Sage Journals: Discover world-class research

Abstract

Optimization technologies have drawn considerable interest in power system research. The success of an optimization process depends on the efficient selection of method and its parameters based on the problem to be solved. Firefly algorithm is a suitable method for power system operation scheduling. This paper presents a modified firefly algorithm to address unit commitment issues. Generally, two steps are involved in solving unit commitment problems. The first step determines the generating units to be operated, and the second step calculates the amount of demand-sharing among the units (obtained from the first step) to minimize the cost that corresponds to the load demand and constraints. In this work, the priority list method was used in the first step and the second step adopted the modified firefly algorithm. Ten generators were selected to test the proposed method, while the values of the cost function were regarded as criteria to gauge and compare the modified firefly algorithm with the classical firefly algorithm and particle swarm optimization algorithms. Results show that the proposed approach is more efficient than the other methods in terms of generator and error selections between load and generation.

Keywords

Unit commitment modified firefly priority list method particle swarm optimization

Introduction

Changes in water level in sea coasts due to the tide, changes in wind power, and fluctuations in prices of other power resources are important reasons for selecting optimal economic units for specific power generation, which is dependent on the specific time of day or year.¹ Unit commitment (UC) is responsible for the selection of the generation units that can work economically.^2,3 UC is also involved in calculating the power for each unit on the basis of the total power demand.^1,2 The differences in features of generating units make UC a considerable problem.^1,4 UC is a tool for economic power dispatch.⁵ The nonlinearity and large scale of power systems compound the solution to UC because it involves two steps, that is, examining the unit to be operated and determining the power-sharing quantity of the demand in each unit.⁶

By contrast, power system optimization is an active research area in the field of electrical engineering.^7,8 An optimal table of the generating units corresponding to the load demand with minimal fuel and transition costs is an important task in power systems.⁹

The intended outcome of an optimization process can be achieved only by complex or calculation methods; thus, researchers have focused on the optimal solution.¹⁰ Both classical and intelligent methods have been suggested to solve this problem.¹¹ The ability of the intelligent methods to optimize multi-range local optimal points renders these algorithms a considerable choice in the field of engineering techniques.^12,13

Many works have been conducted using intelligent algorithms to enhance the quality, reliability, stability, scheduling, and marketing of power systems.^14,15

Pirhayati and Mazlumi¹⁶ proposed the handling of UC problem using an intelligent binary particle swarm optimization (PSO) and evaluation definition methods with the lambda-iteration algorithm. Ladumor et al.¹⁷ proposed a solution to a four-generating unit for single area commitment problems. Whale, the optimization algorithm was used to determine UC.

Yiran et al.¹⁸ developed an improved PSO in optimizing UC problems. In this method, the on/off status of the units is first restricted to feasible schedules through the provision of a novel method related to a new time order. Maryam and Hamdi¹⁹ proposed the handling of UC using a hierarchical combination algorithm which can minimize the operational cost and simulation time. Hourly production of the binary decision variables determines the state of the unit (on/off) with respect to the demand and spinning reserve requirements.

However, all the preview methods have a high error in the calculation of power demand and the determination of the on/off unit mostly is not acceptable by the operation control center; therefore, the cost maybe increased during the control action to follow the power demand.

The modified firefly algorithm (M-FA) was suggested and examined for multiple optimization functions, such as Sphere Griewank and Ackley functions.²⁰ In the same work, the M-FA was successfully applied in power system scheduling and forecasting.²⁰ The original FA was modified using the quaternion’s representation of individuals (QFAs) by Fister et al.²¹ The position formula of the standard FA was modified to optimize the controller parameters of the Smith predictor structure.²²

In this paper, the combination of the priority list (PL) method and the M-FA is proposed to increase the economic gains in power generation units. PL selection depends on the minimum values of incremental fuel cost, whereas the M-FA refers to the process of sending random suggested solutions to estimate the optimal value of each unit power. Three intelligent methods (i.e. PSO, FA, and M-FA) are compared to determine the economic power of each unit. Another benchmark (i.e. the total error between demand and total generation) is added to compare the three algorithms. The results show the advantage of using the M-FA in UC problems.

