*3.2. ED Model for the AC*/*DC Hybrid Microgrid*

#### 3.2.1. Objective Function

For the isolated microgrid, the dynamic economic dispatch strategy is based on the fact that all the distributed power supply equipment costs are fixed; meanwhile, the energy storage units can absorb or release power through their charging or discharging process. The objective function is formulated as

$$\min \left( \sum\_{t=1}^{T} \left[ \sum\_{i \in S\_{TG}} \mathbb{C}\_{i}(P\_{TG,i}(t)) + \sum\_{j \in S\_{WT}} \mathbb{C}\_{j}(P\_{WT,j}(t)) + \sum\_{k \in S\_{SL}} \mathbb{C}\_{k}(P\_{SL,k}(t)) + \sum\_{r \in S\_{BS}} \mathbb{C}\_{r}(P\_{BS,r}(t)) \right] \right) \tag{1}$$

where *T* is the number of time periods in the daily dispatching cycle. *STG* and *SWT* are the sets of TG units and WT units, respectively. *SSL* is the set of PV units, and *SBS* represents the set of BS units. *P*TG,i(t), *PWT,j(t)*, *PSL,k(t)*, and *PBS,r(t)* are the active output power of the *i*th TG, *j*th WT, *k*th PV, and *r*th BS over a time period *t*, respectively. *Ci*(*PTG,i(t)*), *Cj*(*PWT,j(t)*), *Ck*(*PSL,k(t)*), and *Cr*(*PBS,r(t)*) are the generation cost/penalty functions of the corresponding unit set, respectively.

Traditional Generator Set Power Generation Cost Function

Without loss of generality, for a traditional generator set, the operating cost function can be modeled as the following quadratic function form [43]

$$\mathbb{C}\_{i}(P\_{\text{TG},i}(t)) = a\_{\text{TG},i}(P\_{\text{TG},i}(t))^2 + b\_{\text{TG},i}P\_{\text{TG},i}(t) + c\_{\text{TG},i}, \quad i \in \mathcal{S}\_{\text{TG}} \tag{2}$$

where *aTG*,*i*, *bTG*,*i*, and *cTG*,*<sup>i</sup>* represent the cost function coefficients.

Wind Generator and Photovoltaic Unit Abandonment Wind Penalty Function

The wind and solar penalty functions are formulated as

$$C\_j(\boldsymbol{P\_{WT,j}}(t)) = a\_{WT,j} [\boldsymbol{P\_{WT,j}}(t) - \boldsymbol{P\_{WT,j}}(t)]^2 + b\_{WT,j} [\boldsymbol{P\_{WT,j}}(t) - \boldsymbol{P\_{WT,j}}(t)] + c\_{WT,j}, \quad j \in \text{SWT} \tag{3}$$

$$\mathbb{C}\_{k}(P\_{\rm SL,k}(t)) = a\_{\rm SL,k}[P\_{\rm SL,k}(t) - P\_{\rm SL,k}^{\rm st}(t)]^2 + b\_{\rm SL,k}[P\_{\rm SL,k}(t) - P\_{\rm SL,k}^{\rm st}(t)] + c\_{\rm SL,k} \quad k \in \text{S}\_{\rm SL} \tag{4}$$

where *aSL*,*k*, *bSL*,*k*, *cSL*,*<sup>k</sup>* and *aSL*,*k*, *bSL*,*k*, *cSL*,*<sup>k</sup>* are the cost coefficients of abandoned wind and light respectively, and *Pst WT*,*j* (*t*) and *Pst SL*,*k* (*t*) are the maximum value of the adjustable power of the *j*th WT and the *k*th PV at the moment *t*, respectively.

Energy Storage Unit Operating Cost Function

The operating cost function for the energy storage unit is represented as an over-origin quadratic function with an opening up of

$$\mathbb{C}\_{r}(P\_{\rm BS,r}(t)) = a\_{\rm BS,r}(P\_{\rm BS,r}(t))^2 + b\_{\rm BS,r}P\_{\rm BS,r}(t) + c\_{\rm BS,r}, \quad r \in \mathcal{S}\_{\rm BS} \tag{5}$$

where *aBS*,*r*, *bBS*,*r*, *cBS*,*<sup>r</sup>* denotes the coefficient of cost function.
