# Utilizing Energy Storage for Operational Adequacy of Wind-Integrated Bulk Power Systems
## 3 Operational Adequacy Evaluation Framework
### Dynamic System State Probability Evaluation Approach  
The system state probability is a dynamic parameter that depends on the exposure time of the operating system in consideration and the stochastic transitions between system states within the evaluation period. The transition probabilities are assessed using the component outage replacement rate, \(ORR=\lambda\times dt\), as such transitions are the results of component failures. The repair process is ignored as it lasts significantly longer than the unit commitment period. Equation (12) is used to identify the different states between which transition is possible. For instance, if \(s_{y}^{sy},y^{th}\) bit of state transition vector, \(S^{sy}\), is 1, it represents the possibility of state transition from the \(s^{th}\) system state to the \(v^{th}\) system state due to the failure of component represented by the \(y^{th}\) bit of state transition vector. The inter-state transition probability (\(\delta_{sy}\)) and state transition matrix (\(Q_{k}\)), as shown in Equations (13) and (14), are used to update the dynamic system state probabilities.  
\[S^{sy}=z^{s}\oplus z^{v}\quad\text{ where }s,v\in 1:m\text{ }\delta,\text{ }s<v \tag{12}\]  
\[\delta_{sy}=\left\{\begin{array}{ccccc}\sum\limits_{y=1}^{N_{w}+N_{c}+N_{c} +N_{T}}ORR_{y}s_{y}^{sy}\text{ if }s<v\\ 1-\sum\limits_{q=s+1}^{m}\delta_{sq}&\text{ if }s=v\\ 0,&\text{ if }s>v\end{array}\right. \tag{13}\]  
\[\mathbf{Q}_{k}=\left[\begin{array}{ccccc}\delta_{11}&\delta_{12}&\ldots&\delta _{1v}&\ldots&\delta_{1m}\\ 0&\delta_{22}&\ldots&\delta_{2v}&\ldots&\delta_{2m}\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ 0&0&\ldots&\delta_{sy}&\ldots&\delta_{sm}\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ 0&0&\ldots&\ldots&\ldots&\delta_{mm}\end{array}\right] \tag{14}\]  
The system state probability vector of the \(i^{th}\) time interval of the \(k^{th}\) unit commitment period, \(P_{k,i}=\left[p_{k,i}^{1}\ p_{k,i}^{2}\ldots\ p_{k,i}^{j}\ldots\ p_{k,i}^{m}\right]\), gives the corresponding dynamic probabilities of each system state of \(Z_{k}=\left[z^{1},z^{2},\ldots,z^{j},\ldots,\ z^{m}\right]\) to represent the changing system status along with the time intervals in the evaluation period. All the system components, wind generators, ESS conventional generators, and transmission lines are assumed to be operating at the start of the evaluation period. Hence, \(z^{1}\) is the initial system state of the evaluation period and the initial system state probability vector (\(P_{1,0}\)) of the evaluation period, and the first UCP is therefore given by Equation (15).  
\[P_{1,0}=\left[1\ 0\ldots\ 0\ldots\ 0\right] \tag{15}\]  
System state probability vector, \(P_{k,i}\), is updated at the end of each \(i^{th}\) interval within the same \(k^{th}\) UCP. However, \(Z_{k}\) and \(\mathbf{Q}_{k}\) are only updated at the start of the new UCP. The system state probability vector of the preceding interval, \(P_{k,i-1}\), is used as an initial condition to evaluate the system state probability vector of the succeeding interval, \(P_{k,i}\), as shown in Equation (16).  
\[P_{k,i}=P_{k,i-1}\ \mathbf{Q}_{k} \tag{16}\]