Theoretical background

Objective function

Electricity should be provided at any moment in all parts of a power system. In addition to minimizing the total cost of power system generation, minimizing load demand, spinning reserve, and other constraints are the other objectives of UC.²³ The scheduled time horizon can be determined to satisfy these objectives as follows

\cos t_{nh} = \sum_{k = 1}^{n} \sum_{i = 1}^{h} [F C_{i} (P_{ih}) + ST C_{i} (1 - U_{i (h - 1)})] U_{ih}

(1)

where h is the number of hours, n is the number of units, and $F C_{i} (P_{ih})$ is the fuel cost of the ith unit with power output (P_ih) at hth which is represented as follows

F C_{i} (P_{ih}) = a_{i} + b_{i} P_{ih} + c_{i} P_{ih}^{2}

(2)

where $ST C_{i}$ is the start-up cost of the ith unit and $U_{ih}$ is the on/off status of the ith unit at hth, where 1 represents on, whilst 0 represents off

ST C_{i} = (\begin{matrix} Cs c_{i} when X_{i}^{off} > M D_{i} + C_{s} \\ Hs c_{i} when X_{i}^{off} \leq M D_{i} + C_{s} \end{matrix})

(3)

where $Cs c_{i}$ is the cold start cost, $M D_{i}$ is the minimum downtime, $C_{s}$ is the cold start-up h, and $Hs c_{i}$ is the hot start cost. The constraints of UCP are assumed as

Power equality

\sum_{i = 1}^{n} P_{ih} U_{ih} = D_{h}

(4)

where D_h is the power demand at the time (h)

Spinning reserve

\sum_{i = 1}^{n} P_{i (\max)} U_{ih} \geq D_{h} + R_{h}

(5)

Generation limit

P_{i (\min)} \leq P_{ih} \leq P_{i (\max)}

(6)

where $P_{i (\min)}$ is the minimum power generated by each unit, and $P_{i (\max)}$ is the maximum power generated by each unit. From the derivative of equation (2), the power generated by each unit is calculated as follows²⁴

\frac{\partial F C_{i} (P_{ih})}{\partial P_{ih}} = λ_{ph} = 2 α_{i} P_{ih} + β_{i}

(7)

P_{ih} = \frac{λ_{ph} - β_{i}}{2 α_{i}}

(8)

where $λ_{ph}$ is the incremental fuel cost for each h. Initially, $λ_{ph}$ was calculated using equation (9) and adapted at each iteration until the final optimum value was reached

λ_{ph} = unifrnd (λ_{i (\max)}, λ_{i (\min)})

(9)

where $λ_{i (\max)}$ and $λ_{i (\min)}$ are the maximum and minimum incremental fuel costs for all units at each h, respectively.

error = \sum_{i = 1}^{n} P_{ih} - D_{h}

(10)

FA

X-S Yang²⁵ developed FA in 2008 as an optimization algorithm inspired by the light flashing pattern of fireflies.²⁶ Fireflies attract one another through the intensity of the light they emit. FA is composed of three basic principles, as follows:

Fireflies are attracted to one another because they are unisexual.

The degree of attractiveness of a firefly is a function of the light it emits.

The brightness of the light emitted by each firefly decreases with the increase in its distance from other fireflies.

In the absence of a bright firefly, the swarm moves randomly. FA, as an evolutionary technique, can be used to determine the control parameters. In FA, each firefly is a potential solution that can be defined on the basis of its position. The current position of the ith firefly in a d-dimensional vector space is given by $X_{i} = (X_{i 1}, \dots, X_{in}, \dots, X_{id})$ . However, the initialization of the random positions of m fireflies must be carried out within a specified range. Equation (11) is used to update the position of firefly i, which is attracted to a bright firefly (j), whereas equation (12) is used to update the position of the brightest firefly

\begin{matrix} X_{i} (t + 1) = X_{i} (t) + β_{0} \exp (- γ ri j^{2}) (x_{j} - x_{i}) \\ + α (rand - 0.5) \end{matrix}

(11)

xbes t_{i} (t + 1) = xbes t_{i} (t) + α (rand - 0.5)

(12)

