Solving unit commitment problem using a novel version of harmony search algorithm

Abstract

In this context, a novel structure was proposed for improving harmony search (HS) algorithm to solve the unit comment (UC) problem. The HS algorithm obtained optimal solution for defined objective function by improvising, updating and checking operators. In the proposed improved self-adaptive HS (SGHS) algorithm, two important control parameters were adjusted to reach better solution from the simple HS algorithm. The objective function of this study consisted of operation, start-up and shut-down costs. To confirm the effectiveness, the SGHS algorithm was tested on systems with 10, 20, 40 and 60 generating units, and the obtained results were compared with those of the simple HS algorithm and other related works.

Keywords

generation scheduling / harmony search (HS) algorithm / intelligent technique / unit commitment

Introduction

Because of the ever increasing and changing demands in the power system, power generating units should appropriately meet the load demands and changes so as to avoid power outages. The main goal in the unit commitment (UC) problem is to determine the start-up and shut-down schedules of generating units such that minimum cost is to be imposed on the system. The UC problem is formulated as a large-scale non-convex problem under some constrains.

Many researchers suggested different techniques and algorithms to solve the UC problem. In this section, a review is conducted on related past works focusing on artificial intelligence methods because the proposed approach in the present paper is based on these methods. For this, in this study, the genetic algorithm (GA), evolutionary techniques and Tabu search (TS) were analyzed and discussed.

GAs were proposed to solve the UC problem [1–4]. The ON/OFF states of the power generating units were determined and the costs were minimized by considering short-term scheduling where the time horizon was 24 h [1]. A novel varying fitness function technique for GA was suggested to solve the cutting stock and the UC problems [2]. The minimum up and down time constraints, start-up cost and spinning reserve were considered in proposed objective function [3]. The integer coded genetic algorithm (ICGA) was utilized to solve the UC problem.

Evolutionary programming (EP) and evolutionary algorithm (EA) are evolutionary techniques proposed to solve the UC problem [5–8]. The used EP was employed to encode the operating schedule of each unit with regard to its minimum up/down time restrictions. Attempt was made to find the generation scheduling such that the total operating cost could be minimized [5]. The quantum-inspired EA (QEA) was proposed and employed to the UC based on the concept and principles of quantum computing [7]. An improved method was proposed based on quantum-inspired evolutionary algorithm for UC (IQEA-UC) using new initialization method based on unit priority list [8]. The TS was suggested for solving the UC problem [9], and other techniques were proposed based on TS [10–12].

This paper suggested a self-adaptive global best harmony search (SGHS) algorithm to solve the UC problem. For this, the most comprehensive objective function was defined, in which, the operation, start-up and shut-down costs were considered. Besides, each cost term was formulated with proper parameters. Moreover, the structure and operation of simple and improved harmony search (HS) algorithms were introduced, followed by a step by step solution of UC using the SGHS algorithm. Finally, the numerical test results were offered, the ability of the SGHS algorithm was confirmed, and the results were compared with the simple HS algorithm and other recent techniques.

Unit commitment problem

The main goal in the UC process is to minimize the total cost. The objective function (OF) of the UC problem consists of operation cost (OC), start-up cost (SuC), and shut-down cost (SdC), expressed as

(1)\mathrm{O}\mathrm{F}=\min {\displaystyle \sum _{i=\mathrm{1}}^{N_{\mathrm{G}}}{\displaystyle \sum _{t=\mathrm{1}}^{N_{\mathrm{T}}}\left\{\mathrm{O}\mathrm{C}+\mathrm{S}\mathrm{u}\mathrm{C}+\mathrm{S}\mathrm{d}\mathrm{C}\right\}}},

where N_G and N_T are the number of generating units and time periods under study (24 h), respectively.

Operation cost

OC is the most important part of the UC problem. This term is the function of production cost (PC), maintenance cost (MC), and emission cost (EC). The OC is zero if the unit is not involved in production.

(2)\mathrm{O}\mathrm{C}=\{\begin{array}{ll}\mathrm{P}\mathrm{C}+\mathrm{M}\mathrm{C}+\mathrm{E}\mathrm{C}\hfill & \mathrm{if}\mathrm{Production}=1,\hfill \\ \mathrm{0}\hfill & \mathrm{if}\mathrm{Production}=0.\hfill \end{array}

where PC_ci is defined as a quadratic function which is illustrated using a curve.

(3)\mathrm{P}{\mathrm{C}}_{\mathrm{c}i}={\alpha }_i+{\beta }_i{P}_{it}+{\mathit{\gamma }}_i{P}_{it}^{\mathrm{2}},

where α_i, β_i and γ_i are cost factors, and P_it is the output of unit i at instant t.

