## 1 Introduction
Optimization algorithms based on stochastic searching methods have been widely used for the unit commitment problem, including exhaustive attack methods, Lagrangian relaxation methods, conventional quadratic programming (QP), branch and bound (B&B), artificial neural networks (ANN), simulated annealing (SA), and improved particle swarm optimization (IPSO) [10; 11; 12; 13; 14; 15; 16; 17]. The exhaustive attack method is a brute-force algorithm, which is used to obtain all the possible solutions for comparison and then further determine the optimal solution [10]. However, if all possible combinations are used for calculation, the curse of dimensionality may occur, preventing the search from discovering the optimal solution and compromising solution quality and calculation time. According to Lagrangian relaxation method, the constraint for the minimum problem is multiplied by the Lagrange multipliers, and the product is incorporated into the objective function to convert the problem into a dual maximization problem that can be easily solved. Although the solution can be easily obtained, the problem-solving requires a simplified cost curve and thus may produce inaccurate solutions. Inappropriate selection of the initial value may also result in oscillation or even divergence [11]. Conventional quadratic programming can convert a complex nonlinear restriction programming problem into an easily solved sequence quadratic programming subproblem, but is limited to problems written in terms of linear equations and inequality constraints. In addition, the objective function can only be a quadratic function with a constant coefficient [12]. Branch and bound involves the enumeration of candidate solutions through branching and delimiting. The original problem is divided into numerous small problems, which are compared using the upper and lower limits to exclude impossible problems, thus reducing the range of solutions, to accelerate the problem-solving process and minimize the use of computer memory. However, the real optimal solution combination may be discarded during the process of eliminating branches, resulting in the determination of only a regional solution [13]. Artificial neural networks have the abilities of parallel processing, learning, and memorizing. Problems can be solved using different network structures combined with learning algorithms. However, this method requires a large number of samples for training and a long-term training process, and the quality of solutions is affected by network parameters [14]. SA is a randomized optimization algorithm simulating the process of crystal annealing. The solution space of a problem is represented as the crystalline state of particles. However, because it is an absolutely stochastic method, its convergence speed is slow [15]. IPSO is an improved PSO algorithm, based on the concept of PSO original proposed by [PERSON] and [PERSON] in 1995, which focuses on the adjustment of inertia weight factor \(\omega\) (velocity update). This method regards a large number of candidate solutions as a swarm and uses random numbers to initialize a group of random particles; an objective function is then used to evaluate the system [16; 17]. This study investigated the optimization of flexible dispatch of existing units according to the ramp rate during a short time after wind power and solar energy had been integrated into a system. The actual electrical system in Taiwan and the generator parameters were used as the analysis target. The optimal UC problem was solved using the proposed hybrid algorithm (HA), which combined the advantages of both a generalized Lagrangian relaxation algorithm and a random feasible directions algorithm. In addition, the advanced priority list (APL) method was employed to solve the large-scale nonlinear mixed-integer programming UC problem in the power system to avoid local solutions during the searching process and increase the problem-solving accuracy.