The current positions of the less bright and brightest fireflies are represented by the first terms, namely, $X_{i} (t)$ and $xbes t_{i} (t)$ , of equations (12) and (13), respectively. Meanwhile, the second term in equation (12) determines the fireflies’ attractiveness to light. β₀ represents the initial attractiveness at r = 0, whereas γ refers to the absorption parameter in the range of [0, 1]. $r_{i, j}$ represents the distance between any two fireflies (i.e. i and j) at positions x_i and x_j, respectively. This distance can be defined as a Cartesian or Euclidean distance, as follows

r_{ij} = \sqrt{\sum_{n = 1}^{d} {(x_{i, n} - x_{j, n})}^{2}}

(13)

where x_i and x_j represent the position vectors for fireflies i and j, respectively; and x_i(n) represents the positional value for the nth dimension. The third term in equations (11) and (12) is used for randomness reduction, that is, a gradual reduction in the motion of the fireflies via α = α₀δt, where α₀ is within the range of [0, 1]. δ represents the randomness reduction parameter, where 0 < δ < 1 and t is the number of iterations.

Proposed method

In the first step of the proposed method, the on/off cases of the units are provided by the PL method, whereas the load partitioning between the units is achieved by the M-FA in the second step. The result of the PL method was inputted in the second step.

First step

PL is a simpler and faster method compared with other classical methods because it commits the generating units in ascending order, in terms of their average production costs. The lack of storage capacity or ability to optimize the solutions for large power systems is the main disadvantage of PL compared with other traditional methods.²³ The PL vector is obtained using equation (3)

Priority vector = \frac{P_{(\max), vec}}{\max [P_{(\max), vec}]} + \frac{M D_{vec}}{\max [M D_{vec}]}

(14)

The PL method is used to denote the on/off status as an initial solution for UC without the ramp rate. The main condition of the first step is the regulation of the total capacity of the committed generating unit, the spinning reserve, and the total power required.

Second step

M-FA determines the economic power dispatch per unit. Many researchers advise the use of FA as an optimization tool in a wide range of power system applications. To reduce the local optimal trapping possibility and improve the FA search ability, a M-FA was used in this study. In the proposed method M-FA, randomization parameter was not kept fixed and was linearly decreased with iterations, where $X_{q 1}$ was the initial value and $X_{qn}$ was the final value. This strategy could keep balance between the exploration and exploitation abilities of the proposed algorithm. In the early stage, larger $Δ$ provided better global searching ability; and in the later stage, smaller $Δ$ offered better convergence. The improvement in the M-FA can be achieved using two steps:20 first, every two mutations deal with three crossover processes; second, the total firefly generations is led to move toward the optimal promising local or global objective.²⁰ To complete the two steps, a modification in each iteration should be carried out as follows

X_{Mute 1} = X_{q 1} + Δ \times (X_{q 2} - X_{q 3})

(15)

X_{Mute 2} = X_{Mute 1} + Δ \times (X_{Best}^{iter} - X_{Worst}^{iter})

(16)

where $Δ$ is a random number in the range of [0, 1] and $X_{q 1}, X_{q 2}, and X_{q 3}$ are three random fireflies

X_{Best, 1} = [x_{Best, 1}, x_{Best, 2}, . . . ., x_{Best, 2}]

(17)

x_{improve 1, j} = {\begin{matrix} x_{Mute 1, j}, & if k_{1} \leq k_{2} \\ x_{Best 1, j}, & if k_{1} > k_{2} \end{matrix}}

(18)

x_{improve 2, j} = {\begin{matrix} x_{Mute 1, j}, & if k_{2} \leq k_{3} \\ x_{j}, & if k_{2} > k_{3} \end{matrix}}

(19)

x_{improve 3, j} = {\begin{matrix} x_{Best 1, j}, & if k_{3} \leq k_{4} \\ x_{j}, & if k_{3} > k_{4} \end{matrix}}

(20)

x_{improve 4, j} = {\begin{matrix} x_{Mute 1, j}, & if k_{4} \leq k_{5} \\ x_{Mute 2, j}, & if k_{4} > k_{5} \end{matrix}}

(21)

where $X_{Best}^{iter} and X_{Worst}^{iter}$ are the best and worst populations in each iteration, respectively; and $k_{1} - k_{5}$ are random values between 0 and 1. All fireflies are used to find the objective function in determining the optimal value for the running iteration. Figure 1 shows the flowchart of the algorithm.

Figure 1.

M-FA algorithm flowchart.

