF23、λF24、λF25、λF26、λF27、 λF28、λF29、λF30、λF31、λF32、λF33、λF34、λF35、λF36、λF37、λF38、λF39、λF40、λF41、λF42、λF43是三级马尔科夫模型状态转移率；
利用公式：
Figure PCTCN2015099103-appb-000030
解得开关磁阻电机系统处于有效状态的概率矩阵P(t)：
Figure PCTCN2015099103-appb-000031
式(31)中PA1(t)、PA2(t)、PA3(t)和PA4(t)分别表示A1子模型、A2子模型、A3子模型和A4子模型中有效状态的概率，如式(32)到(35)所示：
Figure PCTCN2015099103-appb-000032
Figure PCTCN2015099103-appb-000033
Figure PCTCN2015099103-appb-000034
Figure PCTCN2015099103-appb-000035
式(32)到(35)中，exp表示指数函数，t表示时间，A代表状态转移矩阵；
由公式(31)计算有效状态概率矩阵P(t)各元素之和，得到开关磁阻电机系统的可靠度函数R(t)：
R(t)＝0.0018exp(-3.96t)+0.0184exp(-3.95t)+8.7e-4exp(-3.83t)
-0.004exp(-3.83t)-1.74exp(-0.476t)+0.332exp(-0.237t)+5.14e
-4exp(-3.74t)-0.0142exp(-3.73t)+8.85e-4exp(-3.67t)
+0.0029exp(-1.83t)+0.01exp(-3.64t)-0.035exp(-3.64t)
+0.004exp(-1.82t)-0.003exp(-3.63t)-0.011exp(-3.55t)
+0.0544exp(-1.73t)-0.026exp(-3.44t)+0.119exp(-1.72t)
+0.005exp(-3.43t)-0.0046exp(-3.42t)-0.0211exp(-3.32t)
-0.108exp(-3.24t)-0.0595exp(-3.24t)+0.00269exp(-0.404t)
-0.245exp(-3.22t)+0.145exp(-3.19t)-0.0952exp(-3.11t)
-0.0662exp(-3.08t)+0.024exp(-1.54t)+0.005exp(-3.04t)
-0.166exp(-2.99t)+0.0345exp(-2.96t)-0.0231exp(-2.95t)
+2.05exp(-0.364t)+0.04exp(-0.36t)-2.59exp(-4.81t)
+3.48e-5exp(-4.57t)+1.34e-5exp(-4.43t)+3.54e-4exp(-4.39t)
+3.3exp(-4.38t)+1.28e-5exp(-4.27t)+1.36e-5exp(-4.27t)
+0.0013exp(-4.26t)-3.08e-4exp(-4.14t)+0.01exp(-4.07t)
+0.023exp(-4.07t)+7.97e-4exp(-4.04)+0.001exp(-2.01t)(36)
由可靠度函数R(t)计算出开关磁阻电机系统的平均无故障时间：
Figure PCTCN2015099103-appb-000036
从而实现了三级马尔科夫模型定量分析开关磁阻电机系统可靠性评估。
有益效果：三级马尔科夫模型定量分析的开关磁阻电机系统可靠性评估方法不仅有效地提高了可靠性评估精度，而且若开关磁阻电机系统能容忍三级及三级以上故障，三级马尔科夫模型能够表征开关磁阻电机系统在三级故障下的所有可能运行状态，如果系统的输出在允许的范围内，此时的状态可在马尔科夫模型中体现出来，最大程度地表征了开关磁阻电机系统的容错性能；同时马尔科夫建模过程中基于状态转移的方法，以所有可能发生的故障对开关磁阻电机系统最终影响为状态，极大程度的减少了状态数，提高了可靠性定量评估的快速性，可靠性评估精度和速度都能满足高可靠性开关磁阻电机系统的要求。三级马尔科夫模型计算精度最高，适用于等效故障数较多和失效判别条件相对宽松的环境。
附图说明
图1是本发明的开关磁阻电机系统三级故障下的马尔科夫状态转移图；
图2是本发明的A1马尔科夫子模型；
图3是本发明的A2马尔科夫子模型；
图4是本发明的A3马尔科夫子模型；
图5是本发明的A4马尔科夫子模型；
图6是本发明的一台由三相12/8结构开关磁阻电机和三相双开关式功率变换器组成的开关磁阻电机系统示意图；
图7是本发明的开关磁阻电机系统马尔科夫可靠性模型解得的可靠度函数曲线。
具体实施方式
下面结合附图中的实施例对本发明作进一步的描述：
依据开关磁阻电机系统一级故障发生后系统的表现形式，
依据开关磁阻电机系统一级故障发生后系统的表现形式，将开关磁阻电机系统17种一级故障等效为马尔科夫空间中的4个有效状态和1个失效状态。4个有效状态为电容开路、匝间短路、缺相、下管短路存活状态，分别用A1、A2、A3、A4表示，失效状态用A5表示，一级故障进入5种马尔科夫状态转化的转移率如表1所示。
表1  第一级故障下马尔科夫模型状态转移率
