# Kernel Clustering with a Differential Harmony Search Algorithm for Scheme Classification
## 4 Experiments
### Case Study  
East Lake is the largest scenery-related tourist attraction in Wuhan China, located on the south bank of the Yangtze River. The East Lake Network covers an area of 436 km\({}^{2}\), consisting of East Lake, Sha Lake, Yangchun Lake, Yanxi Lake, Yandong Lake, and Bei Lake. A map of the East Lake is shown in Figure 3. In recent years, climate change and human activities have influenced the lakes significantly. Increasing sewage has led to serious eutrophication. Most of the sub-lakes in the East Lake Network are highly polluted. Chemical oxygen demand (COD), total nitrogen (TN) and total phosphorus (TP) of the lakes are shown in Table 11.  
In recent years, the government has invested 15.8 billion yuan (RMB) to build the Ecological Water Network Project. The Water Network Connection Project is one of the subprojects; this project tries to transfer water from the Yangtze River through the diversion channels to improve the water quality in the lakes. In this project, Water diversion scheduling (WDS) is a new scheduling method combining with hydrodynamics. Previous works, including a hydrodynamic simulation model and water quality model, have already been done. Using these models, water quality of the lakes at the terminal time can be simulated. The purpose of WDS is to find a suitable scheme which has a better water quality result and lower economic cost. This is a multi-objective problem but, unfortunately, multi-objective evolutionary algorithms (MOEA) cannot be adopted because the water quality simulation is time-consuming. After a variety of simulations have been done, several feasible schemes and their simulation results had been made. However, the difference among the results is small. Thus, we use cluster analysis to summarize these schemes. The water diversion scheduling in the East Lake Network is a multi-objective problem. To reduce the construction cost, the existing diversion channels include Zengjiaxiang, Qingshangang, Jiufengqu, Luojiagang, and Beihu pumping stations are considered. Water is brought from the Yangtze River to the East Lake Network through the Zengjiaxiang and Qingshangang channels, while water in the lakes is drained out through Luojiagang and Beihu pumping stations (Figure 4). The objective functions are shown as:  
\[\left\{ \begin{array}{l}\mathbf{I}=f_{1}\big{(}\mathbf{q}_{z},\mathbf{q}_{q},\mathbf{q}_{l}\big{)}\\ C=f_{2}\big{(}\mathbf{q}_{z},\mathbf{q}_{q},\mathbf{q}_{l}\big{)}\\ Q=f_{3}\big{(}\mathbf{q}_{z},\mathbf{q}_{q},\mathbf{q}_{l}\big{)}\end{array} \right. \tag{17}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Lake** & **COD (mg/L)** & **TN (mg/L)** & **TP (mg/L)** \\ \hline East Lake & 24 & 2.32 & 0.196 \\ Sha Lake & 50 & 6.11 & 0.225 \\ Yangchun Lake & 26 & 1.14 & 0.085 \\ Yanxi Lake & 34 & 3.82 & 0.2 \\ Yandong Lake & 10 & 0.7 & 0.05 \\ Bei Lake & 32 & 3.81 & 0.122 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Initial water qualities of the six lakes. Figure 4: Diversion channels of the East Lake Network. where **I** is the water quality index vector including TP, TN, and COD information, which is obtained by the water quality model. \(C\) is the total amount of water, \(Q\) is the economic cost, \(\mathbf{q}_{z}\) is the inflow vector of the Zengjiaxiang channel, \(\mathbf{q}_{q}\) is the inflow vector of the Qinshanggang channel, and \(\mathbf{q}_{l}\) is the outflow vector of the Luojiagang channel. Then the outflow of Jiufengqu and Beihu pumping stations follows:  
\[\mathbf{q}_{j}=\mathbf{q}_{b}=\mathbf{q}_{z}+\mathbf{q}_{q}-\mathbf{q}_{l} \tag{18}\]  
The initial water qualities of the six lakes are shown in Table 11. The maximum inflow of the Qinshangang channel is 30 m\({}^{3}\)/s, the maximum inflow of the Zengjiaxiang channel is 10 m\({}^{3}\)/s, the total diversion time is 30 days. The water quality of Yandong Lake has already reached the standard, so it is not considered in this paper.