Case study

In this test case, 10 thermal units with an hourly supply demand of over 24 h^9,27 were considered. Tables 1 and 2 show the unit information and hourly demand, respectively.

Table 1.

Unit data.

Unit no.	1	2	3	4	5
$P_{i (\max)} (MW)$	455.00	455.00	130.00	130.00	162.00
$P_{i (\min)} (MW)$	150.00	150.00	20.00	20.00	25.00
$a_{i} (Rs / h)$	1000.0	970.00	700.00	680.00	450.0
$β_{i} (Rs / MWh)$	16.1900	17.2600	16.600	16.500	19.70
$c_{i} (Rs / M W^{2} h)$	0.00048	0.00031	0.002	0.00211	0.00398
$M D_{i} (h)$	8.00	8.00	5.00	5.00	6.00
$M U_{i} (h)$	8.00	8.00	5.00	5.00	6.00
$Hs c_{i} (Rs / h)$	4500.0	5000.0	550.0	560.0	900.0
$Cs c_{i} (Rs / h)$	9000	10000	1100	1120	1800
$Cs (h)$	5.00	5.00	4.00	4.00	4.00
Initial status	8.00	8.00	−5.00	−5.00	−6.00
Unit no.	6.00	7.00	8.00	9.00	10.00
$P_{i (\max)} (MW)$	80.00	85.00	55.00	55.00	55.00
$P_{i (\min)} (MW)$	20.00	25.00	10.00	10.00	10.00
$a_{i} (Rs / h)$	370.00	480.00	660.0	665.0	670.0
$β_{i} (Rs / MWh)$	22.260	27.740	25.920	27.270	27.79
$c_{i} (Rs / M W^{2} h)$	0.00712	0.0079	0.00413	0.00222	0.00173
$M D_{i} (h)$	3.000	3.00	1.00	1.00	1.00
$M U_{i} (h)$	3.000	3.00	1.00	1.00	1.00
$Hs c_{i} (Rs / h)$	170.00	260.00	30.00	30.00	30.00
$Cs c_{i} (Rs / h)$	340.0	520.00	60.00	60.00	60.00
$Cs (h)$	2.000	2.000	0.00000	0.0000	0.000
Initial status	−3.000	−3.000	−1.000	−1.000	−1.00

Table 2.

Hourly load.

Time (h)	H1st	H2nd	H3rd	H4th	H5th	H6th
Load (MW)	700.0	750.0	850.0	950.0	1000.0	1100.0
Time (h)	H7th	H8th	H9th	H10th	H11th	H12th
Load (MW)	1150	1200.0	1300	1400.0	1450	1500
Time (h)	H13th	H14th	H15th	H16th	H17th	H18th
Load (MW)	1400	1300.0	1200	1050	1000.0	1100.0
Time (h)	H19th	H20th	H21st	H22nd	H23rd	H24th
Load (MW)	1200	1400.0	1300	1100.0	900.00	800.0

Results

Three different intelligent methods are compared to evaluate the effect of modification on the original FA, especially on the second step of UC. The power system of the 10 units was simulated using MATLAB. Many issues are discussed for this comparison. The error between the actual data and calculated total generated power is considered. Figure 2 illustrates the differences in the daily load curve/hour among the three algorithms, and Figure 3 shows the error values for 20 iterations. The main FA parameters were set to the optimal settings (i.e. β₀ = 0.275, γ = 1, and α = 0.30), and the number of fireflies was set to 25. For the PSO algorithm, bird step = 25, ω = 0.75, and acceleration constants c₁ and c₂ were both 1.85, as previously recommended.^12,13

Figure 2.

Differences between demand and actual power generation as calculated by each method.

Figure 3.

Error comparison for each method.

Figure 2 shows that the calculated power demand through the FA from 18th-h to the 20th-h does not match the actual power. The power demands calculated by the PSO for the interval of time between the 6th-h to the 8th-h and 18th-h to the 20th-h deviate from the actual power demand. Minimum deviation is obtained using the M-FA. A small mismatch value was only observed from 22:00 to 24:00 using the M-FA method in the calculation of power demand.

The total errors of M-FA, PSO, and FA were 47.25176, 74.05321, and 115.7395 MW, respectively. Table 3 provides the error details for each hour.

Table 3.

Estimated demand error.

H	M-FA	PSO	FA	H	M-FA	PSO	FA
1	0.093	0.3240	13.250	13	−0.365	−0.342	−0.012
2	−0.098	1.0669	−5.083	14	−0.353	0.105	−0.040
3	−1.499	−4.7245	10.29	15	0.009	−5.000	−2.033
4	−0.365	0.3413	−0.137	16	0.393	2.040	−2.916
5	0.045	0.0763	−0.461	17	−1.206	−2.707	−24.73
6	−0.013	−0.0521	0.105	18	−3.280	−1.012	−1.669
7	−0.998	45.0000	45.00	19	−0.323	−5.000	−5.010
8	2.693	2.4690	0.874	20	−0.132	−0.092	0.297
9	−0.091	0.5080	0.003	21	0.130	−0.509	−0.085
10	−0.110	0.456	−0.059	22	−0.016	−0.069	−0.320
11	−3.000	0.162	−0.964	23	30.00	0.199	0.448
12	1.844	1.665	−0.187	24	−0.193	0.133	−1.773

Table 3 shows that the maximum errors for PSO, FA, and M-FA are 45, 45, and 30 kW, respectively.

The optimal cost of the 24th-h is shown in Figure 4, which explains the local point behavior toward the optimal point for each optimization method.

Figure 4.

Total generation cost as obtained by each method.

The power generated by each unit was deduced by the three intelligent algorithms that represent the highly complex component of UC. Tables 4 and 5 show the generation units and total cost, respectively, which were calculated using the suggested method for 24 h.

Table 4.

Power matrix for 10 units system for 24 h using M-FA.

	Generation										PD
H	1	2	3	4	5	6	7	8	9	10
1	455	217.9	0	0	0	0	0	0	0	0	700
2	455	301.2	0	0	0	0	0	0	0	0	750
3	873.7	850	0	0	0	0	0	0	0	0	850
4	455	455	0	0	39.4	0	0	0	0	0	950
5	455	455	0	0	89.8	0	0	0	0	0	1000
6	455	455	0	130	59.1	0	0	0	0	0	1100
7	455	394.4	130	130	25	0	0	0	0	0	1150
8	455	455	130	130	30.9	0	0	0	0	0	1200
9	455	455	130	130	109.2	20	0	0	0	0	1300
10	455	455	130	130	162	41.2	25	0	0	0	1400
11	455	455	130	130	162	80	25	10	0	0	1450
12	455	455	130	130	162	80	25	53.7	10	0	1500
13	455	455	130	130	162	43.6	25	0	0	0	1400
14	455	455	130	130	110.2	20	0	0	0	0	1300
15	455	455	130	130	29.7	0	0	0	0	0	1200
16	455	333.5	130	130	25	0	0	0	0	0	1050
17	455	237.7	130	130	25	0	0	0	0	0	1000
18	455	341.2	130	130	25	0	0	0	0	0	1100
19	455	455	130	130	28.6	0	0	0	0	0	1200
20	455	455	130	130	162	43	25	0	0	0	1400
21	455	455	130	130	162	75.6	25	0	0	0	1300
22	455	455	0	0	143.9	20	25	0	0	0	1100
23	455	424.5	0	0	0	20	0	0	0	0	900
24	455	353.5	0	0	0	0	0	0	0	0	800
Total cost = $55,7873

Table 5.

Generation cost for 10 units’ system for 24 h usingM-FA.

Time (h)	Fuel cost ($)	Start-upcost ($)	Time(h)	Fuel cost ($)	Start-upcost ($)
1	13,211.8	0.00	13	29,385.5	0
2	14,662.9	0.00	14	26,594.5	0
3	16,716.9	0.00	15	24,145.8	0
4	18,585	900	16	21,924.5	0
5	19,604.3	0.00	17	20,253.6	0
6	21,842.3	1120	18	22,058.8	0
7	22,989.2	1100	19	24,124	0
8	24,168.1	0.00	20	29,370.8	690
9	26,572.8	340	21	27,233.4	0
10	29,330.6	520	22	22,719	0
11	31,146.5	60.0	23	17,637.7	0
12	33,228.1	60.0	24	15,576.6	0
Total	55,3083		4790

Table 4 indicates that units 8–10 have short operation times when the proposed method is used, but the operation cost of these units is very high. Units 1 and 2 operate for 24 h but have the lowest operation cost.

