# Decentralized Coordination of DERs for Dynamic Economic Emission Dispatch
## 3 Problem Reformulation
### Convex Equivalency of DEED  
We first address the nonconvexity of the charging/discharging efficiency model (8). Define \(p_{i,t}^{+}\in[0,p_{i}^{\max}]\) and \(p_{i,t}^{-}\in[0,-p_{i}^{\min}]\), indicating that ES \(i\) charges and discharges at time slot \(t\), respectively. We have \(p_{i,t}=p_{i,t}^{+}-p_{i,t}^{-},\forall t\in T\). Then, \(p_{i,t}^{batt}\) (8) can be replaced by  
\[p_{i,t}^{batt}=\frac{1}{\eta_{i}^{+}}p_{i,t}^{+}-\eta_{i}^{-}p_{i,t}^{-}, \forall t\in T,i\in\mathcal{S} \tag{12}\]  
Thus, the nonconvex problem (11) is equivalent to  
\[\min F(p_{i,t})=\omega\sum_{t=1}^{T}\sum_{i=1}^{m+s}C_{i}(p_{i,t}) +(1-\omega)\sum_{t=1}^{T}\sum_{i=1}^{m}E_{i}^{emi}(p_{i,t})\] (13) s.t. constraints (3)-(7), (9), (10), and (12)  
Since storages cannot be charged and discharged simultaneously, either \(p_{i,t}^{+}\) or \(p_{i,t}^{-}\) needs to be zero, i.e.,  
\[p_{i,t}^{+}p_{i,t}^{-}=0,\forall t\in T,i\in\mathcal{S} \tag{14}\]  
According to Theorem 1 in [20], since \(\eta_{i}^{+}\) and \(\eta_{i}^{-}\) are strictly less than \(1\), they can be neglected.  
For simplicity of expression, we rewrite problem (13) in a compact form. For each ES, \(i\in\mathcal{S},\forall t\in T\), we define the constraint set  
\[J_{i}^{\mathcal{S}}=\left\{p_{i}\in R^{T}|p_{i,t}\text{ satisfies \ \eqref{eq:p_i,t},\eqref{eq:p_i,t},and \ \eqref{eq:p_i,t}}\right\} \tag{15}\]where \(p_{i}=[p_{i,1},p_{i,2},\ldots,p_{i,T}]^{\prime}\). For each DG, \(i\in\mathcal{M},\forall t\in T\), we define the constraint set  
\[J_{i}^{\mathcal{M}}=\left\{p_{i}\in R^{T}|p_{i,t}\text{ satisfies }\left(5 \right)\text{ and }\left(6\right)\right\} \tag{16}\]  
Define \(J_{i}=\left\{p_{i}|J_{i}^{\mathcal{M}}\cup J_{i}^{S},i\in\mathcal{M}\cup \mathcal{S}\right\}\), \(D=[D_{1},D_{2},\ldots,D_{T}]^{\prime}\), and \(P_{loss}=[P_{loss,1},P_{loss,2},\ldots,P_{loss,T}]^{\prime}\), \(\forall i\in\mathcal{M}\cup\mathcal{S},t\in\mathcal{T}\). By supplementing the emission functions for ESs with \(E_{i}^{emi}(p_{i,t})=0,i\in\mathcal{S}\), problem (13) can be rewritten as follows,  
\[\min_{p_{1},\ldots,p_{m+s}} \sum_{i=1}^{m+s}\tilde{C}_{i}(p_{i})\] (17a) s.t. \[\sum_{i=1}^{m+s}p_{i}-P_{loss}=D \tag{17b}\] \[p_{i}\in J_{i} \tag{17c}\]  
where \(\tilde{C}_{i}(p_{i})=\sum_{t=1}^{T}\omega C_{i}(p_{i,t})+(1-\omega)\sum_{t=1}^ {T}E_{i}^{emi}(p_{i,t})\). Note that the set \(J_{i}\) is convex polyhedra since all constraints therein are affine. Then, we proceed to convert problem (17) to an equivalent convex one by relaxing the equality constraint (17b) to an inequality one, as follows  
\[\min_{p_{1},\ldots,p_{m+s}} \sum_{i=1}^{m+s}\tilde{C}_{i}(p_{i})\] (18a) s.t. \[\sum_{i=1}^{m+s}p_{i}-P_{loss}\geq D \tag{18b}\] \[p_{i}\in J_{i} \tag{18c}\]  
The following lemma provides a sufficient condition to ensure that problem (17) and (18) have the same solution.  
**Lemma 1** ([37]).: _Suppose that \(D_{t}\geq\sum_{i=1}^{m+s}\left(p_{i}^{\min}-B_{i}p_{i}^{\min^{2}}\right)\). Then, the optimal solution to problem (18) is the same as problem (17)._  
**Remark 1**.: _The detailed proof of Lemma 1 refers to [37]. It is shown that if the total demand is larger than or equal to the total minimal power generation subtracting power losses at each time slot \(t\), the optimal solution to problem (18) satisfies \(D=\sum_{i=1}^{N}p_{i}^{\star}-P_{loss}\), where \(p_{i}^{\star}\) is the optimal power generation schedule for each committed units \(i\in\mathcal{M}\cup\mathcal{S}\). That is, problem (18) is an equivalent convex of problem (17). Although each DG has a lower bound of power generation \(p_{i}^{\min}>0,i\in\mathcal{M}\), due to the introduction of ESs, the sufficient condition is easy to satisfy since each ES can charge power with \(p_{i}^{\min}<0,i\in\mathcal{S}\) to relax the sufficient condition._  
Next, we will transform the DEED problem (18) through a Lagrangian duality reformulation into an equivalent consensus optimization problem.