MC has two terms. The first term is constant maintenance cost of unit i at instant t (CM_it), and the second term is incremental maintenance cost (IM_it) multiplied by P_it.

(4)\mathrm{M}{\mathrm{C}}_{\mathrm{c}i}=\mathrm{C}{\mathrm{M}}_{it}+\mathrm{I}{\mathrm{M}}_{it}\times {\mathrm{P}}_{it}.

The behavior of EC is expressed, similar to PC, as

(5)\mathrm{E}{\mathrm{C}}_{\mathrm{c}i}={\omega }_i+{\mathit{\lambda }}_i{P}_{it}+{\mu }_i{P}_{it}^{\mathrm{2}},

where ω_i, λ_i and µ_i are emission factors.

Startup cost (SuC)

To start up thermal units, some fuel should be consumed which do not produce any effective power. The cost resulted from this energy is called SuC. In this section, a step by step technique is used by transition hour h_i^{\mathrm{off}} (transition from OFF to ON) from hot to cold start-up to describe time dependent SuC.

(6)\mathrm{S}\mathrm{u}{\mathrm{C}}_{it}=\{\begin{array}{ll}H-\mathrm{c}\mathrm{o}\mathrm{s}{t}_{it}\hfill & T_i^{\mathrm{o}\mathrm{f}\mathrm{f}}\le Z_{it}^{\mathrm{o}\mathrm{f}\mathrm{f}}\le h_i^{\mathrm{o}\mathrm{f}\mathrm{f}},\hfill \\ C-\mathrm{c}\mathrm{o}\mathrm{s}{t}_{it}\hfill & Z_{it}^{\mathrm{o}\mathrm{f}\mathrm{f}}>h_i^{\mathrm{o}\mathrm{f}\mathrm{f}},\hfill \end{array}

where the first and second scenarios are hot and cold startups of unit i at time t, respectively. T_i^{\mathrm{off}}is the minimum stop time of unit i and Z_i^{\mathrm{off}}is the time duration at the stop of unit i at time t. The cost at the transition hour and cold start-up is expressed as

(7)h_i^{\mathrm{off}}=T_i^{\mathrm{off}}+\left(c-\cos t-{\mathrm{hour}}_i\right),

where the second term is the duration of the cold start-up of unit i.

(8)$$\begin{gathered} c-\cos {\mathrm{t}}_{it}=I_{it}\left(\mathrm{1}-I_{i\left(t-\mathrm{1}\right)}\right) \\ \left[{\delta }_i+{\mathrm{\Omega }}_i\left(\mathrm{1}-\exp \left(-{\displaystyle \frac{X_{it}^{\mathrm{off}}}{t_i}}\right)\right)+{\theta }_i\right], \end{gathered}$$

where δ_i and Ω_i are constant values (consist of costs of crew and maintenance in start-up) and boiler start-up costs, respectively; θ_i and t_i are the turbine start-up cost and thermal time constant of unit i; I_it determines the partnership status of unit i at time t. If the unit state is ON, I_it = 1; if not I_it = 0.

Finally, the hot startup cost is obtained by

(9)H-\cos t_{it}={\delta }_i+\left(C_{ti}\times f_{\mathrm{c}i}\right)Z_{it}^{\mathrm{off}},

where C_{hi}=C_{ti}{f}_{\mathrm{c}i}, in which C_ti and f_ci are hot startup cost at each hour and fuel cost, respectively.

Shut-down cost (SdC)

The SdC for each unit is a constant value which, in standard systems, is equal to zero. This cost is calculated using

(10)\mathrm{S}\mathrm{d}{\mathrm{C}}_{it}=K_i{P}_{it}\left(\mathrm{1}-I_{it}\right)I_{i\left(t-\mathrm{1}\right)},

where K_i is the shut-down incremental cost.

Constraints

The above said objective function is subjected to the following constraints.

Load demand constraints: In the UC problem, power generation is, perhaps, less than the demand power (PD_i) and reserve, which allows more flexibility in UC schedule.

(11){\displaystyle \sum _{i=\mathrm{1}}^N{P}_{it}{U}_{it}\le {\mathrm{PD}}_i},\mathrm{1}\le t\le T.

Spinning reserve (SR_i) constraints: SR_i is the term used to describe the total amount of generation available from all units synchronized (i.e., spinning) on the system, minus the present load and losses being supplied.