Table 6 shows that the total cost using M-FA was less than that using PSO by >0.5% after 20 iterations. This finding indicates that M-FA can achieve higher economic results compared with PSO and FA. The total cost for all methods after 70 iterations was $5,53,897, which was achieved after 45, 55, and 60 iterations for M-FA, PSO, and FA, respectively (Figure 5).

Table 6.

Execution time and total cost for all the cases.

Method	Total time (h)	Total cost ($)
PSO	0.0899773	5,59,750
FA	0.0935586	5,60,904
M-FA	0.117938	5,57,873

Figure. 5.

A total of 70 iterations of optimisation methods.

The selected units for all intelligent methods that supply the demand evolve using the classical PL method during the determination of the on/off units. Table 7 shows the first step solution to the UC problem.

Table. 7.

On/off status matrix.

Hour	Unit no. on/off status
Hour	1	2	3	4	5	6	7	8	9	10
1	1	1	0	0	0	0	0	0	0	0
2	1	1	0	0	0	0	0	0	0	0
3	1	1	0	0	0	0	0	0	0	0
4	1	1	0	0	1	0	0	0	0	0
5	1	1	0	0	1	0	0	0	0	0
6	1	1	0	1	1	0	0	0	0	0
7	1	1	1	1	1	0	0	0	0	0
8	1	1	1	1	1	0	0	0	0	0
9	1	1	1	1	1	1	0	0	0	0
10	1	1	1	1	1	1	1	0	0	0
11	1	1	1	1	1	1	1	1	0	0
12	1	1	1	1	1	1	1	1	1	0
13	1	1	1	1	1	1	1	0	0	0
14	1	1	1	1	1	1	0	0	0	0
15	1	1	1	1	1	0	0	0	0	0
16	1	1	1	1	1	0	0	0	0	0
17	1	1	1	1	1	0	0	0	0	0
18	1	1	1	1	1	0	0	0	0	0
19	1	1	1	1	1	0	0	0	0	0
20	1	1	1	1	1	1	1	0	0	0
21	1	1	1	1	1	1	1	0	0	0
22	1	1	0	0	1	1	1	0	0	0
23	1	1	0	0	0	1	0	0	0	0
24	1	1	0	0	0	0	0	0	0	0

Conclusion

UC is a crucial term in control power strategy, which is responsible for the selection of generation units that can work economically. A rapid solution for unit selection was proposed using the PL method. Following this selection, three optimization algorithms (i.e. PSO, FA, and the proposed M-FA) were compared by determining the power of each unit. Ten units with deference operation cost characteristics have been simulated to supply multi-power level loads for 24 h using MATLAB. The results prove that the M-FA has high convergence characteristics in reducing error and commercial operation. The reliability of this method was evaluated by repeating the simulation run several times.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Balasim M Hussein

Aqeel S Jaber

References

Abedinia

Barazandeh

Amjady

Solving optimal unit commitment problem based on wind power effects using harmony search algorithm. J Basic Appl Sci Res 2012; 2(11): 11764–11773.

Abdou

Tkiouat

Unit commitment problem in electrical power system: a literature review. Int J Electr Comput Eng 2018; 3(11): 1357–1372.

Gögler

Dorfner

Hamacher

. Hybrid robust/stochastic unit commitment with iterative partitions of the continuous uncertainty set. Front Energ Res J 2018; 6: 71–86.

Zhu

Liu

Zhai

Monthly unit commitment model and algorithm with renewable energy generation considering system reliability. Math Prob Eng 2019; 9: 1–9.

Thakshaayene

Kavitha

Unit commitment using a hybrid genetic algorithm with differential evolution. In: Proceedings of the innovations in power and advanced computing technologies (i-PACT), Vellore, India, 21–22 April 2017, pp.1–6. New York: IEEE.

Bertsimas

Litvinov

Sun

, et al. Adaptive robust optimization for the security constrained unit commitment problem. IEEE Trans Power Syst 2013; 28(1): 52–63.

