WO2017041392A1 - 二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
二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 PDFInfo
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- the invention relates to a quantitative evaluation method, in particular to a two-dimensional Markov model for reliability analysis of a switched reluctance motor system of various types and various phases, and a reliability evaluation method for a switched reluctance motor system.
- the switched reluctance motor has a solid structure, no winding on the rotor, small phase-to-phase coupling, and excellent fault tolerance. Good fault tolerance ensures high reliability of switched reluctance motor systems.
- the existing reliability quantitative assessment method can not effectively characterize the fault tolerance of the system and cannot meet the requirements of industrial applications.
- Reliability block diagram modeling and fault tree modeling ignoring the fault tolerance of switched reluctance motor systems, can not indicate that the switched reluctance motor system between normal and failed states has fault-effective operating state, based on Markov reliability
- the model can characterize the intermediate state between normal and failure
- the conventional Markov modeling method only considers the various faults of the switched reluctance motor system to be equivalent to a Markov state, without considering a fault. Access to different Markov states can result in erroneous reliability quantitative assessment results.
- the good fault tolerance of the switched reluctance motor system makes the system in an effective operating state in the case of a secondary fault. Therefore, it is necessary to effectively represent the operating capability of the switched reluctance motor system under the secondary fault to realize the switching magnetic Quantitative evaluation of the secondary Markov model for reliability of the resistance motor system.
- the object of the present invention is to overcome the deficiencies of the prior art, and provide a second-order Markov model with high accuracy and accuracy, simple steps, fast evaluation speed and wide application range, and quantitative analysis of the reliability of the switched reluctance motor system. evaluation method
- the second-order Markov model of the present invention quantitatively analyzes the reliability evaluation method of the switched reluctance motor system, and the steps are as follows:
- the 17 first-order faults that may occur in the normal state of the switched reluctance motor system are equivalent to the four active states and one failure state of the Markov space under the first-order fault, and the second-level is based on the four valid states.
- the system expression is equivalent to 14 effective states and 4 failure states.
- there are 19 effective states and 6 failure states under the system secondary fault using 19 effective states and The state transition diagram of the six failure states establishing the system under the secondary fault, and the effective state transition matrix A under the secondary fault is obtained:
- the state transition matrix A is a square matrix of 19 rows and 19 columns, the row of the state transition matrix A is in an effective state, the column of the state transition matrix A is the next state to be transferred, and the corresponding transition rate is corresponding to the state transition matrix A.
- the transfer rate of its own state is the opposite of the transition probability sum of the state to all states;
- A1, A11, A12, A13, A2, A3, and A4 are non-zero matrices, and O represents a zero matrix.
- A11, A12, and A13 are non-zero except for the first element. Both are 0 elements, and the seven sub-matrices are:
- ⁇ A1 , ⁇ A2 , ⁇ A3 , ⁇ A4 , ⁇ A5 , ⁇ B1 , ⁇ B2 , ⁇ B3 , ⁇ B4 , ⁇ B5 , ⁇ B6 , ⁇ B7 , ⁇ B8 , ⁇ B9 , ⁇ B10 , ⁇ B11 , ⁇ B12 , ⁇ B13 , ⁇ B14 , ⁇ B15 , ⁇ B16 , ⁇ B17 , ⁇ B18 , ⁇ F1 , ⁇ F2 , ⁇ F3 , ⁇ F4 , ⁇ F5 , ⁇ F6 , ⁇ F7 , ⁇ F8 , ⁇ F9 , ⁇ F10 , ⁇ F11 , ⁇ F12 , ⁇ F13 , ⁇ F14 are state transition rates of the second-order Markov model.
- exp represents an exponential function
- t represents time
- A represents a state transition matrix
- the establishment of the second-level Markov reliability quantitative evaluation model solves the problem that the reliability block diagram modeling and fault tree modeling cannot characterize the effective state of the switched reluctance motor system with fault operation.
- the first-level fault and the second-level fault stratification of the switched reluctance motor system greatly improve the accuracy and accuracy of the reliability quantitative evaluation; at the same time, a Markov state and a fault can be entered for multiple faults.
- a variety of Markov states have been considered, so that the established reliability evaluation model is consistent with the actual operation of the switched reluctance motor system, and the quantitative evaluation results meet the requirements of industrial applications.
- the Markov quantitative analysis of the reliability of the switched reluctance motor drive system under the condition of two-level faults improves the accuracy of the quantitative evaluation results of reliability and the equivalent number of faults while maintaining the rapid requirements of reliability modeling. And the case of failure determination conditions.
