*4.2. The ADMM-Based Voltage Compensation Negotiation*

With the voltage–load variation relationship information, a distributed negotiation is triggered to compensate the voltage violation at BUS K in a distribution feeder branch amongst all the agents in the feeder branch. The voltage compensation negotiation aims to maximally recover the voltage back to the allowable range with least sacrifice of PEV charging requirements.

The microgrid agents want to minimize further power curtailment from their critical charging power point. The objective function of microgrid agents are defined in Equation (10).

$$\|f\_k(\Delta \mathcal{P}\_{\text{Cap},k}) = \left\| \Delta \mathcal{P}\_{\text{Cap},k} \right\|\_{2'}^2 \text{: } k = 1, \dots, \text{N} \tag{10}$$

The objective of the grid-level agent is to meet the distribution voltage compensation target as much as possible. Define the distribution RMS voltage square compensation target as Δ|VK| 2 <sup>T</sup> = |VK| 2 <sup>T</sup> −|VK| 2 ins, which is the difference between the target RMS voltage square and the instant RMS voltage square at the violation bus. The value of the target RMS voltage is selected to be a little higher than the distribution voltage lower bound, 0.95 p.u. The objective function of the grid-level agent is represented as Equation (11)

$$\left\| f\_0(\Delta |\mathbf{V\_K}|^2) = \left\| \Delta |\mathbf{V\_K}|^2 - \Delta |\mathbf{V\_K}|\_{\mathbf{T}}^2 \right\|\_2^2 \tag{11}$$

The voltage compensation problem is converted to a coordination problem that balances the objectives of the distribution grid voltage requirement and the PEV charging requirements in multiple microgrids as shown in Equation (12).

$$\begin{aligned} \min\_{\mathbf{x}} & \left[ f \mathfrak{d} (\Delta |\mathbf{V\_{K}}|^{2}) + \sum\_{k=1}^{N} f\_{k} (\Delta \mathbf{P\_{Cap,k}}) \right] \\ \text{s.t.} & \mathbf{c\_{0}} \Delta |\mathbf{V\_{K}}|^{2} + \sum\_{k=1}^{N} \mathbf{c\_{k}} \Delta \mathbf{P\_{Cap,k}} = \mathbf{0} \end{aligned} \tag{12}$$

This minimization problem is solved using ADMM iteratively. The augmented Lagrangian is expressed as Equation (13) in the first step.

$$\begin{aligned} L\_{\mathbb{P}}(\left[\Delta|\mathbf{V\_{K}}|^{2}, \Delta \mathbf{P\_{Cap}}\right], \lambda) &= f\_{\mathbb{0}}(\Delta|\mathbf{V\_{K}}|^{2}) + \sum\_{k=1}^{N} f\_{k}(\Delta \mathbf{P\_{Cap,k}}) + \\ \lambda \Big(\mathbf{c\_{0}} \Delta |\mathbf{V\_{K}}|^{2} + \sum\_{k=1}^{N} \mathbf{c\_{k}} \Delta \mathbf{P\_{Cap,k}}\Big) + \frac{\rho}{2} \left\| \mathbf{c\_{0}} \Delta |\mathbf{V\_{K}}|^{2} + \sum\_{k=1}^{N} \mathbf{c\_{k}} \Delta \mathbf{P\_{Cap,k}} \right\|\_{2}^{2} \end{aligned} \tag{13}$$

<sup>Δ</sup>**PCap** <sup>=</sup> <sup>Δ</sup>PCap,k|<sup>k</sup> <sup>=</sup> 1, ... , N is a vector of microgrid PEV charging power curtailments. The symbol λ is the Lagrangian multiplier. ρ is called the penalty parameter. The optimal solution set of the voltage compensation and the microgrid PEV charging curtailments Δ|VK| 2, <sup>Δ</sup>**PCap** can be found through an iterative optimization process as shown in Equation (14).

$$\begin{aligned} \left[\Delta|\mathbf{V}\_{\mathbf{K}}|^{2}, \Delta \mathbf{P}\_{\mathbf{Cap}}\right]^{m+1} &:= \operatorname\*{argmin}\_{\left[\Delta|\mathbf{V}\_{\mathbf{K}}|^{2}, \Delta \mathbf{P}\_{\mathbf{Cap}}\right]} L\_{\rho}(\left[\Delta|\mathbf{V}\_{\mathbf{K}}|^{2}, \Delta \mathbf{P}\_{\mathbf{Cap}}\right], \lambda^{m});\\ \lambda^{m+1} &= \lambda^{m} + \rho \Big(\mathbf{c}\_{0} \Delta |\mathbf{V}\_{\mathbf{K}}|^{2} + \sum\_{k=1}^{N} \mathbf{c}\_{k} \Delta \mathbf{P}\_{\mathbf{Cap},k}\Big);\end{aligned} \tag{14}$$

Due to the separability of the minimization objective (Equation (12)), the update of decision variable can be conducted by each negotiation participant in parallel using the Jacobian type of method. This method fixes variables that are not directly related to a microgrid to the last iteration decisions, therefore simplifies the calculation. Furthermore, a proximal term φ 2 ||Δ|VK| <sup>2</sup> <sup>−</sup> <sup>Δ</sup> Vm K 2 ||2 <sup>2</sup> <sup>+</sup> *<sup>N</sup> <sup>k</sup>*=<sup>1</sup> ||ΔPCap,k <sup>−</sup> <sup>Δ</sup>P*<sup>m</sup>* Cap,k||<sup>2</sup> 2 is added to strengthen the convexity of the augmented Lagragian and accelerate the convergence. The distributed solution of the distribution voltage compensation problem can be achieved through the following iterations.

### **Distributed Voltage Compensation Negotiation Process**