编号 一级故障类型 A1 A2 A3 A4 A5
1 电容开路(CO) 1 0 0 0 0
2 电容短路(CS) 0 0 0 0 1
3 下管短路(DMS) 0 0 0.34 0.54 0.12
4 下管开路(DMO) 0 0 0.88 0 0.12
5 上管短路(UMS) 0 0 0.43 0 0.57
6 上管开路(UMO) 0 0 0.88 0 0.12
7 上二极管短路(UDS) 0 0 0.88 0 0.12
8 上二极管开路(UDO) 0 0 0 0 1
9 下二极管短路(DDS) 0 0 0.88 0 0.12
10 下二极管开路(DDO) 0 0 0 0 1
11 匝间短路(TTS) 0 0.1 0 0 0.9
12 极间短路(POS) 0 0 0 0 1
13 相对地短路(PGS) 0 0 0 0 1
14 相间短路(PHS) 0 0 0 0 1
15 匝间开路(TTO) 0 0 0.88 0 0.12
16 位置传感器短路(PPS) 0 0 0.34 0.54 0.12
17 位置传感器开路(PPO) 0 0 0.88 0 0.12
在一级马尔科夫状态的基础上，考虑可能发生的第二级故障，将可能发生的第二级故障类型总结为电容开路、匝间短路、上管短路、下管短路、缺相故障和失效故障六种情况。A1状态下可能发生5种故障：匝间短路、上管短路、下管短路、缺相故障和失效故障，故障发生后的状态总结为B1到B4共4种马尔科夫状态。A1状态第二级故障下马尔科夫模型状态转移率如表2所示。
表2  A1状态第二级故障下马尔科夫模型状态转移率
编号 第二级故障类型 B1 B2 B3 B4
1 匝间短路(TTS) 0.1 0 0.9 0
2 上管短路(UMS) 0 0.43 0 0.57
3 下管短路(DMS) 0 0.34 0.54 0.12
4 缺相(DPH) 0 0.88 0 0.12
5 失效故障(F) 0 0 0 1
A2状态下可能发生6种故障：电容开路、匝间短路、上管短路、下管短路、缺相故障和失效故障，对应马尔科夫状态为B5到B9。A2状态第二级故障下马尔科夫模型状态转移率如表3所示。
表3  A2状态第二级故障下马尔科夫模型状态转移率
编号 第二级故障类型 B5 B6 B7 B8 B9
1 电容开路(CO) 1 0 0 0 0
2 匝间短路(TTS) 0 0.1 0.9 0 0
3 上管短路(UMS) 0 0 0.43 0 0.57
4 下管短路(DMS) 0 0 0.34 0.54 0.12
5 缺相(DPH) 0 0 0.88 0 0.12
6 失效故障(F) 0 0 0 0 1
A3状态下可能发生6种故障：电容开路、匝间短路、上管短路、下管短路、缺相故障和失效故障，等效为马尔科夫状态B10到B13。A3状态第二级故障下马尔科夫模型状态转移率如表4所示。
表4  A3状态第二级故障下马尔科夫模型状态转移率
编号 第二级故障类型 B10 B11 B12 B13
1 电容开路(CO) 1 0 0 0
2 匝间短路(TTS) 0 0.1 0.9 0
3 上管短路(UMS) 0 0 0 1
4 下管短路(DMS) 0 0 0.4 0.6
5 缺相(DPH) 0 0 0 1
6 失效故障(F) 0 0 0 1
A4状态下可能发生6种故障：电容开路、匝间短路、上管短路、下管短路、缺相故障和失效故障，等效为马尔科夫状态B14到B18。A4状态第二级故障下马尔科夫模型状态转移率如表5所示。
表5  A4状态第二级故障下马尔科夫模型状态转移率
编号 第二级故障类型 B14 B15 B16 B17 B18
1 电容开路(CO) 1 0 0 0 0
2 匝间短路(TTS) 0 0.1 0.9 0 0
3 上管短路(UMS) 0 0 0.35 0 0.65
4 下管短路(DMS) 0 0 0.4 0.45 0.15
5 缺相(DPH) 0 0 0.4 0.38 0.22
6 失效故障(F) 0 0 0 0 1
在二级马尔科夫状态的基础上，考虑可能发生的第三级故障，同样可总结为六种故障类型电容开路、匝间短路、上管短路、下管短路、缺相故障和失效故障情况。B1状态下可能发生5种故障：匝间短路、上管短路、下管短路、缺相故障和失效故障，故障发生后的状态总结为C1到C4共4种马尔科夫状态。对应转移率如表6所示。
表6  B1状态第三级故障下马尔科夫模型状态转移率
编号 第三级故障类型 C1 C2 C3 C4
1 匝间短路(TTS) 0.1 0.9 0 0
2 上管短路(UMS) 0 0.43 0 0.57
3 下管短路(DMS) 0 0.34 0.54 0.12
4 缺相(DPH) 0 0.88 0 0.12
5 失效故障(F) 0 0 0 1
B2状态下可能发生5种故障：匝间短路、上管短路、下管短路、缺相故障和失效故障，故障发生后的状态总结为C5到C7共3种马尔科夫状态。对应转移率如表7所示。
表7  B2状态第三级故障下马尔科夫模型状态转移率
Figure PCTCN2015099103-appb-000037
Figure PCTCN2015099103-appb-000038
B3状态下可能发生5种故障：匝间短路、上管短路、下管短路、缺相故障和失效故障，故障发生后的状态总结为C8到C11共4种马尔科夫状态。对应状态转移率如表8所示。
