In the model shown in (16), since no detailed output power of each generator is included, the complexity of the model is reduced. In order to further simplify the model, it is assumed that when new generator is started, there is no generators being shut down in the same time interval. In other words, it is not the case to start some generators and shut them down in the same time interval. Based on this assumption, the mathematical description with the number of generators can be simplified in further using a single variableN sij, which represents the number of Type #i generators running in the jth time interval.
It should be noted that the newly started generators in the jth time interval is shown as:
\(\frac{{sign({N_{sij}} - {N_{si,j - 1}}) + 1}}{2}({N_{sij}} - {N_{si,j - 1}})\)(17)
Here, sign(∙) is a sign function. When x is larger than zero, sign(∙) equals 1; when x is smaller than zero, sign(∙) equals -1.
When j = 1, (17) can be rewritten as:
\(\frac{{sign({N_{si1}} - {N_{si7}}) + 1}}{2}({N_{si1}} - {N_{si7}})\)(18)
Similar to the second model derived above, the upper and lower constraints of output power of a generator is defined as:
\({P_{i\min }}{N_{sij}} \le {P_{ij}} \le {P_{i\max }}{N_{sij}}\)(19)
The number of generators running in the given time interval is limited as:
\(0 \le {N_{sij}} \le {N_{i\max }}\)(20)
In order to satisfy the load requirements, it yields that:
\(\sum\limits_{i = 1}^4 {{P_{ij}}} {\rm{ = }}{P_{dj}}\)(21)
Given that the maximum output power should be no less than 120% of the power rating:
\(\sum\limits_{i = 1}^4 {{P_{i\max }}} {N_{sij}} \ge 1.2{P_{dj}}\)(22)
Therefore, the starting cost in the daily generation cost can be shown as:
\(\begin{array}{c}\sum\limits_{j = 2}^7 {\sum\limits_{i = 1}^4 {[{F_{si}} \cdot \frac{{sign({N_{sij}} - {N_{si,j - 1}}) + 1}}{2}({N_{sij}} - {N_{si,j - 1}})]} } \\ + \sum\limits_{i = 1}^4 {[{F_{si}} \cdot \frac{{sign({N_{si1}} - {N_{si7}}) + 1}}{2}({N_{si1}} - {N_{si7}})]} \end{array}\)(23)
The fixed and marginal costs are shown as:
\(\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 {{F_{1i}}} } {T_j}{N_{sij}}\)(24)
\(\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 {{F_{2i}}({P_{ij}} - {P_{i\min }}{N_{sij}}){T_j}} }\)(25)
Hence, the final derived model can be shown as:
\(\)(26)
Therefore, the derived model in (26) is linear and much simplified, which can be used to enhance the computational efficiency.
4.Numerical Experiment
Based on the data shown in Table 1 and Table 2, the derived optimization problem is solved in order to validate the accuracy of the obtained model. The three optimization problems all give the same optimal. By using MATLAB m-scripts, the results can be obtained and summarized in Table 4 and Table 5. Meanwhile, the time stamps during numerical tests can be extracted, as shown in Table 6, where is can be seen that the simplified model features higher computational efficiency.
Table 4. Output power of each type of generators in each time interval.