## 3 Short-Term Hydrothermal Scheduling Problem  
Short-term hydrothermal scheduling is a type of economic dispatch problem in which a parallel operation of hydel power generating units is combined with thermal power generating units on a generation bus. This problem can have several forms varying from single objective to multi-objective, cascaded to non-cascaded, and pumped storage to non-pumped storage problems. The generation model of the STHTS problem is presented in Figure 2, as taken from [24]. The complexities of these problems can be further enhanced by considering the valve point loading effects of thermal generating units. The detailed analysis of all these problem types was presented in [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25]. This paper considers two types of these problems as case studies, as already considered in [19; 20; 21; 22; 23; 24; 25], to implement the improved APSO algorithm and its comparison with the previous implementations of APSO algorithm in [24; 25], in a formal quantitative hypothesis testing. The two problem cases belong to the following two types of STHTS problem.  
1. Non-cascaded pumped storage short-term hydrothermal scheduling [24].
2. Non-cascaded non-pumped storage short-term hydrothermal scheduling while considering transmission losses [25].  
It seems more logical to check the performance of one type of algorithm by implementing it on as many types of problems as possible. However, according to the \"no free lunch theorem\", as given in [29; 30; 31; 32; 33; 34], if one algorithm performs best on a Type A problem, it does not necessarily mean that it performs the best for a Type B problem, i.e., an algorithm cannot be proven to be the best optimization algorithm for all types of problems. Therefore, it is required to find a good performing algorithm for every type of optimization problem. Problems 1 and 2 belong to the same class of single objective and non-cascaded STHTS problems and are different in terms of structure from the remaining types of STHTS problems. Any algorithm, providing the best optimum to these problems, will perform well on all similar types of STHTS problems.  
APSO algorithms are the simplest and most elegant metaheuristic algorithms, which, without having a lot of tuning parameters and having just a single-step particle update equation, are able to solve non-linear and multi-modal objective functions like Michaelwicz 2D functions, as given in [1; 2]. Therefore, the elegance and robustness of these APSO algorithms make them the first choice out of many metaheuristic algorithms to implement on optimization problems. The other types of STHTS problems, for example the cascaded STHTS of a single objective-type problem, requires metaheuristic algorithms with more tuning parameters instead of the APSO algorithm, like PSO and improved PSO. Another type is the multi-objective STHTS problem, which requires variants of metaheuristic  
Figure 2: Generation model of the Short-Term Hydrothermal Scheduling (STHTS) problem [25].  
algorithms specially designed for multi-objective problems, like multi-objective PSO, ant colony and genetic algorithms, cuckoo search, the gravitational search algorithm, teaching-learning-based algorithms, and water cycle algorithms [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19].  
The problems taken into consideration in this paper have been solved using the APSO algorithm by the authors of this paper already in [23; 24], which outperformed the other algorithms implemented on the same problems given in [22; 26; 27]. In this enhanced research, the improved APSO algorithm has been implemented on the same problems that used dynamic search space squeezing, and this modified APSO has shown even better results than the APSO algorithm, as presented in the Results Section. Figure 2 presents the basic generation model of the STHTS problem, which was also considered in [23; 24].