(12){\displaystyle \sum _{i=\mathrm{1}}^N{R}_{it}{U}_{it}\le {\mathrm{SR}}_i},\mathrm{1}\le t\le T.

Power balance constraints:

(13)P_i^{\min }\le P_i\le P_i^{\max },i=\mathrm{1},\mathrm{2},\mathrm{...},N.

Minimum up (MU)/down (MD) time constraints: A thermal unit can undergo only gradual temperature changes. The MU time means that once the unit is running, it should not be turned off immediately. The MD time means that once the unit is decommitted, there must be a minimum time before it can be recommitted.

(14)T_i^{\mathrm{on}}\ge {\mathrm{MU}}_i,i=\mathrm{1},\mathrm{2},\mathrm{...},N\mathrm{;}

(15)T_i^{\mathrm{off}}\ge {\mathrm{MD}}_i,i=\mathrm{1},\mathrm{2},\mathrm{...},N.

Ramp rate limits: The ramp rate limits confine the power output movement of a generating unit between adjacent hours, which are given below.

(16)\begin{array}{l}{P}_i^{\max }\left(t\right)=\min \left\{P_i^{\max },P_i\left(t-\mathrm{1}\right)+\tau {\mathrm{RU}}_i\right\},\hfill \\ P_i^{\min }\left(t\right)=\max \left\{p_i^{\max },P_i\left(t-\mathrm{1}\right)+\tau {\mathrm{RD}}_i\right\},\hfill \end{array}

where τ = 60 min is the UC time step.

Crew constraints: The crew constraints are restricted to two or more units to be turned ON at the same time [13].

HS algorithm

The HS algorithm is proposed based on musical improvisation process [14].

HS algorithm definitions

Before starting the HS algorithm, a number of parameters should be defined:

IN: the number of iterations. IN is determined by complexity and nature of problem.

DVN: the number of decision variables. The set of these DVNs constitute a harmony.

SHS: the size of harmony search. SHS is used to specify the number of harmonies that will be stored in the harmony memory.

PAR: pitch adjustment rate. PAR determines the probability value to get decision values. By adding a determined value, these values are copied from a harmony in memory.

HMCR: harmony memory considering rate. HMCR determines the rate at which decision variables in the harmony are considered as elements of a new harmony that will be created. This parameter is determined between [0,1].

Simple HS structure

In classic structure, the HS algorithm has five steps:

Step 1 Initializing the problem and parameters: The main goal to use this algorithm is to minimize the objective function, f (U), which is a vector with the size of decision variables, N, thenwhere U_i is a possible vector of values for each decision variable with the size of PAR.

Step 2 Initializing harmony memory: HS is a population based algorithm. This algorithm is started using an initial matrix. Dimensions of this matrix are determined by the number of decision variables and harmony memory size. Thus, harmony memory (HM) is obtained by

Step 3 Improvising a new harmony: A new harmony is generated according to memory consideration with a probability of HMCR. Then, the new harmony is produced as {\mathit{U}}^{\prime }=\left[u_{\mathrm{1}}^{\prime },u_{\mathrm{2}}^{\prime },\mathrm{...},u_N^{\prime }\right]. The PAR adjusts the parameter from memory consideration. Three important rules i.e. memory consideration, pitch adjustment, and random selection are considered to generate the new harmony vector.

To generate the new vector, the value of the first decision variable (U_{\mathrm{1}}^{\mathrm{N}\mathrm{E}\mathrm{W}}) and other decision variables (U_{\mathrm{2}}^{\mathrm{N}\mathrm{E}\mathrm{W}}, U_{\mathrm{3}}^{\mathrm{N}\mathrm{E}\mathrm{W}}, …, U_N^{\mathrm{N}\mathrm{E}\mathrm{W}}) are selected from the values in the specified HM range based on HMCR,

(19)U_i^{\mathrm{NEW}}=\{\begin{array}{lll}{U}_i^{\mathrm{NEW}}\in \left\{U_i^{\mathrm{1}},U_i^{\mathrm{2}},\mathrm{...},U_i^{\mathrm{HMS}}\right\}\hfill & \mathrm{if}\hfill & \mathrm{HMCR,}\hfill \\ U_i^{\mathrm{NEW}}\in U_i\hfill & \mathrm{if}\hfill & \mathrm{1}-\mathrm{HMCR}.\hfill \end{array}

Any obtained component by the memory consideration is examined to determine whether it should be pitch-adjusted or not. This operation uses the PAR parameter, which is the rate of pitch adjustment as Eq. (20):

