**Exit**

In the negotiation process shown in Figure 4, each participant agent places an initial value for its decision variable, Δ Vm=<sup>0</sup> K <sup>2</sup> for the grid-level agent and <sup>Δ</sup>P<sup>m</sup> Cap,k, k = 1, ... , N, for each of the microgrid agents, and notifies all other participant agents. Then, with the knowledge of all other agents' proposed decisions, and the voltage–load variation relationship coefficients, ck, each participant agent optimizes the augmented Lagrangian with the proximal, considering its own decision as the only variable. Due to the separability of the objective in Equation (12), the optimization conducted by each negotiation participant is closed to its target value with the constraint of the voltage–load variation relationship. In addition, all components of the individual optimization functions are square errors, which are quadratic. This ensures the strict convexity of these functions and guarantees that a unique vector of minimizers can be found in each round of negotiation. The grid-level agent collects all the optimization results from all negotiation participants. The residual is calculated and used to judge if the negotiation agreement is obtained. If the residual is beyond a pre-specified range, a new round of negotiation starts with the updated Lagrangian multiplier, λm<sup>+</sup>1. The negotiation iteration continues until the residual converges to a small enough value. Notice that the selection of the initial values, such as the initial Lagrangian multiplier, λ, the penalty factor ρ, the proximal factor φ and VGI microgrid load curtailments can all have the effects on the negotiation convergence. The investigation of these impacts is conducted to ensure the performance of the control scheme.

**Figure 4.** The development and message flow of the distributed voltage compensation negotiation and VGI microgrid control.