- 1 is a Markov state transition diagram of a switched reluctance motor system of the present invention under a secondary fault
- 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 state after the fault is summarized as four Markov states from B1 to B4.
- the second-order Markov state transfer rate in 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
- the secondary Markov state transition rate in the A3 state is shown in Table 4:
- the secondary Markov state transfer rate in the A4 state is shown in Table 5:
- 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.
- the second effective state is based on the four active states.
- the system performance of the fault is equivalent to 14 effective states and 4 failure states.
- the state transition diagram under the secondary Markov model of the switched reluctance motor system is obtained, as shown in Figure 1.
- the Markov space state is represented by a circle.
- 00 is the first valid state
- A1 corresponds.
- the second valid state, B1 to B3 are valid states 3 to 5
- A2 corresponds to the sixth valid state
- B5 to B8 correspond to valid states 7 to 10
- the eleventh valid state is A3, and B10 to B12 correspond to the valid state 12 to 14,
- A4 corresponds to the active state 15, and the 16th through 19th active states are B14 to B17.
- the remaining states A5, B4, B9, B13, B18 and F in the state transition diagram are all in a failed state.
- Figure 1 shows the Markov space state symbols and their meanings as shown in Table 6:
- ⁇ 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 two-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.
- the second effective state is based on the four active states.
- the system performance of the fault is equivalent to 14 effective states and 4 failure states.
- the state transition diagram under the fault the effective state transition matrix A under the secondary fault is obtained:
- the state transition matrix A is a square matrix of 19 rows and 19 columns, the row of the state transition matrix A is in an effective state, the column of the state transition matrix A is the next state to be transferred, and the corresponding transition rate is corresponding to the state transition matrix A.
- the element's transfer rate of its own state is the inverse of the transition probability sum of the state to all states.
- A1, A11, A12, A13, A2, A3, and A4 are non-zero matrices, and O represents a zero matrix.
- A11, A12, and A13 are non-zero except for the first element. Both are 0 elements, and the seven sub-matrices are:
- ⁇ A1 , ⁇ A2 , ⁇ A3 , ⁇ A4 , ⁇ A5 , ⁇ B1 , ⁇ B2 , ⁇ B3 , ⁇ B4 , ⁇ B5 , ⁇ B6 , ⁇ B7 , ⁇ B8 , ⁇ B9 , ⁇ B10 , ⁇ B11 , ⁇ B12 , ⁇ B13 , ⁇ B14 , ⁇ B15 , ⁇ B16 , ⁇ B17 , ⁇ B18 , ⁇ F1 , ⁇ F2 , ⁇ F3 , ⁇ F4 , ⁇ F5 , ⁇ F6 , ⁇ F7 , ⁇ F8 , ⁇ F9 , ⁇ F10 , ⁇ F11 , ⁇ F12 , ⁇ F13 , ⁇ F14 are state transition rates of the second-order Markov model.
- the application range of the second-order Markov model is larger than that of the third-order Markov model, which can be used to trade the calculation accuracy and speed. Under the general conditions, it is recommended to use the second-order Markov model for reliability prediction, taking into account the equivalent number of failures. And the case of failure determination conditions.
- a switched reluctance motor system consisting of a three-phase 12/8 structure switched reluctance motor and a three-phase double-switching power converter, as shown in FIG. 6, passes through the switched reluctance motor system shown in FIG.
- the Markov state transition diagram under the fault is established, and the state transition matrix A under the two-level fault is established, and the probability matrix P(t) of the switched reluctance motor system is determined, and the effective state probability matrix P(t) is calculated.
- 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 achieves a quantitative evaluation of the reliability of the two-phase Markov model of the three-phase switched reluctance motor system.
- the average time between failures reflects the reliability function curve R(t) and the area enclosed by the coordinate axis. The larger the area, the more reliable the system.
