## 4 Case Study
### MOPSO-Based Problem Solution  
#### 4.3.1 Comparison of Bi- and Mono-Objective Modeling  
In order to verify the effectiveness of bi-objective modeling and the advantages over the mono-objective counterparts, we investigate the following three scenarios:  
* Scenario A: \"Money-oriented\", a mono-objective model that solely saves EC.
* Scenario B: \"User-first\", a mono-objective model that solely enhances PQ.
* Scenario C: \"Pareto-optimal\", a bi-objective model that compromises EC and PQ.  
The running data of Hongfeng Eco-town in a typical day are utilized for simulation. The scenarios formulated by mono-objective models are solved by using classical PSO algorithm, and Scenario C with a bi-objective model is solved by using MOPSO algorithm. The particle population is 200 and the maximal generation is 150. Each scenario is tested by 20 trials of simulation, and the optimization processes are depicted in Figure 12.  
In Scenario A, EC monotonically decreases with the generation counts since the optimization is \"money-oriented\". On the other hand, the curves of PQ are fluctuant and eventually converge to low levels. It is revealed that pursuing economies blindly will face the risk of losing users' satisfaction upon power quality. On the contrary, the PQ curves of Scenario B are monotonically increasing due to the \"user-first\" doctrine in optimization. Nonetheless, higher cost is inevitable. As a result, it is necessary to consider Scenario C, which reconciles the two conflicting objectives and derives the Pareto-optimal solutions.  
Figure 11: Comparison of prediction errors of Grey Model and LSTM: (**a**) root mean square error; (**b**) maximum error.  
Figure 10: Prediction results of Grey model.  
Figure 12d demonstrates the converged positions of 200 particles in EC-PQ plane after 150 generations of optimization. Each particle of Pareto frontier, which is denoted by a red star, is a non-inferior solution to the bi-objective optimization problem. Without loss of generality, one Pareto optimum is selected from the median position of Pareto frontier and its convergence history is depicted in Figure 12c. Neither EC nor PQ are monotonic functions of generation count due to the metaheuristic nature of MOPSO. However, in a global view, EC and PQ respectively converge to low and high levels, i.e., EC and PQ are simultaneously improved. Numeric results of three scenarios are listed in Table 3. The Pareto frontier provides a visible and flexible way for decision guidance. The operator ME-VPP can select dispatch solution from Pareto frontier according to various specifications of scenarios. One noninferior solution of the Pareto frontier is shown in Figure 13, which contains power output and the amount of energy storage inside the storage devices.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Scenario** & **Description** & **Optimization Objective** & **EC (10\({}^{6}\)****CNY)** & **PQ (Score in 100)** \\ \hline A & Money-oriented & Economic cost (EC) & 2.55 & 58.30 \\ B & User-first & Power quality (PQ) & 4.82 & 90.38 \\ C & Pareto-optimal & EC and PQ & 4.26 & 84.36 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Optimization results of three scenarios.  
Figure 12: Optimization results of mono-objective and bi-objective: (**a**) “money-oriented” scenario; (**b**) “user-first” scenario; (**c**) “Pareto-optimal” scenario; (**d**) economic cost and power quality (EC-PQ) plane after 150 generations.