# Incentive Price-Based Demand Response in Active Distribution Grids
## 4 Proposed IDR Method for Control of the BESS
### Proposed Optimization Problem of the LVGC  
The typical objectives of the LVGC are minimization of the grid active power losses, optimum use of DERs and optimum active and reactive power drawn/exported to the MV grid subjected to the constraints of the LV grid such as node voltage limits, branch current limits, and power limits of the DERs etc. In this paper, the discussion is limited to the optimum flexible power to be provided by DERs for maximizing the self-power consumption at each node by providing incentive price signals to each DER. Although the same formulation can be extended to include other objectives as well, it is out of scope of this paper and will be taken into consideration in our future works. Let us assume that the DSO can provide an incentive price signal \(\alpha_{ji}^{\kappa}\) to each customer who owns the BESS denoted by \(i\) at each time instant \(\kappa\). Each customer may react to the incentive price signal differently and the DSO cannot influence the customer's response to provide flexibility. As a worst case, the customer may ignore the incentive price and will not provide any flexibility. Alternatively, their response could be a linear variation of BESS power consumption with respect to the incentive price signal over a certain range and beyond that there will not be any increase in power consumption due to limitation in the power rating of the BESS. Let us formulate the above dependency by an unknown function \(p_{fi}^{\kappa}=f_{i}(\alpha_{ji}^{\kappa})\). A simplistic assumption, as mentioned earlier, is that it could be a ramp function with bounds on both ends as stated in [18], but this assumption will make the function non-affine. The above unknown function could also be nonlinear depends on configuration of the DER controller's response to the incentive price signal. The objective of the LVGC is defined as the minimization of the costs of net power consumption \(\mathbf{p}_{n}\) at each node \(n\) over the control horizon, which indirectly means maximization of the self-consumption of power at each node, by the following expression  
\[\min_{\mathbf{u}}J=\sum_{\kappa=1}^{N_{\varsigma}}\mathbf{p}_{n}^{\kappa} \mathbf{Q}^{\kappa}\mathbf{p}_{n}^{\kappa}\] (12a) subject to the following constraints in which \[\kappa\in\mathcal{N}_{c}\]. The matrix \[\mathbf{Q}^{\kappa}\] is a diagonal matrix containing the costs of electricity power consumption at the time instant \[\kappa\]. The constraint specifying the node voltages as a linear function of node active and reactive powers based on the derivation in Section 3.1 is provided below. \[\mathbf{v}^{\kappa}=\mathbf{v}_{s}^{\kappa}+\mathbf{B}_{sp}(\mathbf{C}_{f} \mathbf{\tilde{p}}_{f}^{\kappa}-\mathbf{p}_{d}^{\kappa})+\mathbf{B}_{sq}( \mathbf{C}_{f}\mathbf{q}_{f}^{\kappa}-\mathbf{q}_{d}^{\kappa}) \tag{12b}\]  
The minimum and maximum limits of node voltages are specified in the below constraint whose typical values are 0.9 and 1.1 pu respectively. \[\mathbf{v}_{\min}\leq\mathbf{v}^{\kappa}\leq\mathbf{v}_{\max} \tag{12c}\]  
The power limits of each DER is forced in the below constraint. \[\mathbf{p}_{f\min}\leq\mathbf{p}_{f}^{\kappa}\leq\mathbf{p}_{f\max} \tag{12d}\]  
The below constraint specifies the dependency of the power consumption of the \(i\)th DER on the incentive price signal \(\alpha_{i}\) provided by the DSO by an unknown function \(f_{i}\) at each time instant \(\kappa\). \[p_{fi}^{\kappa}=f_{i}^{\kappa}(\alpha_{fi}^{\kappa})\quad\forall i\in \mathcal{N}_{f} \tag{12e}\]  
This function need not be an affine function because the incentive price calculated for each BESS need not have a linear relationship with its power consumption. This property makes the optimization problem of the LVGC formulated in (12) to be non-convex. A convex relaxation of the above optimization problem is derived in [18] in which the constraint (12e) is removed and the modified formulation is provided below.