表8  B3状态第三级故障下马尔科夫模型状态转移率
编号 第三级故障类型 C8 C9 C10 C11
1 匝间短路(TTS) 0.1 0.9 0 0
2 上管短路(UMS) 0 0.35 0 0.65
3 下管短路(DMS) 0 0.4 0.45 0.15
4 缺相(DPH) 0 0.4 0.38 0.22
5 失效故障(F) 0 0 0 1
B5状态下可能发生5种故障：匝间短路、上管短路、下管短路、缺相故障和失效故障，故障发生后的状态总结为C12到C15共4种马尔科夫状态。对应状态转移率如表9所示。
表9  B5状态第三级故障下马尔科夫模型状态转移率
编号 第三级故障类型 C12 C13 C14 C15
1 匝间短路(TTS) 0.1 0.9 0 0
2 上管短路(UMS) 0 0.43 0 0.57
3 下管短路(DMS) 0 0.34 0.54 0.12
4 缺相(DPH) 0 0 0.88 0.12
5 失效故障(F) 0 0 0 1
B6状态下可能发生6种故障：电容开路、匝间短路、上管短路、下管短路、缺相故障和失效故障，等效为马尔科夫状态C16到C19。B6状态第三级故障下马尔科夫模型状态转移率如表10所示。
表10  B6状态第三级故障下马尔科夫模型状态转移率
编号 第三级故障类型 C16 C17 C18 C19
1 电容开路(CO) 1 0 0 0
2 匝间短路(TTS) 0 0 0 1
3 上管短路(UMS) 0 0.43 0 0.57
4 下管短路(DMS) 0 0.34 0.54 0.12
5 缺相(DPH) 0 0.88 0 0.12
6 失效故障(F) 0 0 0 1
B7状态下可能发生6种故障：电容开路、匝间短路、上管短路、下管短路、缺相故障和失效故障，等效为马尔科夫状态C20到C23。B7状态第三级故障下马尔科夫模型状态转移率如表11所示。
表11  B7状态第三级故障下马尔科夫模型状态转移率
Figure PCTCN2015099103-appb-000039
Figure PCTCN2015099103-appb-000040
B8状态下可能发生6种故障：电容开路、匝间短路、上管短路、下管短路、缺相故障和失效故障，等效为马尔科夫状态C24到C28。B8状态第三级故障下马尔科夫模型状态转移率如表12所示。
表12  B8状态第三级故障下马尔科夫模型状态转移率
编号 第三级故障类型 C24 C25 C26 C27 C28
1 电容开路(CO) 1 0 0 0 0
2 匝间短路(TTS) 0 0.1 0.9 0 0
3 上管短路(UMS) 0 0 0.35 0 0.65
4 下管短路(DMS) 0 0 0.4 0.45 0.15
5 缺相(DPH) 0 0 0.4 0.38 0.22
6 失效故障(F) 0 0 0 0 1
B10状态下可能发生5种故障：匝间短路、上管短路、下管短路、缺相故障和失效故障，等效为马尔科夫状态C29到C31。B10状态第三级故障下马尔科夫模型状态转移率如表13所示。
表13  B10状态第三级故障下马尔科夫模型状态转移率
编号 第三级故障类型 C29 C30 C31
1 匝间短路(TTS) 0.1 0 0.9
2 上管短路(UMS) 0 0 1
3 下管短路(DMS) 0 0.38 62
4 缺相(DPH) 0 0 1
5 失效故障(F) 0 0 1
B11状态下可能发生6种故障：电容开路、匝间短路、上管短路、下管短路、缺相故障和失效故障，等效为马尔科夫状态C32到C35。B11状态第三级故障下马尔科夫模型状态转移率如表14所示。
表14  B11状态第三级故障下马尔科夫模型状态转移率
编号 第三级故障类型 C32 C33 C34 C35
1 电容开路(CO) 1 0 0 0
2 匝间短路(TTS) 0 0.1 0 0.9
3 上管短路(UMS) 0 0 0 1
4 下管短路(DMS) 0 0 0.38 0.62
5 缺相(DPH) 0 0 0 1
6 失效故障(F) 0 0 0 1
B12状态下可能发生6种故障：电容开路、匝间短路、上管短路、下管短路、缺相故障和失效故障，等效为马尔科夫状态C36到C39。B12状态第三级故障下马尔科夫模型状态转移率如表15所示。
表15  B12状态第三级故障下马尔科夫模型状态转移率
Figure PCTCN2015099103-appb-000041
Figure PCTCN2015099103-appb-000042
B14状态下可能发生5种故障：匝间短路、上管短路、下管短路、缺相故障和失效故障，等效为马尔科夫状态C40到C43。B14状态第三级故障下马尔科夫模型状态转移率如表16所示。
表16  B14状态第三级故障下马尔科夫模型状态转移率
编号 第三级故障类型 C40 C41 C42 C43
1 匝间短路(TTS) 0 0.1 0.9 0
2 上管短路(UMS) 0 0.35 0 0.65
3 下管短路(DMS) 0 0.4 0.45 0.15
4 缺相(DPH) 0 0.4 0.38 0.22
5 失效故障(F) 0 0 0 1
B15状态下可能发生6种故障：电容开路、匝间短路、上管短路、下管短路、缺相故障和失效故障，等效为马尔科夫状态C44到C48。B15状态第三级故障下马尔科夫模型状态转移率如表17所示。
