3.Model Implementation

The constraints in the above problem formulation are comprised of the following aspects: 1) Maximum and minimum constraints of generator output power; 2) Number of particular type of generators in each given time interval; 3) Load power is satisfied in each given time interval; 4) The maximum output power is no less than 120% of the power rating. Meanwhile, the total generation cost is comprised of: 1) Total starting cost of all the generators; 2) Total fixed cost of all the generators; 3) Total marginal cost of all the generators.

The constraints and objective in this optimization formulation are detailed as follows:

Define that and.

The upper and lower limits of output power of each generator is shown as:

(1)

The total number of Type #i generator is limited by:

(2)

In the given time interval, the total output power of all the generators should meet the load requirements, which is shown as:

(3)

The maximum output power should not be lower than 120% of the load requirements:

(4)

The daily starting cost is shown as:

(5)

The fixed generation cost can be represented as:

(6)

Meanwhile, the marginal cost can be derived as:

(7)

Therefore, the system model can be formulated as:

(8)

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:

(9)

The total number of generators in each given time interval is also limited as:

(10)

Meanwhile, the total generation power should satisfy the load power,

(11)

The total generation power should be less than 120% of the total load power, which yields that:

(12)

The start cost in each daily cost is shown as:

(13)

The fixed cost is shown as:

(14)

Meanwhile, the marginal cost is represented as:

(15)

Therefore, the final model can be obtained:

(16)

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:

(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:

(18)

Similar to the second model derived above, the upper and lower constraints of output power of a generator is defined as:

(19)

The number of generators running in the given time interval is limited as:

(20)

In order to satisfy the load requirements, it yields that:

(21)

Given that the maximum output power should be no less than 120% of the power rating:

(22)

Therefore, the starting cost in the daily generation cost can be shown as:

(23)

The fixed and marginal costs are shown as:

(24)

(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.