The daily starting cost is shown as:
\(\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 }{{F_{si}N_{ij}}'}\)\(\)(5)
The fixed generation cost can be represented as:
\(\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 {{F_{1i}}} } {T_j({N_{ij}} + {N_{ij}}')}\)(6)
Meanwhile, the marginal cost can be derived as:
\(\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 {\sum\limits_{k = 1}^{{N_{ij}} + {N_{ij}}'} {{F_{2i}}} } } ({P_{ijk}} - {P_{i\min }}){T_j}\)(7)
Therefore, the system model can be formulated as:
\({min {W}}{\rm{=}}\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 }{{F_{si}N_{ij}}'}+\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 {{F_{1i}}} } {T_j({N_{ij}} + {N_{ij}}')}\)\(\)\(\)\(\)
\(+ \sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 {\sum\limits_{k = 1}^{{N_{ij}} + {N_{ij}}'} {{F_{2i}}} } } ({P_{ijk}} - {P_{i\min }}){T_j}\)
\(\)s.t. (8)
\({P_{i\min }} \le {P_{ijk}} \le {P_{i\max }}\)
\(0 \le{N_{ij}} + {N_{ij}}'\le{N_{i\max}}\)
\(\sum\limits_{i = 1}^4 {\sum\limits_{k = 1}^{{N_{ij}} + {N_{ij}}'} {{P_{ijk}}} } {\rm{ = }}{P_{dj}}\)
\(\)\(\sum\limits_{i = 1}^4{{P_{imax}} }({N_{ij}} + {N_{ij}}'){\ge}{1.2P_{dj}}\)
As it can be seen from (8), the derived model is complicated with
multiple decision variables. Meanwhile, since only the number of
generators is needed to be confirmed with minimum daily generation cost
and the detailed generation power of each generator is not needed, the
above model in (8) can be simplified. By setting the total output power
as the decision variables, it can be derived as follows. First, the
upper and lower limits of the generator output power can be obtained as:
\(\)\(\)\({P_{i\min }({N_{ij}} + {N_{ij}}'})\le{P_{ij}}\le{P_{i\max }}({{N_{ij}} + {N_{ij}}'})\)(9)
The total number of generators in each given time interval is also
limited as:
\(\)\(0\le {{N_{ij}}+{N_{ij}}'}\le {N_{i\max }}\)(10)
Meanwhile, the total generation power should satisfy the load power,
\(\sum\limits_{i = 1}^4 {{P_{ij}}} {\rm{ = }}{P_{dj}}\)(11)
The total generation power should be less than 120% of the total load
power, which yields that:
\(\)\(\sum\limits_{i = 1}^4 {{P_{i\max}}{({N_{ij}} + {N_{ij}}')}} {\ge}{1.2P_{dj}}\)(12)
The start cost in each daily cost is shown as:
\(\)\(\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 }{{F_{si}N_{ij}}'}\)(13)
The fixed cost is shown as:
\(\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 {{F_{1i}}} } {T_j({N_{ij}} + {N_{ij}}')}\)\(\)\(\)\(\)(14)
Meanwhile, the marginal cost is represented as:
\(\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 }{F_{2i}}{[{P_{ij}-{P_{i\min}}{({N_{ij}} + {N_{ij}}')}]}}{T_{j}}\)\(\)\(\)(15)
Therefore, the final model can be obtained:
\(\)\({min {W}}{\rm{=}}\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 }{{F_{si}N_{ij}}'}+\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 {{F_{1i}}} } {T_j({N_{ij}} + {N_{ij}}')}\)
\(\)\(\) \(+\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 {{F_{2i}}} } {T_j{{[{P_{ij}}-}{P_{i\min}}({N_{ij}} + {N_{ij}}')}]}\) \(\)
s.t. (16)
\(\)\({{P_{i\min}}}({N_{ij}} + {N_{ij}}')\le {P_{ij}}\le {{P_{i\max}}}({N_{ij}} + {N_{ij}}')\)
\(0 \le {N_{ij}} + {N_{ij}}'\le{N_{i\max}}\)
\(\)\(\sum\limits_{i = 1}^4 {{P_{ij}}} {\rm{ = }}{P_{dj}}\)
\(\)\(\)\(\sum\limits_{i = 1}^4 {{P_{i\max}}{({N_{ij}} + {N_{ij}}')}} {\ge}{1.2P_{dj}}\)
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:
\(\)\(\)\(\min W = \sum\limits_{j = 2}^7 {\sum\limits_{i = 1}^4 {[{F_{si}} \cdot \frac{{sign({N_{sij}} - {N_{si,j - 1}}) + 1}}{2}({N_{sij}} - {N_{si,j - 1}})]} } \)
\(\) \(+ \sum\limits_{i = 1}^4 {[{F_{si}} \cdot \frac{{sign({N_{si1}} - {N_{si7}}) + 1}}{2}({N_{si1}} - {N_{si7}})]} \)
\(\) \(\)\(+\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 {{F_{1i}}} } {T_j({N_{ij}} + {N_{ij}}')}\)
\(\)\(\) \(+\sum\limits_{j = 1}^7 {\sum\limits_{i = 1}^4 {{F_{2i}}} } {T_j{{[{P_{ij}}-}{P_{i\min}}({N_{ij}} + {N_{ij}}')}]}\)
s.t. (26)
\({P_{i\min }}{N_{sij}} \le {P_{ij}} \le {P_{i\max }}{N_{sij}}\)
\(0 \le {N_{sij}} \le {N_{i\max }}\)
\(\sum\limits_{i = 1}^4 {{P_{ij}}} {\rm{ = }}{P_{dj}}\)
\(\sum\limits_{i = 1}^4 {{P_{i\max }}} {N_{sij}} \ge 1.2{P_{dj}}\)
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.