Abstract
二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，通过将开关磁阻电机系统正常状态下可能发生的17种一级故障等效为一级故障下马尔科夫空间的4个有效状态和1个失效状态，以4个有效状态为基础将第二级故障下系统表现形式等效为14个有效状态和4个失效状态，同时考虑初始正常状态和最终失效状态得到系统二级故障下共有19个有效状态和6个失效状态，建立系统在二级故障下的状态转移图，得到二级故障下的有效状态转移矩阵，计算有效状态概率矩阵各元素之和，得到开关磁阻电机系统的可靠度函数，计算出开关磁阻电机系统的平均无故障时间，从而采用二级马尔科夫模型定量分析实现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
本发明涉及一种定量评估方法，尤其适用于各种类型、各种相数的开关磁阻电机系统可靠性的二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。
开关磁阻电机结构坚固，转子上无绕组，相间耦合小，容错性能优良。良好的容错性能确保了开关磁阻电机系统具有较高的可靠性。但是现有可靠性定量评估方法无法有效表征系统容错能力，无法满足工业应用的要求。可靠性框图建模和故障树建模，忽略了开关磁阻电机系统容错能力，无法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，基于马尔科夫的可靠性建模虽然能够对正常和失效之间的中间状态进行表征，但是常规的马尔科夫建模方法只考虑将开关磁阻电机系统多种故障等效为一种马尔科夫状态，没有考虑一种故障可以进入不同的马尔科夫状态，会产生错误的可靠性定量评估结果。开关磁阻电机系统良好的容错能力使系统在二级故障的情况下，也有可能处于有效运行状态，因此，需对开关磁阻电机系统在二级故障下运行能力进行有效的表示，实现开关磁阻电机系统可靠性的二级马尔科夫模型定量评估。
发明内容
本发明的目的是克服已有技术的不足之处，提供一种估精度和准确性高，步骤简单，评估速度快，使用范围广的二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
为实现上述技术目的，本发明的二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，其步骤如下：
将开关磁阻电机系统正常状态下可能发生的17种一级故障等效为一级故障下马尔科夫空间的4个有效状态和1个失效状态，以4个有效状态为基础将第二级故障下系统表现形式等效为14个有效状态和4个失效状态，同时考虑初始正常状态和最终失效状态得到系统二级故障下共有19个有效状态和6个失效状态，利用19个有效状态和6个失效状态建立系统在二级故障下的状态转移图，得到二级故障下的有效状态转移矩阵A:
状态转移矩阵A为19行19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，七个非零矩阵中A11，A12，A13除了第一个元素非零外，其余均为0元素，七个子矩阵分别是：
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、λF1、λF2、λF3、λF4、λF5、λF6、λF7、λF8、λF9、λF10、λF11、λF12、λF13、λF14是二级马尔科夫模型状态转移率。
利用公式：
计算得开关磁阻电机系统处于有效状态的概率矩阵P(t)：
式中，exp表示指数函数，t表示时间，A代表状态转移矩阵；
利用公式(10)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)＝0.649exp(-0.476t)-0.0379exp(-3.86t)+0.684exp(-3.75t)-0.0857exp(-3.68t)
-0.401exp(-3.67t)+2.33exp(-3.54t)+0.0308exp(-3.08t)+0.351exp(-2.99t)
+3.46exp(-4.81t)-0.0375exp(-4.69t)-2.43e-4exp(-4.57t)-0.00499exp(-4.39t)
-7.08exp(-4.38t)+0.281exp(-4.26t)+0.87exp(-4.07t) (11)
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：
从而实现了二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：二级马尔科夫可靠性定量评估模型的建立，解决了可靠性框图建模和故障树建模无法对开关磁阻电机系统带故障运行下有效状态进行表征的问题。开关磁阻电机系统第一级故障和第二级故障分层考虑，极大程度提高了可靠性定量评估精度和准确性；同时对多种故障对应一种马尔科夫状态和一种故障可进入多种马尔科夫状态进行了考虑，使建立的可靠性评估模型与开关磁阻电机系统实际运行情况相符，定量评估结果满足工业应用要求。二级故障下的马尔科夫定量分析开关磁阻电机驱动系统可靠性评估方法在保持可靠性建模快速性要求的前提下，有效提高了可靠性定量评估结果的精确性，兼顾等效故障数和失效判别条件的场合。
图1是本发明的开关磁阻电机系统二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。
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，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 第一级故障下开关磁阻电机系统马尔科夫状态转化转移率
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状态下二级马尔科夫状态转移率如表2所示：
表2 A1状态下二级马尔科夫状态转移率
| 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 | 0.43 | 0.57 |
| 3 | 下管短路(DMS) | 0 | 0.34 | 0.54 | 0.12 |
| 4 | 缺相(DPH) | 0 | 0.88 | 0.88 | 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，对应二级马尔科夫状态为B5到B9，A2状态下二级马尔科夫状态转移率如表3所示：
表3 A2状态下二级马尔科夫状态转移率
| 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状态下二级马尔科夫状态转移率如表4所示：
表4 A3状态下二级马尔科夫状态转移率
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状态下二级马尔科夫状态转移率如表5所示：
表5 A4状态下二级马尔科夫状态转移率
