*4.2. Robust Equal Conversion*

On the basis of the sets of uncertainties introduced in Sections 2.1 and 2.2, the spinning reserve constraint (Equation (18)) can be transformed to

$$\sum\_{i=1}^{I} P\_{i,t}^{\max} + \sum\_{j=1}^{J} PMT\_{j,t}^{\max} + P\_{grid,t}^{\max} + \sum\_{l=1}^{L} \left( \overline{PV\_{l,t}} + P\overline{V}\_{l,t} \right) \ge (1 + L\_t) \left[ P\_{load,t} + \sum\_{k=1}^{K} \left( \overline{PEV\_{k,t}} + PE\overline{V}\_{k,t} \right) \right] \tag{19}$$

RO deals with uncertain data in the worst-case scenario. The worst-case scenario can be defined as

$$F = \max\limits\_{l=1}^{L} P\hat{V}\_{l,t} - \left(1 + L\_{l}\right) \sum\_{k=1}^{K} P E \hat{V}\_{k,t}\big]$$

$$\begin{array}{c} \underbrace{P\hat{V}\_{l,t}}\_{\in P\hat{V}\_{k,t}} \le P\hat{V}\_{l,t} \le \overline{P\hat{V}\_{l,t}}\\ \underbrace{PE\hat{V}\_{k,t}}\_{} \le PE\hat{V}\_{k,t} \le \overline{PE\hat{V}\_{k,t}}\end{array} \tag{20}$$

The schedule objective function is monotonically increasing, strictly convex, and differentiable. Its dual problem is also feasible and bounded, and the objective values coincide according to strong duality [19]. Therefore, the dual problem becomes

$$\min \left( -\sum\_{l=1}^{L} \underbrace{P\hat{V}\_{l,t}^{G}}\_{l,t} \alpha\_{t} + \sum\_{l=1}^{L} \overbrace{P\hat{V}\_{l,t}^{G}}\_{l,t} \beta\_{t} - \sum\_{k=1}^{K} \underbrace{PE\hat{V}\_{k,t}\gamma\_{t}}\_{k=1} + \sum\_{k=1}^{K} \overbrace{PE\hat{V}\_{k,t}\delta\_{t}}\_{k} \right) \tag{21}$$
 
$$\begin{array}{c} -\alpha\_{t} + \beta\_{t} \ge 1 \\ -\gamma\_{t} + \delta\_{t} \ge -1 - L\_{t} \\ \alpha\_{t}, \beta\_{t}, \gamma\_{t}, \delta\_{t} \ge 0 \end{array} \tag{22}$$

where α*t*, β*t*, γ*t*, δ are the dual coefficients. Then the original spinning reserve constraint is converted to

$$\begin{aligned} \sum\_{i=1}^{I} P\_{i,t}^{\max} &+ \sum\_{j=1}^{I} PMT\_{j,t}^{\max} + P\_{grid,t}^{\max} + \sum\_{l=1}^{L} \overline{PV\_{l,t}} - (1 + L\_{l}) \sum\_{k=1}^{K} \overline{PEV\_{k,t}} - \sum\_{l=1}^{L} \underline{P\hat{V}\_{l,t} \alpha\_{l}} \\ &+ \sum\_{l=1}^{L} \overline{PV\_{l,t}} \beta\_{l} - \sum\_{k=1}^{K} \underline{PE\hat{V}\_{k,t} \gamma\_{l}} + \sum\_{k=1}^{K} \overline{PE\hat{V}\_{k,t}} \delta\_{l} \ge (1 + L\_{l}) P\_{load,t} \end{aligned} \tag{22}$$

## *4.3. Robust Economic Dispatch Model*

The purpose of this study was to minimize system operating costs and environmental protection costs while meeting the load on the microgrid system:

$$\begin{array}{l}\min\left\{\begin{array}{l}\sup\limits\_{P V\_{l,t} \in \mathcal{V} \mathcal{V}\_{k,t}} \mathcal{C}\_{1} + \mathcal{C}\_{2} \\ \text{s.t. } (12)-(17), (20)-(22) \end{array} \right\} \tag{23}$$

The probability of spinning reserve constraint violated (POV) is expressed as,

$$P\_I \left[ \sum\_{l=1}^{I} P\_{lj}^{\max} + \sum\_{j=1}^{I} PMT\_{j,l}^{\max} + P\_{grid,t}^{\max} + \sum\_{l=1}^{L} PV\_{l,l}^G < (1 + L\_l)(P\_{load,t} + \sum\_{k=1}^{K} PEV\_{k,l}^G) \right] \le P\_r \left[ \sum\_{m \in V} \eta\_{m,l} \omega\_{m,t} \ge \Gamma\_t \right] \tag{24}$$

where Γ*<sup>t</sup>* is the number of the uncertain variables. Furthermore, according to literature [29], the robustness of the system can be expressed by Formula (25):

$$P\_r\left[\sum\_{m\in V} \eta\_{m,t} \alpha\_{m,t} \ge \Gamma\_t\right] \le \exp\left[-\frac{\Gamma\_t^2}{2|I\_t|}\right] \tag{25}$$

where η*m*,*t*, ω*m*,*t*, *Jt* are the coefficients of POV.
