# Performance Analysis of Partitioned Step Particle Swarm Optimization in Function Evaluation
## 2 Partitioned Step Particle Swarm Optimization
### Performance Tests
#### 2.3.3 Unit Commitment Problem  
The PSPSO is, likewise, applied to solve the IEEE 10-unit unit commitment (UC) problem, which is a non-convex, large-scale, non-linear, and mixed-integer combinatorial problem. There are several attempts to solve the IEEE10-unit UC problem in literature using NICPSO [18], hybrid genetic-imperialist competitive algorithm (HGICA) [25], GA [26], quantum-inspired binary grey wolf optimizer (QBGWO) [27], binary artificial sheep algorithm (BASA) [28], binary grey wolf optimizer (BGWO) [29], and MILP-based stochastic multi-objective UC (SMOUC) using one scenario and single objective base case [30].  
The UC problem deals with the 24-h scheduling of generators, deciding on when a generator turns on or off (unit commitment decision) and what amount of power should be scheduled for each generator (economic dispatch decision) in order to have the least operating cost. The generator constants and load are listed in Tables 4 and 5. Table 6 lists the objective function and constraints of the UC problem. The objective, as found in Table 6-a, is to minimize sum of the fuel cost and start-up (SU) cost of the generators for the entire period. It is subject to seven system constraints. The total fuel cost at each hour \(t\) is equal the summation of the fuel cost of each generator \(i\) given their fuel constants (a, b, and c) (Table 6-b). \(UC_{i}\) is the commitment of generator \(i\); a \(UC\) value of 1 means that the generator is turned on, and 0 means the generator is turned off. The power balance of the system must be maintained at each hour such that the summation of the power generated by each generator \(P_{DG}\) must be equal to the load demand \(P_{L}\) at a particular hour (Table 6-c). The generator power output should not exceed the maximum (\(UB\)) and minimum (\(LB\)) operating limits of each generator (Table 6-d). The ramping up and down of power should not exceed the generator's ramping limits \(Rmp_{DG}\) (Table 6-e). The generators have a minimum up time (MUT) and minimum down time (MDT) where the generator cannot turn off and turn on, respectively (Table 6-f). The shutdown costs of each generator depend on the cold start hour (CSH) of each generator (Table 6-g). Finally, the scheduled reserve (10% of the load) at each hour (Table 6-h) should be available.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Gen**} & **MAX** & **MIN** & **a** & **b** & **c** & **Hot Start** & **Cold Start** & **Cold Start** & **MUT** & **MDT** & **t\({}_{\text{li}}\)** & **Ramp Up/** \\  & **OP** & **OP** & **(5)** & **(S/MW)** & **(S/MW)** & **(S/MW2)** & **(HSC)** & **(CSC)** & **(CSH)** & **(H)** & **(h)** & **(h)** & **Down** \\ \hline
1 & 455 & 150 & 1000 & 16.19 & 0.00048 & 4500 & 9000 & 5 & 8 & 8 & 8 & 160 \\
2 & 455 & 150 & 970 & 17.26 & 0.00031 & 5000 & 10,000 & 5 & 8 & 8 & 8 & 160 \\
3 & 130 & 20 & 700 & 16.60 & 0.00200 & 550 & 1100 & 5 & 5 & 4 & \(-\)5 & 100 \\
4 & 130 & 20 & 680 & 16.50 & 0.00211 & 560 & 1120 & 5 & 5 & 4 & \(-\)5 & 100 \\
5 & 162 & 25 & 450 & 19.70 & 0.00398 & 900 & 1800 & 6 & 6 & 4 & \(-\)6 & 100 \\
6 & 80 & 20 & 370 & 22.25 & 0.00712 & 170 & 340 & 2 & 3 & 3 & \(-\)3 & 60 \\
7 & 85 & 25 & 480 & 27.74 & 0.00079 & 260 & 520 & 2 & 3 & 3 & \(-\)3 & 60 \\
8 & 55 & 10 & 660 & 25.92 & 0.0413 & 30 & 60 & 0 & 1 & 1 & \(-\)1 & 40 \\
9 & 55 & 10 & 665 & 27.27 & 0.00222 & 30 & 60 & 0 & 1 & 1 & \(-\)1 & 40 \\
10 & 55 & 10 & 670 & 27.79 & 0.00173 & 30 & 60 & 0 & 1 & 1 & \(-\)1 & 40 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Generator data.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**No.**} & **Function Name** & **Dimension** \\ \hline c1 & [PERSON] [31] & 2 \\ c2 & Bird Problem [32] & 2 \\ c3 & Townsend [33] & 2 \\ c4 & G01 [34] & 13 \\ c5 & G02 [34] & 20 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Benchmark functions (Constrained).