| 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 |
一级状态不同发生二级故障后，得到不同的有效状态。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。
综上所述，依据一级马尔科夫状态的不同，可得到18种二级马尔科夫状态。
通过将开关磁阻电机系统正常状态下可能发生的17种一级故障等效为一级故障下马尔科夫空间的4个有效状态和1个失效状态，以4个有效状态为基础将第二级故障下系统表现形式等效为14个有效状态和4个失效状态，同时考虑初始正常状态和最终失效状态得到系统二级故障下共有19个有效状态和6个失效状态。若开关磁阻电机系统有第三级以上故障发生，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二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到B3是有效状态3到5，A2对应第6个有效状态，B5至B8对应有效状态7到10，第11个有效状态是A3，B10到B12对应有效状态12到14，A4对应有效状态15，第16到19个有效状态是B14到B17。状态转移图中其余的状态A5，B4，B9，B13，B18和F均为失效状态。图1种马尔科夫空间状态符号及含义如表6所示：
表6马尔科夫状态符号
状态转移率符号及计算公式如表7所示：
表7二级马尔科夫模型状态转移率
表7中用到的符号含义如表8所示：
表8转移率符号含义
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)
二级马尔科夫模型由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。
通过将开关磁阻电机系统正常状态下可能发生的17种一级故障等效为一级故障下马尔科夫空间的4个有效状态和1个失效状态，以4个有效状态为基础将第二级故障下系统表现形式等效为14个有效状态和4个失效状态，同时考虑初始正常状态和最终失效状态得到系统二级故障下共有19个有效状态和6个失效状态，建立系统在二级故障下的状态转移图，得到二级故障下的有效状态转移矩阵A:
状态转移矩阵A是19行19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，七个非零矩阵中A11，A12，A13除了第一个元素非零外，其余均为0元素，七个子矩阵分别是：
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、λF1、λF2、λF3、λF4、λF5、λF6、λF7、
λF8、λF9、λF10、λF11、λF12、λF13、λF14是二级马尔科夫模型状态转移率。
利用公式：
计算得开关磁阻电机系统处于有效状态的概率矩阵P(t)：
式中，exp表示指数函数，t表示时间；
由公式(10)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)＝0.649exp(-0.476t)-0.0379exp(-3.86t)+0.684exp(-3.75t)-0.0857exp(-3.68t)
-0.401exp(-3.67t)+2.33exp(-3.54t)+0.0308exp(-3.08t)+0.351exp(-2.99t)
+3.46exp(-4.81t)-0.0375exp(-4.69t)-2.43e-4exp(-4.57t)-0.00499exp(-4.39t)
-7.08exp(-4.38t)+0.281exp(-4.26t)+0.87exp(-4.07t) (11)
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：
从而实现了二级马尔科夫模型定量分析开关磁阻电机系统可靠性评估。
二级马尔科夫模型的适用范围大于三级马尔科夫模型，能够将计算精度和速度经行折衷，一般条件下，推荐使用二级马尔科夫模型进行可靠性预测，同时兼顾等效故障数和失效判别条件的场合。
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二级故障下的马尔科夫状态转移图，建立二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，从而实现了该三相开关磁阻电机系统二级马尔科夫模型可靠性的定量评估。平均无故障时间反映了可靠度函数曲线R(t)与坐标轴所围面积的大小，面积越大，系统越可靠
Claims (1)
- 一种二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：将开关磁阻电机系统正常状态下可能发生的17种一级故障等效为一级故障下马尔科夫空间的4个有效状态和1个失效状态，以4个有效状态为基础将第二级故障下系统表现形式等效为14个有效状态和4个失效状态，同时考虑初始正常状态和最终失效状态得到系统二级故障下共有19个有效状态和6个失效状态，利用19个有效状态和6个失效状态建立系统在二级故障下的状态转移图，得到二级故障下的有效状态转移矩阵A:
状态转移矩阵A为19行19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，七个非零矩阵中A11，A12，A13除了第一个元素非零外，其余均为0元素，七个子矩阵分别是：
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、λF1、λF2、λF3、λF4、λF5、λF6、λF7、λF8、λF9、λF10、λF11、λF12、λF13、λF14是二级马尔科夫模型状态转移率。利用公式：
计算得开关磁阻电机系统处于有效状态的概率矩阵P(t)：
式中，exp表示指数函数，t表示时间，A代表状态转移矩阵；利用公式(10)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)＝0.649exp(-0.476t)-0.0379exp(-3.86t)+0.684exp(-3.75t)-0.0857exp(-3.68t)-0.401exp(-3.67t)+2.33exp(-3.54t)+0.0308exp(-3.08t)+0.351exp(-2.99t)+3.46exp(-4.81t)-0.0375exp(-4.69t)-2.43e-4exp(-4.57t)-0.00499exp(-4.39t)-7.08exp(-4.38t)+0.281exp(-4.26t)+0.87exp(-4.07t) (11)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：
从而实现了二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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