Operation cost reduction in unit commitment problem using improved quantum binary PSO algorithm

Received May 31, 2019 Revised Oct 10, 2019 Accepted Oct 18, 2019 Unit Commitment (UC) is a nonlinear mixed integer-programming problem. UC used to minimize the operational cost of the generation units in a power system by scheduling some of generators in ON state and the other generators in OFF state according to the total power outputs of generation units, load demand and the constraints of power system. This paper proposes an Improved Quantum Binary Particle Swarm Optimization (IQBPSO) algorithm. The tests have been made on a 10-units simulation system and the results show the improvement in an operation cost reduction after using the proposed algorithm compared with the ordinary Quantum Binary Particle Swarm Optimization (QBPSO) algorithm


PROBLEM FORMULATION
The objective function of UC problem was formulated to minimize the total operation cost over the time horizon [1]. This minmization may be done by selecting the combination of the generation units that satisfies all the constraints of the power system and the generation unit itself and these combinations 0-1 that represented the status of each generator ON/OFF. The formulation of cost function for the UC problem that must be minimized by the sum of the starting cost of the generators and operational cost for each unit over a specefied period of time and it can be expressed by the following equation: (1) where f gk is the fuel cost function, U gk is the state of unit g which can be 0 or 1 at the hour k, N is the number of generation units,T is the time horizon, g is the index of the unit, k is the index of time, P gk is the power delivered from the unit g at the hour k and STC gk is the start-up cost of the unit g at the hour k.
The fuel cost function is calculated as follows f gk (P gk ) = c g (P gk )² + b g (P gk ) + a g (2) where c g, b g , a g are the fuel cost coeficients and the start-up are represented by the following equation: where (HSC g , CSC g ) are the hot and cold start-up cost of the unit g; CSH g is the cold start hours for the unit g; (MUT g , MDT g ) are the minimum up and downtime of the unit g and (T g on , T g off ) are the time of the unit g is continuously ON or OFF The objective function of UC problem is resterercted by some constraints and these constraints are the system constraints and the generation unit constraints [1]. 1. The demand must be supplied by the generators at each hour.
2. The constraint of spinning reserve in case of increase the demand or loss generator unit from the group.
3. The generator can produce power in the range between the maximum and minimum values.
4. The generation unit must be operated at least for a time equals to the minimum up time.
T g on ≥ MUT g 5. The generation unit must be shut-down or in the OFF state at least for a time equals to the minimum downtime.
where D k is the load demand of the system at the hour k; R k is the spinning reserve of the system at the hour k and (P g max , P g min ) are the maximum and minimum power that can be supplied from the unit g.
ISSN: 2088-8708  Operation cost reduction in unit commitment problem using improved quantum … (Ali Nasser Hussain) 1151

PARTICLE SWARM OPTIMIZATION AGORITHM
Particle Swarm Optimization (PSO) is introduced firstly by James Kennedy and Russell Eberhart in 1995 [21]. PSO algorithm is a heuristic optimization method based on the parallel experience of the individuals to search for the optimum solution. The PSO particles spread in a search space D of the problem and each of them has a position vector and speed vector [21]. In this algorithm, the particles are guided using the personal experience for each particle which is known as and the overall or the global experience among all particles that termed as . Then, the velocity and location of each particle in the population are modified by using the calculation of the current particle velocity and the distance from location and location [22]. Furthermore, the experience can be accelerated by two set of the acceleration factors, ( 1 , 2 ) are the cognitive and asocial acceleration constant factors respectively; ( 1 , 2 ) are two random numbers generated between [0, 1]. The movement is also can be controlled by multiplying it by inertia factor that lies in the range of [⍵ , ⍵ ] and the typical range is ⍵ = 0.9 to ⍵ = 0.4. The velocity update is described by the following equation: where m is the current itration. The position of the particles can be updated as follows equation: the inertia factor is represented by the following equation: where (⍵ , ⍵ ) are the initial and final weights, is the maximum iteration number and is the current iteration number. The binary version of the PSO (BPSO) has been presented by James Kennedy and Russell Eberhart to be used in discrete spaces [23]. The update proces of the position for the particles can be achieved by using a new variable known as Sigmoid Limiting Transformation and can be written as By using the sigmoid function, the position update of the particle in the binary version of a PSO algorithm is done as the following equation where is a random number distributed uniformly between [0, 1].

HYBRIDIZATION OF QUANTUM COMPUTING WITH BPSO ALGORITHM
The quantum bit is known as the smallest unit of information that store in the quantum computer [24]. The quantum bit can be in two states, the first state is 0 and the second is 1 .These states may be written as |0⟩ and |1⟩ and the quantum bit state can be reproduced as follows: where and are two complex numbers that identifies the probability amplitude of the relative conditions. The state of the quantum bit can be normalized to unity to guarantee that | | 2 + | | 2 = 1. Quantum gates have been used to change the state of the quantum bit and for examples of these gates, NOT gate, Hadamard gate and rotation gate [25]. The novel QEA has been proposed by Kim and Han as [24]. This QEA is inspired from the quantum-computing concept so the quantum bit has been designed to get the binary solutions. The quantum bit is defined by pair of numbers which are α and β and the quantum bit can be formulated as a string of where | | 2 + | | 2 = 1 and j = 1, 2 ……n .
The rotation gate can be used as a variance factor to update the individual of the quantum bit and the rotation gate is represented by the following equation: where is the jth quantum bit rotation angle that goes to 0 or 1state. A lookup table as shown in Table 1 is used to determine the value of and adjusted as 1 = 0, 2 = 0, 3 = 0.01 , 4 = 0, 5 = − 0.01 ,  [26]. Each element in the particle has a state of 1 or 0 according to the probability of | | 2 + | | 2 = 1. The QBPSO proposes a new way to update the velocity of each particle by the use of Quantum Computing. The inertia factors (⍵ , ⍵ ) and the acceleration factors ( 1 , 2 ) are omitted in the QBPSO and replaced by the rotation angle. The update process of the position vector can be done by using the probability | | 2 that has been stored in the ith quantum bit individual ( ). Therefore, the jth element of the ith particle takes a value of 1 or 0 as in the following equation [26]: The rotation angle can be determined by using the current position and the global position of the swarm as in the following equation: where is the rotation angle magnitude and ( 1 , 2 ) can be found by a comparison among the fitness of the current position of the particle i, the fitness of the best position and the fitness of the global position respectively as in equations (19) and (20): The magnitude of the rotation is decreased monotonously from a maximum value to a minimum value along the iteration by the following equation:

IMPROVED QBPSO ALGORITHM
The QBPSO algorithm may fail in finding the optimum value of the solution, therefor; an improvement is made on the QBPSO to get the better solution. The improvement on QBPSO is to search for the fitness in the personal best for the first half of the iterations and after finding it the fitness in the global best will be searched in the second half of the iterations. This improvement can be expressed as the following equations: and the rotation angle is updated as in (18, 21)

CONCLUSION
The unit commitment problem is a hard optimization process that the power system has to be deal with it to get a minimum operation cost. This paper proposes an improved quantum binary particle swarm optimization (IQBPSO) algorithm to solve the unit commitment problem. The algorithm has been tested on 10 unit simulation system during 24-hour time horizon and a minimized operational cost has been achieved.  ISSN: 2088-8708