表17  B15状态第三级故障下马尔科夫模型状态转移率
编号 第三级故障类型 C44 C45 C46 C47 C48
1 电容开路(CO) 1 0 0 0 0
2 匝间短路(TTS) 0 0.1 0.9 0 0
3 上管短路(UMS) 0 0 0.35 0 0.65
4 下管短路(DMS) 0 0 0.4 0.45 0.15
5 缺相(DPH) 0 0 0.4 0.38 0.22
6 失效故障(F) 0 0 0 0 1
B16状态下可能发生6种故障：电容开路、匝间短路、上管短路、下管短路、缺相故障和失效故障，等效为马尔科夫状态C49到C52。B16状态第三级故障下马尔科夫模型状态转移率如表18所示。
表18  B16状态第三级故障下马尔科夫模型状态转移率
编号 第三级故障类型 C49 C50 C51 C52
1 电容开路(CO) 1 0 0 0
2 匝间短路(TTS) 0 0.1 0 0.9
3 上管短路(UMS) 0 0 0 1
4 下管短路(DMS) 0 0 0.4 0.6
5 缺相(DPH) 0 0 0 1
6 失效故障(F) 0 0 0 1
B17状态下可能发生6种故障：电容开路、匝间短路、上管短路、下管短路、缺相故障和失效故障，等效为马尔科夫状态C53到C57。B17状态第三级故障下马尔科夫模型状态转移率如表19所示。
表19  B17状态第三级故障下马尔科夫模型状态转移率
编号 第三级故障类型 C53 C54 C55 C56 C57
1 电容开路(CO) 1 0 0 0 0
2 匝间短路(TTS) 0 0.1 0.9 0 0
3 上管短路(UMS) 0 0 0.35 0 0.65
4 下管短路(DMS) 0 0 0.4 0.45 0.15
5 缺相(DPH) 0 0 0.4 0.38 0.22
6 失效故障(F) 0 0 0 0 1
上述缺相故障包含下管开路、上管开路、上二极管短路、下二极管短路、匝间开路和位置传感器开路五种情况，电容短路、上二极管开路、下二极管开路、极间短路、相对地短路、相间短路组成失效故障。
通过对开关磁阻电机驱动系统在第一级故障、第二级故障和第三级故障下系统运行情况进行分析，共得到5个一级马尔科夫状态包括4个有效状态和1个失效状态，18个二级马尔科夫状态包括14个有效状态和4个失效状态，57个三级马尔科夫状态包括43个有效状态和14个失效状态，同时考虑初始正常状态和最终失效状态，则三级马尔科夫模型中共有62个有效状态和20个失效状态，
若开关磁阻电机系统有第四级以上故障发生，一般认为开关磁阻电机系统失效。
综上分析得到开关磁阻电机系统三级马尔科夫模型下的状态转移图，如图1所示。马尔科夫空间状态用圆圈表示，状态转移图中00为第1个有效状态，A1对应第2个有效状态，B1对应第3个有效状态，C1到C3对应有效状态4到6，B2为第7个有效状态，C5到C6对应有效状态8到9，B3为第10个有效状态，C8到C10对应有效状态11到13，A2对应第14个有效状态，B5对应第15个有效状态，C12到C14对应有效状态16到18，B6为第19个有效状态，C16到C18对应有效状态20到22，B7为第23个有效状态，C20到C22对应有效状态24到26，B8为第27个有效状态，C24到C27对应有效状态28到31，A3对应第32个有效状态，B10对应第33个有效状态，C29到C30对应有效状态34到35，B11对应第36个有效状态，C32到C34对应有效状态37到39，B12对应第40个有效状态，C36到C38对应有效状态41到43，A4对应第44个有效状态，B14对应第45个有效状态，C40到C42对应有效状态46到48，B15为第49个有效状态，C44到C47对应有效状态50到53，B16为第54个有效状态，C49到C51对应有效状态55到57，B17为第58个有效状态，C53到C56对应有效状态59到62。状态转移图中其余的状态A5，B4，B9，B13，B18，C4，C7，C11，C15，C19，C23，C28，C31，C35，C39，C43，C48，C52，C57和F均为失效状态。
图1中一级和二级马尔科夫空间状态符号及含义如表20所示。
表20  一级和二级马尔科夫空间状态符号
Figure PCTCN2015099103-appb-000043
Figure PCTCN2015099103-appb-000044
图1中三级马尔科夫空间状态及最终失效状态符号及含义如表21所示。
表21  三级马尔科夫空间状态符号
Figure PCTCN2015099103-appb-000045
表22为图1中一级和二级故障的状态转移率符号及计算公式。
表22  一级和二级故障状态转移率
Figure PCTCN2015099103-appb-000046
三级故障状态转移率如表23所示。
表23  三级故障状态转移率
Figure PCTCN2015099103-appb-000047
Figure PCTCN2015099103-appb-000048
Figure PCTCN2015099103-appb-000049
三级有效状态向最终状态的转移率如表24所示。
表24  三级有效状态向最终状态的转移率
Figure PCTCN2015099103-appb-000050
