Practical economic dispatch considering wind turbine uncertainty

Abstract

This paper proposes a new stochastic framework to solve the practical economic dispatch (ED) problem considering the penetration of wind turbines (WTs). The proposed stochastic method is constructed based on krill herd (KH) algorithm to search the problem space optimally. Since the output power of WT is neither stable nor continues, point estimate method (PEM) with 2m scheme is used to model the uncertainty effects. In addition to the traditional constraints, the practical ED includes valve-loading effect, multi-fuel option, prohibited operating zone (POZ), ramp rate limit and spinning reserve. According to the high complexity and nonlinearity of the problem, a new version of KH called Modified Levy Krill Herd (MLKH) is suggested to solve the problem optimally. The satisfying performance of the problem is examined on two standard test systems and is compared with some of the recent well-know methods in the area successfully.

Keywords

Economic dispatch point estimate method Levy Krill Herd practical constraints

Nomenclature

a_i, b_i, c_i, e_i, f_i

Cost coefficients of the ith generating unit.

Transmission loss coefficients matrix.

E(.)

Expectation function.

F _T

Total generation cost ($/h).

F_i (.)

Fuel cost function of ith unit ($/h).

h(.)

Probabilistic function in PEM.

Shape factor of Weibull distribution.

Le′vy(.)

Levy flight function.

Number of estimating points in PEM.

Number of dispatchable units.

N _farm

Number of wind farms.

N _fuel,I

Number of fuel types for ith unit.

N _GP

Number of generators with POZ.

NP _i

Number of POZs of unit i.

N _w

Number of wind-powered generators.

P _i

Active power output of ith unit (MW).

P _D

Total active power demand.

P _Loss

Total active loss of network (MW).

P_{i}^{min}

Minimum active power output of ith unit.

P_{i}^{max}

Maximum active power output of ith unit.

P_{ij}^{min}

Minimum power output of unit i with fuel option j

P_{ij}^{max}

Maximum power output of unit i with fuel option j.

P _0i

Active power output of unit in the previous hour.

P_{i, j}^{LB}

Lower boundary of POZ j of generator i.

P_{i, j}^{UB}

Upper boundary of POZ j of generator i.

P _w,f

Output power of wth wind generator in fth wind farm.

P _r,w,f

Rated output power of wth wind generator in fth wind farm.

Prb _θ

Probability stacks in θth sub-modification.

rand (.)

Random number generator.

RUP _i

Ramp-up rate limit of unit i (MW/h).

RDN _i

Ramp-down rate limit of unit i (MW/h).

Number of concentrations to be calculated in PEM.

s _j,w,f

Slope of jth segment of unit w in wind farm f (MWs/m).

SR _i

Spinning reserve contribution of unit i.

{SR}_{i}^{max}

Maximum spinning reserve contribution of unit i.

SR _R

System required spinning reserve (MW).

SR _r

Total spinning reserve contribution of all units.

T _F

Teaching factor in SAMSFLA.

v _w,f

Wind speed of unit w in wind farm f (m/s).

v _ci,w,f

Cut-in wind speed of unit w in wind farm f (m/s).

v _co,w,f

Cut-out wind speed of unit w in wind farm f (m/s).

v _j,w,f

Breakpoint jth segment of unit w in wind farm f (m/s).

v _r,w,f

Rated wind speed of unit w in wind farm f (m/s).

w _k,s

Weighting factor in PEM.

Vector of control variables.

y _k,s

Location of estimating points in PEM.

Random input variable of point estimate method.

Random output variable of point estimate method.

μ _Y

Mean value of random variable Y.

σ _Y

Standard deviation of random variable Y.

Set of all online units.

Set of online units which have POZs.

ζ _k,s

standard location of the random variables in PEM.

λ _k,3

skewness of the random variable y _k.

Approximation of the levy flight.

φv ₁

Random variable in the range [0,1].

φ _i

random values in the range [0,1].

V_{ind, i}^{m}

induced velocity of ith krill at the mth movement.

V_{ind}^{\max}

maximum induced velocity.

V_{frg, i}^{m}

foraging velocity of ith krill at mth movement.

α _ind,i

attractive/repulsive tendency factor.

small positive number.

N _p

population size.

ρ, κ

empirical constant factors.

V_{diff, i}^{m}

diffusion velocity of ith krill at mth movement.

V_{r, i}^{m}

resultant velocity of ith krill at mth movement.

W _i

weighting factor of ith individual.

ω _ind/frg/diff

inertia of induction/foraging/ diffusion motion.

Iter _max

maximum iteration.

u_j/l_j

upper/lower bound of jth control variable.

1 Introduction

In recent years, the high penetration of different types of renewable energy sources (RESs) has affected most of the available strategies [1, 2]. The main motivations behind the high penetration of RESs are clean energy, modular system and easy installation. Among different types of RESs, wind turbine (WT) is an interesting choice referring to the availability of wind in almost any area. The new technologies of WT can provide energy from residential applications to wide grid connected applications [3]. Regarding the WT, advantages such as reducing the dependence on fossil fuels and enhancing the independence and flexibility of large power grid are the most prominent. Nevertheless, the high volatile nature of wind speed injects much uncertainty in the grid that if not managed optimally can put the analyses far from the reality. In this regard, on of the significant strategies that can be affected by the high penetration of WTs is economic dispatch (ED) [4].

ED is defined as the process of designating the optimal power value to the generation units such that the total cost is minimized [5]. ED problem is imposed by some practical limitations caused either by machines or operation planning to operate the system in a secure condition. For example, Prohibited Operating Zones (POZs) are considered to avoid vibration in the shaft bearing. Some of the techniques proposed for considering POZ in the ED are Genetic Algorithm (GA) [6], advantageous decision space approach [7], Particle Swarm Optimization (PSO) approach [8], Integrated Artificial Intelligence (IAI) technique [5], Decomposition Technique (DT) [8]. The other practical constraint is ramp rate limit that is caused by the limitation in the output power of units [9]. This limitation is significant in ED with several planning horizons like 24 hours. The significant matter that should be preserved in the power systems is spinning reserve limitation that is determined by the reliability group. This issue is more significant in the new systems with much uncertainty. Some of the well-knowntechniques proposed for solving the ED are decomposition technique [8], Bender’s Decomposition (BD) [10], and PSO with the sequential quadratic programming (SQP) [11]. The other constraint is caused by the valve loading process. According to the structure of the valves, loading is limited to a ripple-like curve that is reflected in the cost function and makes it non-convex. Evolutionary Programming (EP) [12], improved fast EP (IFEP) [13], Modified PSO (MPSO) with a dynamic search space reduction strategy [14], GA approach [15, 16], fast EP using the weighted mean of Gaussian and Cauchy mutations called MFEP [13], hybrid EP combined with SQP [17], Hybrid GA (HGA) [18], Hybrid PSO (HPSO) [19] Evolutionary Strategy Optimization (ESO) [20], Improved GA (IGA) [21], and modified PSO (MPSO) with a dynamic search space reduction strategy [14] are some of the techniques that have considered valve-loading effect. Finally, multi-fuel option is another limit that should be met as the result of different fuel costs. Some of the works of this area are Hierarchical Method (HM) [22], and Taguchi Method (TM) [23].

Considering the entire above constraints make the ED a complex nonlinear optimization problem that requires a powerful tool to be solved optimally. