# Decentralized Coordination of DERs for Dynamic Economic Emission Dispatch
## 3 Problem Reformulation
### Lagrange Dual Problem  
Note that both the constraint sets (18b) and (18c) are convex. Recall Assumption 1 and the definition of the emission function. Hence, problem (18) has a unique minimizer denoted \(p_{i}^{\star}\). Moreover, there is a zero duality gap, which allows us to convert the original problem (18) to its Lagrange dual form.  
To facilitate the design of decentralized algorithms, we introduce \(V_{i}\in R^{T}\) with no physical meanings, representing virtual local demands at unit \(i\) for all the time slots such that \(\sum_{i=1}^{m+s}V_{i}=D\). The virtual demand for each unit \(i\) can choose \(V_{i}=D/(m+s)\) for convenience. Define the indicator function \(\delta_{l_{i}}(p_{i})\) as \(\delta_{l_{i}}(p_{i})=0\) if \(p_{i}\in\delta_{l_{i}}\) and \(\delta_{l_{i}}(p_{i})=\infty\), otherwise. Then, problem (18) can be written as the following equivalent formulation \[\min_{p_{1},\ldots,p_{m+s}} \sum_{i=1}^{m+s}\tilde{C}_{i}(p_{i})+\delta_{I_{i}}(p_{i})\] (19a) s.t. \[\sum_{i=1}^{m+s}(p_{i}-p_{i}^{l})\geq\sum_{i=1}^{m+s}V_{i} \tag{19b}\]  
Define the Lagrangian function associated with problem (19) as follows:  
\[\Phi(p,\lambda)=\sum_{i=1}^{m+s}\tilde{C}_{i}(p_{i})+\delta_{I_{i}}(p_{i})+ \lambda^{\prime}\Bigg{(}\sum_{i=1}^{m+s}(p_{i}-p_{i}^{l}-V_{i})\Bigg{)} \tag{20}\]  
where \(p=[p_{1}{}^{\prime},\ldots,p_{m+s}{}^{\prime}]^{\prime}\in R^{(m+s)\times T}\), and \(\lambda\in R^{T}\) is the Lagrangian multiplier corresponding to constraint (19b). Then, the corresponding Lagrangian dual problem of (19) is defined as  
\[\max_{\lambda\in R^{T}}\inf_{p\in R^{(m+s)\times T}}\Phi(p,\lambda) \tag{21}\]  
where  
\[\inf_{p\in R^{(m+s)\times T}}\Phi(p,\lambda) =\inf_{p\in R^{(m+s)\times T}}\Bigg{\{}\sum_{i=1}^{m+s}\tilde{C}_ {i}(p_{i})+\delta_{I_{i}}(p_{i})+\lambda^{\prime}\Bigg{(}\sum_{i=1}^{m+s}(p_{ i}-p_{i}^{l}-V_{i})\Bigg{)}\Bigg{\}}\] \[=-\sum_{i=1}^{m+s}\sup_{p_{i}\in R^{T}}\Big{(}-\tilde{C}_{i}(p_{i })-\delta_{I_{i}}(p_{i})-\lambda^{\prime}(p_{i}-p_{i}^{l})\Big{)}-\sum_{i=1}^{ m+s}\lambda^{\prime}V_{i} \tag{22}\]  
Define the conjugate function \(F_{i}^{\otimes}(\lambda)=\sup_{p_{i}\in R^{T}}\{\lambda^{\prime}(p_{i}-p_{i}^ {l})-\tilde{C}_{i}(p_{i})-\delta_{I_{i}}(p_{i})\}\). Then, we have  
\[\inf_{p\in(m+s)\times T}\Phi(p,\lambda)=-\sum_{i=1}^{m+s}F_{i}^{\otimes}(- \lambda)-\sum_{i=1}^{m+s}\lambda^{\prime}V_{i} \tag{23}\]  
Hence, the dual problem (21) can be rewritten as  
\[-\min_{\lambda\in T}\Bigg{(}\sum_{i=1}^{m+s}F_{i}^{\otimes}(-\lambda)+\sum_{i =1}^{m+s}\lambda^{\prime}V_{i}\Bigg{)} \tag{24}\]  
which is equivalent to the following consensus optimization problem  
\[\min\sum_{i=1}^{m+s}\phi_{i}(\lambda_{i})\] \[\text{s.t. }\lambda_{i}=\lambda_{j},\ \ (i,j)\in\mathcal{E} \tag{25}\]  
where \(\phi_{i}(\lambda)\stackrel{{\Delta}}{{=}}F_{i}^{\otimes}(- \lambda)+\lambda^{\prime}V_{i}\) is convex, continuously differentiable, and \(1/\mu_{i}\)-Lipschitz smooth (see Theorem 4.2.1, [39]). For any given \(\lambda\in R^{T}\), the dual function \(\phi_{i}(\lambda)\) is differentiable at \(\lambda\) and its gradient is \(\
abla\phi_{i}(\lambda)=-\arg\min_{p_{i}\in R^{T}}(\lambda(p_{i}-p_{i}^{l})+ \tilde{C}_{i}(p_{i})+\delta_{I_{i}}(p_{i}))+V_{i}\).  
Remark 2: _Note that the dual problem (25) is a typical consensus-based optimization problem that has been intensively investigated in the field of decentralized optimization on multi-agent systems. Such a formulation caters to a majority of existing decentralized consensus-based algorithms. For conventional undirected communication networks, one may resort to DGD [24], EXTRA [31], AugDGD [34], etc., while for directed or dynamic communication networks, the insightful works like Push-DIGing [33], Frost [32], etc., are worth adoption._