Figure PCTCN2015099103-appb-000051
表22、表23、表24的计算公式中符号含义如表25所示。
表25  状态转移率符号含义
符号 含义 符号 含义
λCO 电容开路故障概率 λTTS 匝间短路故障概率
λCS 电容短路故障概率 λTTO 匝间开路故障概率
λDMS 下管短路故障概率 λPOS 极间短路故障概率
λDMO 下管开路故障概率 λPGS 相对地短路故障概率
λUMS 上管短路故障概率 λPHS 相间短路故障概率
λUMO 上管开路故障概率 λPSS 位置传感器短路故障概率
λDDS 下二极管短路故障概率 λPSO 位置传感器开路故障概率
λDDO 下二极管开路故障概率 λPH 一相故障总失效概率
λUDS 上二极管短路故障概率 λA 系统所有器件故障概率
λUDO 上二极管开路故障概率 λDP 本质缺相概率
λSP1 缺相后系统失效概率 λDP1 等效缺相概率
λSP 系统本质失效概率    
上表中λDP、λDP1、λSP和λSP1的计算公式如下所示：
λDP＝λUMO+λDMO+λUDS+λDDS+λTTO+λPSO
λDP1＝0.88λDP+0.34(λDMS+λPSS)+0.43λUMS+0.9λTTS
λSP＝λCS+3(λUDO+λDDO+λPOS+λPGS+λPHS)
λSP1＝λCS+2(λUDO+λDDO+λPOS+λPGS+λPHS)
三级马尔科夫模型由A1、A2、A3和A4四个子模型构成，分别如图2、图3、图4、图5所示。
由于可靠度是处于有效状态的概率和，因此只需求出有效状态的概率和，即可实现可靠性的定量评估。
通过对开关磁阻电机驱动系统在第一级故障、第二级故障和第三级故障下系统运行情况进行分析，共得到5个一级马尔科夫状态包括4个有效状态和1个失效状态，18个二级马尔科夫状态包括14个有效状态和4个失效状态，57个三级马尔科夫状态包括43个有效状态和14个失效状态，同时考虑初始正常状态和最终失效状态，则三级马尔科夫模型中共有62个有效状态和20个失效状态，建立开关磁阻电机驱动系统在三级故障下的状态转移图，得到三级故障下的有效状态转移矩阵A:
Figure PCTCN2015099103-appb-000052
状态转移矩阵A是62行62列的方阵，状态转移矩阵A的行是所处有效状态，状态转移矩阵A的列是要转移的下一状态，对应的转移率为状态转移矩阵A中对应的元素，自身状态的转移率是该状态向所有状态(包含失效状态)转移的转移概率和的相反数。式(1)中，A1，A11，A12，A13，A2，A3，A4为非零矩阵，O表示零矩阵，子矩阵A1是13行13列的方阵：
Figure PCTCN2015099103-appb-000053
式(2)中，B1，B21，B31，B2，B3为非零矩阵，O表示零矩阵，五个非零矩阵中B21和B31仅有1个非零元素，其余均为0元素，五个子矩阵分别是：
Figure PCTCN2015099103-appb-000054
Figure PCTCN2015099103-appb-000055
Figure PCTCN2015099103-appb-000056
Figure PCTCN2015099103-appb-000057
Figure PCTCN2015099103-appb-000058
子矩阵A2是18行18列的方阵：
Figure PCTCN2015099103-appb-000059
式(8)中，B5，B61，B71，B81，B6，B7，B8为非零矩阵，O表示零矩阵，七个非零矩阵中B61，B71和B81仅有1个非零元素，其余均为0元素，七个子矩阵分别是：
Figure PCTCN2015099103-appb-000060
Figure PCTCN2015099103-appb-000061
Figure PCTCN2015099103-appb-000062
Figure PCTCN2015099103-appb-000063
Figure PCTCN2015099103-appb-000064
Figure PCTCN2015099103-appb-000065
Figure PCTCN2015099103-appb-000066
子矩阵A3是12行12列的方阵：
Figure PCTCN2015099103-appb-000067
式(16)中，B10，B111，B121，B11，B12为非零矩阵，O表示零矩阵，五个非零矩阵中B111和B121仅有1个非零元素，其余均为0元素，五个子矩阵分别是：
Figure PCTCN2015099103-appb-000068
Figure PCTCN2015099103-appb-000069
Figure PCTCN2015099103-appb-000070
Figure PCTCN2015099103-appb-000071
Figure PCTCN2015099103-appb-000072
子矩阵A4是19行19列的方阵：
Figure PCTCN2015099103-appb-000073
式(22)中，B14，B151，B161，B171，B15，B16，B17为非零矩阵，O表示零矩阵，七个非零矩阵中B151，B161和B171仅有1个非零元素，其余均为0元素，七个子矩阵分别是：