(20)\mathrm{Pitch}\mathrm{adjusting}\mathrm{decision}\mathrm{for}{U}_i^{\mathrm{NEW}}\leftarrow \{\begin{array}{lll}\mathrm{Yes}\hfill & \mathrm{if}\hfill & \mathrm{PAR,}\hfill \\ \mathrm{No}\hfill & \mathrm{if}\hfill & \mathrm{1}-\mathrm{PAR}.\hfill \end{array}

The value of (1-PAR) sets the rate of doing nothing. If the pitch adjustment decision for U_i^NEW is YES, U_i^NEW is replaced as Eq. (21):

(21){\mathit{U}}_{\mathit{i}}^{\mathrm{N}\mathrm{E}\mathrm{W}}\leftarrow U_i^{\mathrm{N}\mathrm{E}\mathrm{W}}\pm \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\left(0,1\right)\times \mathrm{B}\mathrm{W},

where BW is an arbitrary distance bandwidth.

Step 4 Updating HM: In this step, the updating process is done based on the solution of the objective function. In other words, if the related solution of the new obtained harmony vector is better than the worst harmony vector in the HM, this vector is selected.

Step 5 Termination criteria: To terminate the algorithm, there are two techniques: reaching the optimal solution and finishing iteration number. In the most optimization problem, the second criterion is used.

Improved HS algorithm

Review of improved HS algorithms

To reach a better solution than the simple HS algorithm, several improvement techniques were suggested, which could mainly be categorized into two groups. The goal of the first group was to adapt the control parameters of the HS algorithm, while the second category focused on changing the structure of the simple HS algorithm operators and/or adding a new operator.

A novel improvisation scheme with a previously developed mechanism for updating the PAR and BW was combined [15]. Two improved HS algorithms called efficient harmony search (EHS) algorithm and self-adaptive harmony search (SAHS) algorithm were proposed for sizing optimization of truss structures [16].

Although the standard HS algorithm is able to solve binary-coded optimization problems, the pitch adjustment operator of HS is degenerated in the binary space, which spoils the performance of the algorithm. Thus an improved adaptive binary harmony search (ABHS) algorithm was proposed to solve the binary-coded problems more effectively [17]. Various adaptive mechanisms were examined and investigated, and a scalable adaptive strategy was developed for ABHS to enhance its search ability and robustness.

The impacts of constant parameters on the HS algorithm are discussed and a strategy for tuning these parameters was presented [18]. To generate new solution vectors that could enhance the accuracy and convergence rate of the HS algorithm, the impacts of constant parameters on the HS algorithm were focused on for generating new solution vectors that enhance the accuracy and convergence rate of HS [19].

A self-adaptive penalty function scheme was used to improve classic HS algorithm [20], and applications of the classical HS, improved harmony search (IHS) and the HS with differential mutation (HSDM) based pith adjustment to PAR core reloading pattern optimization problems were addressed [26].

Self-adaptive global best HS algorithm

In this paper, a method of adapting control parameters of the HS algorithm was used to improve the simple HS algorithm. In the SGHS algorithm, four control parameters harmony memory size (HMS), HMCR, PAR and BW are closely related to the problem being solved and the phase of the search process which may be either exploration or exploitation. In this paper, HCMR and PAR were dynamically adapted to a suitable range by recording their historic values corresponding to generated harmonies entering the HM. It was assumed that the HMCR (PAR) value was normally distributed in the range of [0.9, 1.0] ([0.0, 1.0]) with mean HMCR_m (PAR_m) and standard deviation 0.01 (0.05). Initially, HMCR_m (PAR_m) was set at 0.98 (0.9). And then SGHS started with a HMCR (PAR) value generated according to the normal distribution. During the evolution, the HMCR (PAR) value associated with the generated harmony successfully replacing the worst member in the HM was recorded.

After a specified number of generations LP (i.e., the learning period which was 100 in the experiment), HMCR_m (PAR_m) was recalculated by averaging all the recorded HMCR (PAR) values during this period. With the new mean and the given standard deviation of 0.01 (0.05), the new HMCR (PAR) value was produced and used in the subsequent iterations. The above procedure was repeated. As a result, an appropriate HMCR (PAR) value could be gradually learned to suit the particular problem and the particular phases of the search process.

The parameter BW is a distance bandwidth for the continuous design variable. A large BW value is in favor of the algorithm searching in a large scope, while a small BW value is appropriate for fine-tuning of the best solution vectors. To well balance the exploration and exploitation of the proposed SGHS algorithm, the BW value decreased dynamically with increasing generations (IN) as

(22)

where BW_max and BW_min are the maximum and minimum distance bandwidths, respectively; and N_gen is the number of generation.

