## 3 Electric Vehicle Charging Loads Control Strategy and the Transmission Loss Optimization
### 0-1 Integer Programming for Regulating Electric Vehicle Charging Loads  
There are only two states for vehicles: charging or not charging. In this case, the binary states of vehicles can be represented as 1 for charging and 0 for not charging. Under this condition, the complexity of electric vehicle charging loads optimization can be reduced, if the 0-1 Integer Programming Model is developed to regulate charging loads. As demonstrated at the beginning of this section, to reduce system losses, the fluctuation of feeder loads should be reduced. By discretizing charging loads into \(m\) periods, this paper aims to minimize the variance of daily charging loads based on the 0-1 Integer Programming. Therefore, the mathematical expression of the proposed problem can be expressed as:  
\[f_{1}=min\frac{1}{m}\left\{\sum\limits_{j=1}^{m}\left(L(j)+\sum\limits_{i=1}^{ N}S_{ij}P_{c}-P_{av}\right)^{2}\right\} \tag{6}\]  
where  
\[P_{av}=\frac{\sum\limits_{j=1}^{m}\left(L(j)+\sum\limits_{i=1}^{N}S_{ij}P_{c }\right)}{m} \tag{7}\]  
In (6) and (7),\(f_{1}\) is the objective function; \(L(j)\) is the total electrical demand of the power system without considering the electric vehicle charging loads at time period \(j\), \(j=1\), \(2\), \(3\)\(\ldots\)\(m\); \(S_{ij}\) is the binary charging states of electric vehicle \(i\) at time period \(j\); and \(P_{av}\) is the average electrical demand of a  
Figure 4: An example of daily charging load curves of 2000 electric vehicles. power system. Given that the proposed model only shifts loads, \(P_{av}\) is a constant for a given power system. Therefore, the objective function can be simplified as:  
\[f_{s}=min\left\{\sum_{j=1}^{m}\left(L(j)+\sum_{i=1}^{N}S_{ij}P_{c}\right)^{2}\right\} \tag{8}\]  
The aforementioned 0-1 Integer Programming Model is a quadratic integer programming problem. For this kind of problem, with the increase of samples (electric vehicles connected to the power grid), the number of decision variables and the complexity of calculation are increased. To accelerate computation for a large number of electric vehicles, an equivalent linearization method to linearize the objective function is proposed in [22; 23], which significantly reduces computation complexity. By taking this same method, the total demand of power system at time period \(j\) can be expressed as:  
\[\left(L(j)+\sum_{i=1}^{N}S_{ij}P_{c}\right)^{2}=\sum_{n=1}^{S}\alpha_{n}(j) \delta_{n}(j) \tag{9}\]  
In Equation (9), \(S\) is the segment number; \(\alpha_{n}\) (\(j\)) is the slope of the load of the \(n\)th segment at time period \(j\) after linearization; and \(\delta_{n}\) (\(j\)) is the value of the load of the \(n\)th segment at time period \(j\) after linearization. Therefore, the linearized equivalent simplified objective function of the proposed model can be written as:  
\[f=min\sum_{j=1}^{m}\sum_{n=1}^{S}\alpha_{n}(j)\delta_{n}(j) \tag{10}\]  
To develop the aforementioned model, there are four assumptions that need to be highlighted:  
1. Electric vehicle charging power There are two processes of electric vehicle charging, which are the constant power charging state and the charging power linearly decrease state. The constant power charging state is the main process of electric vehicle charging, and this process takes a relatively long time and has a relatively high efficiency compared with the charging power linearly decrease state. In addition, with the development of electric vehicle charging technologies, the charging power linearly decrease state tends to disappear [24]. Therefore, in this paper, it is assumed that electric vehicles are charged with constant power. 2. Electric vehicle charging time For most domestic users, it is more preferable to charge electric vehicles shortly after the last trip of the day, although a few of them may charge their vehicles when they are at the office. In this paper, it is assumed that domestic users' start charging their vehicles shortly after finishing their last trip and must stop charging vehicles before going to work.