Figure PCTCN2015099103-appb-000074
Figure PCTCN2015099103-appb-000075
Figure PCTCN2015099103-appb-000076
Figure PCTCN2015099103-appb-000077
Figure PCTCN2015099103-appb-000078
Figure PCTCN2015099103-appb-000079
Figure PCTCN2015099103-appb-000080
式中，λA1、λA2、λA3、λA4、λA5、λB1、λB2、λB3、λB4、λB5、λB6、λB7、λB8、λB9、λB10、λB11、λB12、λB13、λB14、λB15、λB16、λB17、λB18、λC1、λC2、λC3、λC4、λC5、λC6、λC7、λC8、λC9、λC10、λC11、λC12、λC13、λC14、λC15、λC16、λC17、λC18、λC19、λC20、λC21、λC22、λC23、λC24、λC25、λC26、λC27、λC28、λC29、λC30、λC31、λC32、λC33、λC34、λC35、λC36、λC37、λC38、λC39、λC40、λC41、λC42、λC43、λC44、λC45、λC46、λC47、λC48、λC49、λC50、λC51、λC52、λC53、λC54、λC55、λC56、λC57、λF1、λF2、λF3、λF4、λF5、λF6、λF7、λF8、λF9、λF10、λF11、λF12、λF13、λF14、λF15、λF16、λF17、λF18、λF19、λF20、λF21、λF22、λF23、λF24、λF25、λF26、λF27、λF28、λF29、λF30、λF31、λF32、λF33、λF34、λF35、λF36、λF37、λF38、λF39、λF40、λF41、λF42、λF43是三级马尔科夫模型状态转移率；
由公式：
Figure PCTCN2015099103-appb-000081
解得开关磁阻电机系统处于有效状态的概率矩阵P(t)：
Figure PCTCN2015099103-appb-000082
式(31)中PA1(t)、PA2(t)、PA3(t)和PA4(t)分别表示A1子模型、A2子模型、A3子模型和A4子模型中有效状态的概率，如式(32)到(35)所示：
Figure PCTCN2015099103-appb-000083
Figure PCTCN2015099103-appb-000084
Figure PCTCN2015099103-appb-000085
Figure PCTCN2015099103-appb-000086
式(32)到(35)中，exp表示指数函数，t表示时间。
由公式(31)计算有效状态概率矩阵P(t)各元素之和，得到开关磁阻电机系统的可靠度函数R(t)：
R(t)＝0.0018exp(-3.96t)+0.0184exp(-3.95t)+8.7e-4exp(-3.83t)
-0.004exp(-3.83t)-1.74exp(-0.476t)+0.332exp(-0.237t)+5.14e
-4exp(-3.74t)-0.0142exp(-3.73t)+8.85e-4exp(-3.67t)
+0.0029exp(-1.83t)+0.01exp(-3.64t)-0.035exp(-3.64t)
+0.004exp(-1.82t)-0.003exp(-3.63t)-0.011exp(-3.55t)
+0.0544exp(-1.73t)-0.026exp(-3.44t)+0.119exp(-1.72t)
+0.005exp(-3.43t)-0.0046exp(-3.42t)-0.0211exp(-3.32t)
-0.108exp(-3.24t)-0.0595exp(-3.24t)+0.00269exp(-0.404t)
-0.245exp(-3.22t)+0.145exp(-3.19t)-0.0952exp(-3.11t)
-0.0662exp(-3.08t)+0.024exp(-1.54t)+0.005exp(-3.04t)
-0.166exp(-2.99t)+0.0345exp(-2.96t)-0.0231exp(-2.95t)
+2.05exp(-0.364t)+0.04exp(-0.36t)-2.59exp(-4.81t)
+3.48e-5exp(-4.57t)+1.34e-5exp(-4.43t)+3.54e-4exp(-4.39t)
+3.3exp(-4.38t)+1.28e-5exp(-4.27t)+1.36e-5exp(-4.27t)
+0.0013exp(-4.26t)-3.08e-4exp(-4.14t)+0.01exp(-4.07t)
+0.023exp(-4.07t)+7.97e-4exp(-4.04)+0.001exp(-2.01t)               (36)
由可靠度函数R(t)计算出开关磁阻电机系统的平均无故障时间：
Figure PCTCN2015099103-appb-000087
