Stochastic hybrid energy system modelling with component failure and repair

Abstract

Hybrid energy framework is the designing plan of hybridizing power supply part or blending them, for instance, organizing different energy assets to work in parallel (equivalent) is exceptionally normal in force. In this way, hybridizing is characterized as the shaping crossbreed of sets of specialists for cooperating to accomplish a reason. Hybrid energy framework is a foundation plan that incorporates different or various energy converts to energy storage, energy conditions, energy management framework. In this paper an independent PV/wind hybrid energy framework is thought of and its reliability quality attributes are talked about by utilizing Markov method. The assumed hybrid energy framework has three type of states i.e., operating, degraded and failed. The framework is assumed to be repaired from degraded and failed states. Various reliability indices such as reliability, availability, mean time to failure (MTTF), expected cost and sensitivity are evaluated. Graphical representation of the reliability characteristics is also illustrated for the considered system. The area of investigation inside the field of sustainable power is worldwide and the aftereffects of this investigation are plentifully useful for the architects, planners, analysts and understudies of different fields.

Appendix

The mathematical model of hybrid renewable energy system can be formulated with the help of Markov process that possess the set of differential equations by the probability of the considerations and continuity arguments. The probability that the system is in states good, degraded and failed at time t, and remains there in the interval $(t,\;t\; + \Delta t)$ or/and if it is in some other state at time then it should be transit to these states in the interval $(t,\;t\; + \Delta t)$ provided that the transition occurs between the states as $\Delta t \to 0.$ Accordingly, the Eqs. (1) to (4) are construed as given below.

The probability of the system to be in good state with probability $P_{0}$ in the interval $\left( {t,t + {\Delta }t} \right)$ is given by

$$\begin{gathered} P_{0} \left( {t + \Delta t} \right) = \left( {1 - 3\lambda_{1} \Delta t} \right)\left( {1 - \lambda_{2} \Delta t} \right)\left( {1 - \lambda_{3} \Delta t} \right)P_{0} \left( t \right) + \mu \Delta tP_{1} \left( t \right) + \mu \Delta tP_{2} \left( t \right) \hfill \\ + \mathop \smallint \limits_{0}^{\infty } \emptyset \left( x \right)P_{3} \left( {x,t} \right)\Delta tdx + \mathop \smallint \limits_{0}^{\infty } \emptyset \left( x \right)P_{4} \left( {x,t} \right)\Delta tdx + \mathop \smallint \limits_{0}^{\infty } \emptyset \left( x \right)P_{5} \left( {x,t} \right)\Delta tdx \hfill \\ \end{gathered}$$

$$\left[ {1 - 3\lambda_{1} \Delta t - \lambda_{2} \Delta t - \lambda_{3} \Delta t} \right]P_{0} \left( t \right) + \mu \Delta t\left[ {P_{1} \left( t \right) + P_{2} \left( t \right)} \right] + \mathop \smallint \limits_{0}^{\infty } \emptyset \left( x \right)\Delta t\left[ {P_{3} + P_{4} + P_{5} } \right]dx$$

$$\begin{gathered} P_{0} \left( {t + \Delta t} \right) = P_{0} \left( t \right) - 3\lambda_{1} \Delta tP_{0} \left( t \right) - \lambda_{2} \Delta tP_{0} \left( t \right) - \lambda_{3} \Delta tP_{0} \left( t \right) + \mu \Delta t\left[ {P_{1} \left( t \right) + P_{2} \left( t \right)} \right] \hfill \\ \quad+ \mathop \smallint \limits_{0}^{\infty } \emptyset \left( x \right)\Delta t\left[ {P_{3} + P_{4} + P_{5} } \right]dx \hfill \\ \end{gathered}$$

$$\begin{gathered} \mathop {\lim }\limits_{\Delta t \to 0} \frac{{P\left( {t + \Delta t} \right) - P_{0} \left( t \right)}}{\Delta t} = - 3\lambda_{1} P_{0} \left( t \right) - \lambda_{2} P_{0} \left( t \right) - \lambda_{3} P_{0} \left( t \right) + \mu \left[ {P_{1} \left( t \right) + P_{2} \left( t \right)} \right] \hfill \\ \,\, + \mathop \smallint \limits_{0}^{\infty } \emptyset \left( x \right)\Delta t\left[ {P_{3} + P_{4} + P_{5} } \right]dx \hfill \\ \end{gathered}$$

$$\frac{d}{dt}[P_{0} \left( t \right) + 3\lambda_{1} P_{0} \left( t \right) + \lambda_{2} P_{0} \left( t \right) + \lambda_{3} P_{0} \left( t \right) = \mu \left[ {P_{1} \left( t \right) + P_{2} \left( t \right)} \right] + \mathop \smallint \limits_{0}^{\infty } \emptyset \left( x \right)\Delta t\left[ {P_{3} + P_{4} + P_{5} } \right]dx$$

$$\left[ {\frac{d}{dt} + 3\lambda_{1} + \lambda_{2} + \lambda_{3} } \right]P_{0} \left( t \right) = \mu \left[ {P_{1} \left( t \right) + P_{2} \left( t \right)} \right] + \mathop \smallint \limits_{0}^{\infty } \emptyset \left( x \right)\Delta t\left[ {P_{3} + P_{4} + P_{5} } \right]dx$$

(34)

Similarly,

$$\left[ {\frac{d}{dt} + 2\lambda_{1} + \lambda_{2} + \lambda_{3} + \mu } \right]P_{1} \left( t \right) = 3\lambda_{1} \Delta tP_{0} \left( t \right)$$

(35)

$$\left[ {\frac{d}{dt} + \lambda_{1} + \lambda_{2} + \lambda_{3} + \mu } \right]P_{2} \left( t \right) = 2\lambda_{1} \Delta tP_{1} \left( t \right)$$

(36)

For failed states

$$\left[ {\frac{\partial }{\partial x} + \frac{\partial }{\partial t} + \emptyset \left( x \right)} \right]P_{3} \left( {x,t} \right) = 0$$

(37)

$$\left[ {\frac{\partial }{\partial x} + \frac{\partial }{\partial t} + \emptyset \left( x \right)} \right]P_{4} \left( {x,t} \right) = 0$$

(38)

$$\left[ {\frac{\partial }{\partial x} + \frac{\partial }{\partial t} + \emptyset \left( x \right)} \right]P_{5} \left( {x,t} \right) = 0$$

(39)

Boundary conditions

$$P_{3} \left( {0,t} \right) = \lambda_{1} P_{2} \left( t \right)$$

(40)

$$P_{4} \left( {0,t} \right) = \lambda_{2} \left[ {P_{0} \left( t \right) + P_{1} \left( t \right) + P_{2} \left( t \right)} \right]$$

(41)

$$P_{5} \left( {0,t} \right) = \lambda_{3} \left[ {P_{0} \left( t \right) + P_{1} \left( t \right) + P_{2} \left( t \right)} \right]$$

(42)

When the system is perfectly good i.e., in the initial state $P_{0}$, then

$$P_{0} \left( 0 \right) = 1\;\,and \, other \, state \, probabilities \, are \, zero \, at\;t = 0$$

(43)