# Operation Scheduling Optimization for Microgrids Considering Coordination of Their Components
## 2 Problem Framework
### Formulation of Target Problem  
As described in Section 2.1, the operation scheduling problem has five optimization variables defined as  
\[u_{i,t}\in[0,\,1],\text{ for }\forall i,\,\forall t, \tag{1}\]  
\[g_{i,t}\in\left[G_{i}^{min},G_{i}^{max}\right],\text{ for }\forall i,\,\forall t, \tag{2}\]  
\[s_{j,t}\in\left[S_{j}^{min},S_{j}^{max}\right],\text{ for }\forall j,\,t\in TS_{j}, \tag{3}\]  
\[v_{k,t}\in\left[V_{k}^{min},0\right],\text{ for }\forall k,\,t\in TV_{k}, \tag{4}\]  
\[e_{t}\in\mathbb{R},\text{ for }\forall t, \tag{5}\]  
where \(t\) is time (\(t=1,\cdots,T\)); \(i\) is the number of CGs (\(i=1,\cdots,NG\)); \(u_{i,t}\) is the ON/OFF state variable of CGs (ON: 1, OFF: 0), which is an element of vectors \(u_{t}\) and \(u\); \(g_{i,t}\) is the output of CGs, which is an element of vectors \(g_{t}\) and \(g\); \(G_{i}^{max}\) and \(G_{i}^{min}\) are the maximum and the minimum outputs of CGs; \(j\) is the number of ESSs (\(j=1,\cdots,NS\)); \(s_{j,t}\) is the output of ESSs and an element of vectors \(s_{t}\) and \(s\); \(S_{j}^{max}\) and \(S_{j}^{min}\) are the maximum and the minimum capable outputs of ESSs (\(S_{j}^{min}<0<S_{j}^{max}\)); \(TS_{j}\) is the set of available time that ESSs are available; \(k\) is the number of CLs (\(k=1,\cdots,NV\)); \(v_{k,t}\) is the consumption of CLs and an element of vectors \(v_{t}\) and \(v\); \(V_{k}^{min}\) is the maximum capable consumption of CLs (\(V_{k}^{min}<0\)); \(TV_{k}\) is the set of available time of CLs; and \(e_{t}\) is the trading electricity, which is an element of vector \(e\). In general, the operation scheduling problem requires the assumption that forecasted values of the electricity consumption and the VREG outputs are given no matter whether they include uncertainty or not. With a view to treat the uncertainty, the authors express the variable representing the changeable net load with  
\[d_{t}\in\left[d_{t}^{min},d_{t}^{max}\right],\text{ for }\forall t, \tag{6}\]  
where \(d_{t}^{max}\) and \(d_{t}^{min}\) are the maximum and the minimum assumable values of the net load, and they can be set by referring to the historical record. Fuel costs of the CGs are normally approximated as quadratic functions by means of their outputs and, thus, can be minimized by controlling the CG outputs as steadily as possible. As opposed to the  
Figure 1: Example of a generalized microgrid model. CG, controllable generation system; ESS, energy storage system; CL, controllable load; VREGs, variable renewable energy-based generation systems. CGs, the ESSs consume no fuel in their operations directly, and this makes it challenging to evaluate their operating cost. In this paper, the objective function is represented as  
\[\min_{u,g,s,v,e}\sum\
olimits_{t=1}^{T}\int_{t_{t}^{min}}^{t_{t}^{max}}\Big{\{} C_{t}(u_{t},\ g_{t})+M_{t}e_{t}+l_{t}e_{t}^{\prime}\Big{\}}f\Big{(}\hat{d}_{t} \Big{)}\text{d}d_{t}, \tag{7}\]  
subject to  
\[C_{t}(u_{t},g_{t})=\sum\
olimits_{i=1}^{\text{NG}}\Big{(}\big{(}A_{i}+B_{i}g_ {i,t}+C_{i}g_{i,t}{}^{2}\big{)}+SC_{i}(1-u_{i,t-1})\Big{)}u_{i,t}, \tag{8}\]  
where \(C(u_{t},\ g_{t})\) is the operation cost of CGs; \(M_{t}\) is the price of electricity trade; \(l_{t}\) is the price of the imbalance penalty; \(e_{t}^{\prime}\) is the imbalance of electricity; \(f\Big{(}\hat{d}_{t}\Big{)}\) is the probability density function for the forecasted net load, \(\hat{d}_{t}\); \(A_{i}\), \(B_{i}\), and \(C_{i}\) are the coefficients of fuel cost of CGs; and \(SC_{i}\) is the start-up cost of CGs. Besides, the microgrid must satisfy the below conditions.