从而实现了三级马尔科夫模型定量分析开关磁阻电机系统可靠性评估。
例如，对一台由三相12/8结构开关磁阻电机和三相双开关式功率变换器组成的开关磁阻电机系统，如图6所示，通过图1所示开关磁阻电机系统三级故障下的马尔科夫状态转移图，建立三级故障下的状态转移矩阵A，解得开关磁阻电机系统处于有效状态的概率矩阵P(t)，计算有效状态概率矩阵P(t)各元素之和，得到开关磁阻电机系统的可靠度函数R(t)，如图7所示，对可靠度函数曲线R(t)在时间域0到无穷上的积分，可计算出该三相开关磁阻电机系统的平均无故障时间MTTF为3637112小时，从而实现了该三相开关磁阻电机系统三级马尔科夫模型可靠性的定量评估。平均无故障时间MTTF反映了可靠度函数曲线R(t)与横坐标轴和纵坐标轴在第一象限所围的面积的大小，面积越大，系统越可靠。

Claims (1)

  1. 一种三级马尔科夫模型定量分析开关磁阻电机系统可靠性评估方法，其特征在于步骤如下：
    通过对开关磁阻电机驱动系统在第一级故障、第二级故障和第三级故障下系统运行情况进行分析，共得到5个一级马尔科夫状态包括4个有效状态和1个失效状态，18个二级马尔科夫状态包括14个有效状态和4个失效状态，57个三级马尔科夫状态包括43个有效状态和14个失效状态，同时考虑初始正常状态和最终失效状态，则三级马尔科夫模型中共有62个有效状态和20个失效状态，建立开关磁阻电机驱动系统在三级故障下的状态转移图，得到三级故障下的有效状态转移矩阵A:
    Figure PCTCN2015099103-appb-100001
    状态转移矩阵A为62行62列的方阵，状态转移矩阵A的行为所处有效状态，状态转移矩阵A的列是要转移的下一状态，对应的转移率为状态转移矩阵A中对应的元素，自身状态的转移率是该状态向所有状态(包含失效状态)转移的转移概率和的相反数；式(1)中，A1，A11，A12，A13，A2，A3，A4为非零矩阵，O为零矩阵，子矩阵A1是13行13列的方阵：
    Figure PCTCN2015099103-appb-100002
    式(2)中，B1，B21，B31，B2，B3为非零矩阵，O表示零矩阵，五个非零矩阵中B21和B31仅有1个非零元素，其余均为0元素，五个子矩阵分别是：
    Figure PCTCN2015099103-appb-100003
    Figure PCTCN2015099103-appb-100004
    Figure PCTCN2015099103-appb-100005
    Figure PCTCN2015099103-appb-100006
    Figure PCTCN2015099103-appb-100007
    子矩阵A2是18行18列的方阵：
    Figure PCTCN2015099103-appb-100008
    式(8)中，B5，B61，B71，B81，B6，B7，B8为非零矩阵，O表示零矩阵，七个非零矩阵中B61，B71和B81仅有1个非零元素，其余均为0元素，七个子矩阵分别是：
    Figure PCTCN2015099103-appb-100009
    Figure PCTCN2015099103-appb-100010
    Figure PCTCN2015099103-appb-100011
    Figure PCTCN2015099103-appb-100012
    Figure PCTCN2015099103-appb-100013
    Figure PCTCN2015099103-appb-100014
    Figure PCTCN2015099103-appb-100015
    子矩阵A3是12行12列的方阵：
    Figure PCTCN2015099103-appb-100016
    式(16)中，B10，B111，B121，B11，B12为非零矩阵，O表示零矩阵，五个非零矩阵中B111和B121仅有1个非零元素，其余均为0元素，五个子矩阵分别是：
    Figure PCTCN2015099103-appb-100017
    Figure PCTCN2015099103-appb-100018
    Figure PCTCN2015099103-appb-100019
    Figure PCTCN2015099103-appb-100020
    Figure PCTCN2015099103-appb-100021
    子矩阵A4是19行19列的方阵：
    Figure PCTCN2015099103-appb-100022
    式(22)中，B14，B151，B161，B171，B15，B16，B17为非零矩阵，O表示零矩阵，七个非零矩阵中B151，B161和B171仅有1个非零元素，其余均为0元素，七个子矩阵分别是：
    Figure PCTCN2015099103-appb-100023
    Figure PCTCN2015099103-appb-100024
    Figure PCTCN2015099103-appb-100025
    Figure PCTCN2015099103-appb-100026
