Advance first order second moment (AFOSM) method for single reservoir operation reliability analysis: a case study

Abstract

Reservoir system reliability is the ability of reservoir to perform its required functions under stated conditions for a specified period of time. In classical method of reservoir system reliability analysis, the operation policy is used in a simple simulation model, considering the historical/synthetic inflow series and a number of physical bounds on a reservoir system. This type of reliability analysis assumes a reservoir system as fully failed or functioning, called binary state assumption. A number of researchers from various research backgrounds have shown that the binary state assumption in the traditional reliability theory is not extensively acceptable. Our approach to tackle the present problem space is to implement the algorithm of advance first order second moment (AFOSM) method. In this new method, the inflow and reservoir storage are considered as uncertain variables. The mean, variance and covariance of uncertain variables are determined using moment values of reservoir state variables. For this purpose, a stochastic optimization model developed based on the constraint state formulation is applied. The proposed model of reliability analysis is used to a real case study in Iran. As a result, monthly probabilities of water allocation were computed from AFOSM method, and the outputs were compared with those from Monte Carlo method. The comparison shows that the outputs from AFOSM method are similar to those from the Monte Carlo method. In term of practical use of this study, the proposed method is appropriate to determine the monthly probability of failure in water allocation without the aid of simulation.

Appendix 1: Derivations of moment equation for E(S j t+1 , S j t )

The approach implemented in this paper involved taking the expectation of E(S ^t+1 _j , S ^t _i ) over the following partitions, that t and t + 1 are two consequent months.

$$ \begin{aligned} S_{i}^{t} = S_{i}^{max(t)} \\ S_{i}^{t} = S_{i}^{min(t)} \\ \left. {S_{i}^{min(t)} \le S_{i}^{t} \le S_{i}^{max(t)} } \right|S_{j}^{t + 1} = S_{j}^{min(t + 1)} \\ \left. {S_{i}^{min(t)} \le S_{i}^{t} \le S_{i}^{max(t)} } \right|S_{j}^{min(t + 1)} \le S_{j}^{t} \le S_{j}^{max(t + 1)} \\ \left. {S_{i}^{min(t)} \le S_{i}^{t} \le S_{i}^{max(t)} } \right|S_{j}^{t + 1} = S_{j}^{max(t + 1)} \\ \end{aligned} $$

1.1 Appendix 1.1: Derivations of moment equation for S ^t _i  = S ^max(t) _i

Taking E( ) over these partitions, the first part is presented as S ^max(t) _i E(S ^t+1 _j )Pr(S ^t _i  = S ^max(t) _i ) which can be extended as follows:

$$ \begin{gathered} S_{i}^{max(t)} E\left( {S_{j}^{t + 1} } \right)Pr\left( {S_{i}^{t} = S_{i}^{max(t)} } \right) = S_{i}^{max(t)} E\left[ {\left( {S_{j}^{t} + I_{j}^{t + 1} - u_{j}^{t + 1} } \right) \cdot 1_{{\left( {S_{j}^{min(t + 1)} < S_{j}^{t + 1} < S_{j}^{max(t + 1)} } \right)}} } \right] \cdot Pr\left( {S_{i}^{t} = S_{i}^{max(t)} } \right) = \end{gathered} $$

Considering the deterministic part of equations, i.e., I ^t+1 _j  − u ^t+1 _j , the second part of equation can be simplified as:

$$ S_{i}^{max(t)} \left[ {E\left( {S_{j}^{t} } \right) - \left( {I_{j}^{t + 1} + u_{j}^{t + 1} } \right)} \right].E\left( {1_{{\left( {S_{j}^{min(t + 1)} < S_{j}^{t + 1} < S_{j}^{max(t + 1)} } \right)}} } \right) \cdot Pr\left( {S_{i}^{t} = S_{i}^{max(t)} } \right) = $$

$$ \begin{gathered} S_{i}^{max(t)} \left[ {E\left( {S_{j}^{t} } \right) - \left( {I_{j}^{t + 1} + u_{j}^{t + 1} } \right)} \right] \cdot E\left\{ {1_{{\left[ {S_{j}^{min(t + 1)} < S_{j}^{t + 1} < S_{j}^{max(t + 1)} } \right]}} } \right\} \cdot E\left\{ {1_{{\left[ {1_{{\left( {S_{i}^{max(t)} , + \infty } \right)}} } \right]}} } \right\} = \end{gathered} $$

where $E\{ 1_{{[S_{j}^{\min (t + 1)} < S_{j}^{t + 1} < S_{j}^{\max (t + 1)} ]}} \}$can be simplified as:

$$ E\left\{ {1_{{\left[ {S_{j}^{min(t + 1)} < S_{j}^{t + 1} < S_{j}^{max(t + 1)} } \right]}} } \right\} = Pr\left\{ {S_{j}^{min(t + 1)} - \left( {I^{t} - u^{t} } \right) < \eta_{I}^{t + 1} + S_{j}^{t} < S_{j}^{max(t + 1)} - \left( {I^{t} - u^{t} } \right)} \right\} $$

$$ = \int_{{S_{j}^{\min (t)} }}^{{S_{j}^{\max (t)} }} {\int_{{S_{j}^{\min (t + 1)} - \left( {I^{t} - u^{t} } \right) - S_{j}^{(t)} }}^{{S_{j}^{\max (t + 1)} - \left( {I^{t} - u^{t} } \right) - S_{j}^{(t)} }} {f_{\eta t}^{t + 1} \left( {\eta_{t}^{t + 1} } \right)f_{{S_{j}^{t} }} \left( {S_{j}^{t} } \right)d\eta_{t}^{t + 1} dS_{j}^{t} } } $$

Using a Taylor’s series first-order approximation and the error function for the numerical computation of the probability function, Fletcher and Ponnambalam (2008) show that $E\left\{ {1_{{\left[ {S_{j}^{min(t + 1)} < S_{j}^{t + 1} < S_{j}^{max(t + 1)} } \right]}} } \right\}$ can be presented as follows:

$$ = \frac{1}{2}\left\{ {erf\left[ {\frac{{S_{j}^{max(t + 1)} - \left( {I_{j}^{t + 1} - u_{j}^{t + 1} } \right) - E\left( {S_{j}^{t} } \right)}}{{2\sqrt {Var\left( {\eta_{I}^{t} } \right)} }}} \right]} \right. - \left. {erf\left[ {\frac{{S_{j}^{min(t + 1)} - \left( {I_{j}^{t + 1} - u_{j}^{t + 1} } \right) - E\left( {S_{j}^{t} } \right)}}{{2\sqrt {Var\left( {\eta_{I}^{t} } \right)} }}} \right]} \right\} $$

As a result, $s_{i}^{max(t)} E\left( {s_{j}^{t + 1} } \right)Pr\left( {s_{i}^{t} = s_{i}^{max(t)} } \right)$ can be determined as follows:

$$ = S_{i}^{max(t)} \left[ {E\left( {S_{j}^{t} } \right) + \left( {I_{j}^{t + 1} - u_{j}^{t + 1} } \right)} \right]\frac{1}{2}\left\{ {erf\left[ {\frac{{S_{j}^{max(t + 1)} - \left( {I_{j}^{t + 1} - u_{j}^{t + 1} } \right) - E\left( {S_{j}^{t} } \right)}}{{2\sqrt {Var\left( {\eta_{I}^{t} } \right)} }}} \right]} \right. - $$

$$ \left. {erf\left[ {\frac{{S_{j}^{min(t + 1)} - \left( {I_{j}^{t + 1} - u_{j}^{t + 1} } \right) - E\left( {S_{j}^{t} } \right)}}{{2\sqrt {Var\left( {\eta_{I}^{t} } \right)} }}} \right]} \right\}\left\{ {\frac{1}{2}\left[ {1 + erf\left( { - \frac{{S_{i}^{max(t)} - \left( {I_{i}^{t} - u_{i}^{t} } \right) - E\left( {S_{i}^{t - 1} } \right)}}{{2\sqrt {Var\left( {\eta_{I}^{t} } \right)} }}} \right)} \right]} \right\} $$

The derivation of moment equation for S ^t _i  = S ^min(t) _i was down in the same manner.

1.2 Appendix 1.2: Derivations of moment equation for .S ^min(t) _i  ≤ S ^t _i  ≤ S ^max(t) _i |S ^t+1 _j  = S ^min(t+1) _j

Taking E( ) over these partitions, the expectation from is approximated by using the product of the marginal probabilities as follows:

$$ S_{j}^{min(t + 1)} E\left( {S_{i}^{t} } \right)Pr\left( {S_{i}^{min(t)} \le S_{i}^{t} \le S_{i}^{max(t)} ,S_{j}^{t + 1} = S_{j}^{min(t + 1)} } \right) = $$

$$ S_{i}^{min(t + 1)} E\left( {S_{i}^{t} } \right)Pr\left( {S_{j}^{t + 1} = S_{j}^{min(t + 1)} } \right) $$

$$ = S_{j}^{min(t + 1)} E\left( {S_{i}^{t} } \right) \cdot \left\{ {\frac{1}{2}\left[ {1 + erf\left( {\frac{{S_{j}^{min(t + 1)} - \left( {I_{j}^{t + 1} - u_{j}^{t + 1} } \right) - E\left( {S_{j}^{t} } \right)}}{{2\sqrt {Var\left( {\eta_{I,j}^{t + 1} } \right)} }}} \right)} \right]} \right\} $$

The derivation of moment equation for $\left. {S_{i}^{min(t)} \le S_{i}^{t} \le S_{i}^{max(t)} } \right|S_{j}^{t + 1} = S_{j}^{max(t + 1)}$ was down in the same manner.