Jaber et al AS. A new parameters identification of single area power system based LFC using Segmentation Particle Swarm Optimization (SePSO) algorithm. In: Proceedings of the IEEE PES Asia-Pacific power and energy engineering conference (APPEEC), Kowloon, China, 8–11 December, pp.1–6. New York: IEEE.

Jaber et al AS. Advance two-area load frequency control using particle swarm optimization scaled fuzzy logic. Adv Mater Res 2013; 622–623: 80–85.

Tiwari

Dwivedi

Dave

A two-stage solution methodology for deterministic unit commitment problem. In: Proceedings of the electrical, computer and electronics engineering (UPCON), Varanasi, India, 9–11 December 2016, pp.317–322. New York: IEEE.

10.

Ting

Rao

Loo

A novel approach for unit commitment problem via an effective hybrid particle swarm optimization. IEEE Trans Power Syst 2006; 21(1): 411–418.

11.

Reddy

Ramana

Reddy

KR.

Detailed literature survey on different methodologies of unit commitment. J Theor Appl Inform Technol 2013; 53(3): 111–116.

12.

Nadtokaa

Al-Zihery Balasim

. Mathematical modelling and short-term forecasting of electricity consumption of the power system, with due account of air temperature and natural illumination, based on support vector machine and particle swarm. Procedia Engineer 2015; 129: 657–663.

13.

Jaber et al AS. A new improvement of conventional PI/PD contrllersfor load frequency control with scaled fuzzy controller. Int J Eng Appl Sci 2015; 2(4): 69–74.

14.

Tong

Shahidehpour

SM.

An innovative approach to generation scheduling in large-scale hydro-thermal power systems with fuel constrained units. IEEE Trans Power Syst 1990; 5(2): 665–673.

15.

Moghimi Haji

Vahidi

. A solution to the unit commitment problem using imperialistic competition algorithm. IEEE Trans Power Syst 2012; 27(1): 117–124.

16.

Pirhayati

Mazlumi

. An intelligent binary particle swarm optimization for unit commitment problem. In: Proceedings of the IEEE international conference on power and energy, Kuala Lumpur, Malaysia, 29 November–1 December 2010, pp.248–252. New York: IEEE.

17.

Ladumor

Trivedi

Jangir

, et al. A whale optimization algorithm approach for unit commitment problem solution. In: Proceedings of the National Conference advancement in electrical & power electronics engineering (AEPEE 2016), 28th–29th JUNE, 2016 Morbi, India, 2016.

18.

Yiran

Guo

Zhang

Jingrui

Fang

Zheng

. An improved particle swarm optimization approach for unit commitment problem. Open Automat Control Syst J 2014; 6: 629–636.

19.

Maryam

Hamdi

A solution to the unit commitment problem applying a hierarchical combination algorithm. J Energ Manag Technol 2017; 1(2): 12–19.

20.

Kavousi-Fard

Samet

Marzbani

A new hybrid modified firefly algorithm and support vector regression model for accurate short-term load forecasting. Expert Syst Appl 2014; 41(13): 6047–6056.

21.

Fister

Yang

X-S

Brest

, et al. Modified firefly algorithm using quaternion representation. Expert Syst Appl 2013; 40(18)): 7220–7230.

22.

Gupta

Padhy

PK.

Modified firefly algorithm based controller design for integrating and unstable delay processes. Eng Sci Technol 2016; 19(1): 548–558.

23.

Govardhan

Solution to unit commitment with priority list approach. Int J Adv Res Electr Electr Instrum Eng 2016; 5(6): 5114–5121.

24.

Narasimha

PSO technique for solving the economic dispatch problem considering the generator constraints. Int J Adv Res Electr Electr Instrum Eng 2014; 3(7): 10439–10454.

25.

Yang

XS.

Nature-Inspired Metaheuristic Algorithms. Luniver Press, 2008.

26.

Bendjeghaba

Boushaki

Zemmour

. Firefly Algorithm for optimal tuning of PID controller parameters. In: Proceedings of the Fourth international conference on power engineering, energy and electrical drives, Istanbul, 13–17 May 2013.

27.

Shantanu

Senjyu

Saber

, et al. Optimal thermal unit commitment integrated with renewable energy sources using advanced particle swarm optimization. IEEJ Trans 2009; 4: 609–617.

Unit commitment based on modified firefly algorithm