    Figure PCTCN2015099103-appb-100027
    Figure PCTCN2015099103-appb-100028
    Figure PCTCN2015099103-appb-100029
    式中，λA1、λA2、λA3、λA4、λA5、λB1、λB2、λB3、λB4、λB5、λB6、λB7、λB8、λB9、λB10、λB11、λB12、λB13、λB14、λB15、λB16、λB17、λB18、λC1、λC2、λC3、λC4、λC5、λC6、λC7、 λC8、λC9、λC10、λC11、λC12、λC13、λC14、λC15、λC16、λC17、λC18、λC19、λC20、λC21、λC22、λC23、λC24、λC25、λC26、λC27、λC28、λC29、λC30、λC31、λC32、λC33、λC34、λC35、λC36、λC37、λC38、λC39、λC40、λC41、λC42、λC43、λC44、λC45、λC46、λC47、λC48、λC49、λC50、λC51、λC52、λC53、λC54、λC55、λC56、λC57、λF1、λF2、λF3、λF4、λF5、λF6、λF7、λF8、λF9、λF10、λF11、λF12、λF13、λF14、λF15、λF16、λF17、λF18、λF19、λF20、λF21、λF22、λF23、λF24、λF25、λF26、λF27、λF28、λF29、λF30、λF31、λF32、λF33、λF34、λF35、λF36、λF37、λF38、λF39、λF40、λF41、λF42、λF43是三级马尔科夫模型状态转移率；
    利用公式：
    Figure PCTCN2015099103-appb-100030
    解得开关磁阻电机系统处于有效状态的概率矩阵P(t)：
    Figure PCTCN2015099103-appb-100031
    式(31)中PA1(t)、PA2(t)、PA3(t)和PA4(t)分别表示A1子模型、A2子模型、A3子模型和A4子模型中有效状态的概率，如式(32)到(35)所示：
    Figure PCTCN2015099103-appb-100032
    Figure PCTCN2015099103-appb-100033
    Figure PCTCN2015099103-appb-100034
    式(32)到(35)中，exp表示指数函数，t表示时间，A代表状态转移矩阵；
    由公式(31)计算有效状态概率矩阵P(t)各元素之和，得到开关磁阻电机系统的可靠度函数R(t)：
    R(t)＝0.0018exp(-3.96t)+0.0184exp(-3.95t)+8.7e-4exp(-3.83t)
    -0.004exp(-3.83t)-1.74exp(-0.476t)+0.332exp(-0.237t)+5.14e
    -4exp(-3.74t)-0.0142exp(-3.73t)+8.85e-4exp(-3.67t)
    +0.0029exp(-1.83t)+0.01exp(-3.64t)-0.035exp(-3.64t)
    +0.004exp(-1.82t)-0.003exp(-3.63t)-0.011exp(-3.55t)
    +0.0544exp(-1.73t)-0.026exp(-3.44t)+0.119exp(-1.72t)
    +0.005exp(-3.43t)-0.0046exp(-3.42t)-0.0211exp(-3.32t)
    -0.108exp(-3.24t)-0.0595exp(-3.24t)+0.00269exp(-0.404t)
    -0.245exp(-3.22t)+0.145exp(-3.19t)-0.0952exp(-3.11t)
    -0.0662exp(-3.08t)+0.024exp(-1.54t)+0.005exp(-3.04t)
    -0.166exp(-2.99t)+0.0345exp(-2.96t)-0.0231exp(-2.95t)
    +2.05exp(-0.364t)+0.04exp(-0.36t)-2.59exp(-4.81t)
    +3.48e-5exp(-4.57t)+1.34e-5exp(-4.43t)+3.54e-4exp(-4.39t)
    +3.3exp(-4.38t)+1.28e-5exp(-4.27t)+1.36e-5exp(-4.27t)
    +0.0013exp(-4.26t)-3.08e-4exp(-4.14t)+0.01exp(-4.07t)
    +0.023exp(-4.07t)+7.97e-4exp(-4.04)+0.001exp(-2.01t)   (36)
    由可靠度函数R(t)计算出开关磁阻电机系统的平均无故障时间：
    Figure PCTCN2015099103-appb-100035
    从而实现了三级马尔科夫模型定量分析开关磁阻电机系统可靠性评估。
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