The SGHS algorithm does not require a precise setting of specific values to critical parameters HMCR, PAR and BW in accordance with the characteristic and complexity of the problem. These parameters are self-adapted by a learning mechanism or dynamically decreased with a generation counter. Therefore, the SGHS algorithm can demonstrate consistently good performance on problems with different properties [21].

Solving UC problem using SGHS algorithm

To solve UC problem by SGHS algorithm, two data sets were applied. The first set is the control parameters of HS algorithm (i.e., the number of iteration, size of HS, adjustment rate and harmony memory considering rate, minimum and maximum of PAR and maximum and minimum BW), and the second set is the system data. With these data, harmony memory is initialized randomly.

To start the UC process, the minimum average production cost of all units was computed, and from the priority list, the units were arranged from the smallest value of the minimum average production cost. If the load was increasing during that hour, the number of the units that could be started up was determined according to the minimum downtime of the unit. Then, the top units for turning on were selected from the priority list according to the amount of load increasing. If the load is dropping during that hour, the number of the units that could be stopped was determined according to the minimum up time of the unit. Then, the last units for stopping were selected from the priority list according to the amount of load dropping. Figure 1 shows the flowchart of solving the UC problem using the proposed technique.

Case study

The SGHS algorithm was used to solve the UC problem. The systems consisting of 10, 20, 40 and 60 generating units were used as the test systems. The simulation results of the HS algorithm were compared with those of five other techniques. The initial data of the SGHS algorithm were listed in Table 1.

To confirm the capability of the HS algorithm in solving the UC problem, this algorithm was tested on the systems with 10, 20, 40 and 60 units in a time horizon of 24 h. The required data of the systems and its load data were presented in Ref. [22]. The total load of this system is 27100 kW.

Simulation results of simple HS and its algorithms

The simple HS and SGHS algorithms have been tested on four test systems, i.e., 10, 20, 40 and 60 units. The simulation was run for 20 times for each test system and the best solution, average of obtained solutions and Best/Iteration (B/I) were listed in Table 2.

It is observed form Table 2 that, in all cases, the proposed SGHS algorithm presented better solution compared to the simple HS algorithm. The best solution of the simple HS algorithm is 122, 361, 568 and 873 $ higher than the related parameter of the improved HS algorithm in Cases 1, 2, 3 and 4, respectively. This reduction in the mean cost of 20 times run are 1319, 3971, 7721 and 11840 $, respectively. In all cases, the SGHS algorithm presents its best solution in less than 10 runs while the best solution of the HS algorithm is obtained in more than 10 runs except for the 20-unit system. Figure 2 illustrates the production percentage of the total demand by one of the 10 units.

From the viewpoint of generation percentage, the units were categorized in Group i (units 1 and 2), Group ii (units 3, 4 and 5), and Group iii (units 6, 7, 8, 9, 10). Groups i, ii and iii fed 76.79%, 20.3506% and 2.8597% of the total demand, respectively. Then, the reliability of units 1 and 2 was more important than that the other units. These units generated more than two-thirds of produced power. These units fed the base load demand of the system and their operation failure had much greater impact than the other units which produced only 23.21% power. The ON/OFF status of this system is presented in Table 3 and Fig. 3.

Comparison

The results of the simple and the improved HS algorithm in the 10-uint system were compared with those of other techniques, i.e., GA, ICGA, simulated annealing (SA), seeded memetic (SM) [4,23,24]. Table 3 demonstrates the comparison of the results of the HS and SGHS algorithms with those of the Lagrangian relaxation (LR) and ICGA techniques in the 10, 20, 40 and 60-unit system.

It is noticed from Table 3 that, in all cases, the proposed technique presents better solutions compared with the other five methods.

Conclusions

This paper intended to solve the UC problem using a novel self-adaptive global best HS algorithm. The HS algorithm was presented by musical process of searching for a perfect state of harmony. Musicians, during a rehearsal or a performance, try to create pleasing sounds and approach the ideal state of harmony. To improve the simple HS algorithm, the pitch adjustment rate and arbitrary distance bandwidth were adjusted by novel and reliable techniques. From simulation results, the following conclusions can be made:

In addition to the optimum solution, other advantage of self-adaptive global best HS algorithm in respect to its simple version is that it can reach the optimal solution in less iteration (the last column in Table 2). Due to the dynamic performance of the UC problem, this feature is of great importance.

The reliability of the 1st and 2nd generating units in the 10-unit system is very important since they meet the base load demand.

The cost-save percentage of SGHS is roughly the same rate for all cases; however this is not the case in the other techniques.

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