1.3 Appendix 1.3: Derivations of moment equation for .S ^min(t) _i  ≤ S ^t _i  ≤ S ^max(t) _i |S ^min(t+1) _j  ≤ S ^t _j  ≤ S ^max(t+1) _j

Taking E( ) over $\left. {S_{i}^{min(t)} \le S_{i}^{t} \le S_{i}^{max(t)} } \right|S_{j}^{min(t + 1)} \le S_{j}^{t} \le S_{j}^{max(t + 1)}$ we have:

$$ E(S_{i}^{t} S_{j}^{t + 1} )Pr(S_{i}^{min(t)} \le S_{i}^{t} \le S_{i}^{max(t)} ,S_{j}^{min(t + 1)} \le S_{j}^{t + 1} \le S_{j}^{max(t + 1)} ) $$

E(S ^t _i S ^t+1 _j ) accounts for the covariance of storage in a recursive manner, and can be approximated as:

$$ \begin{aligned} = Cov\left( {S_{i}^{t - 1} S_{j}^{t} } \right) + E\left( {S_{i}^{t - 1} } \right) \cdot E\left( {S_{j}^{t} } \right) + E\left( {S_{i}^{t - 1} } \right) \cdot \left( {I_{j}^{t + 1} - u_{j}^{t + 1} } \right) + E\left( {S_{j}^{t} } \right) \cdot \left( {I_{i}^{t} - u_{i}^{t} } \right) + \left( {I_{j}^{t + 1} - u_{j}^{t + 1} } \right) \cdot \left( {I_{i}^{t} - u_{i}^{t} } \right) + \\ Cov\left( {\eta_{I,i}^{t} \eta_{I,j}^{t + 1} } \right) \end{aligned} $$

where Cov(η ^t _I,i η ^t+1 _I,j ) represents the covariance between the noise components of the inflow variables at time t and time t + 1. As a result, the part of equation can be presented as:

$$ \begin{gathered} = \left\{ {Cov\left( {S_{i}^{t - 1} S_{j}^{t} } \right) + E\left( {S_{i}^{t - 1} } \right) \cdot E\left( {S_{j}^{t} } \right) + E\left( {S_{i}^{t - 1} } \right) \cdot \left( {I_{j}^{t + 1} - u_{j}^{t + 1} } \right) + E\left( {S_{j}^{t} } \right) \cdot \left( {I_{i}^{t} - u_{i}^{t} } \right)} \right. \hfill \\ \left. { + \left( {I_{j}^{t + 1} - u_{j}^{t + 1} } \right) \cdot \left( {I_{i}^{t} - u_{i}^{t} } \right) + Cov\left( {\eta_{I,i}^{t} \eta_{I,j}^{t + 1} } \right)} \right\} \hfill \\ \end{gathered} $$

$$ \left\{ {\frac{1}{2}\left[ {erf\left( {\frac{{S_{i}^{max(t)} - \left( {I_{i}^{t} - u_{i}^{t} } \right) - E\left( {S_{i}^{t - 1} } \right)}}{{2\sqrt {Var\left( {\eta_{I,i}^{t} } \right)} }}} \right)} \right] - erf\left[ {\frac{{S_{i}^{min(t)} - \left( {I_{i}^{t} - u_{i}^{t} } \right) - E\left( {S_{i}^{t - 1} } \right)}}{{2\sqrt {Var\left( {\eta_{I,i}^{t} } \right)} }}} \right]} \right\} $$

$$ \left\{ {\frac{1}{2}\left[ {erf\left( {\frac{{S_{j}^{max(t + 1)} - \left( {I_{j}^{t + 1} - u_{j}^{t + 1} } \right) - E\left( {S_{j}^{t} } \right)}}{{2\sqrt {Var\left( {\eta_{I,j}^{t + 1} } \right)} }}} \right)} \right] - erf\left[ {\frac{{S_{j}^{min(t + 1)} - \left( {I_{j}^{t + 1} - u_{j}^{t + 1} } \right) - E\left( {S_{j}^{t} } \right)}}{{2\sqrt {Var\left( {\eta_{I,j}^{t + 1} } \right)} }}} \right]} \right\} $$

$$ S_{j}^{max(t + 1)} E\left( {S_{i}^{t} } \right)\left\{ {\frac{1}{2}\left[ {1 - erf\left( {\frac{{S_{j}^{max(t + 1)} - \left( {I_{j}^{t + 1} - u_{j}^{t + 1} } \right) - E\left( {S_{j}^{t} } \right)}}{{2\sqrt {Var\left( {\eta_{I,j}^{t + 1} } \right)} }}} \right)} \right]} \right\} $$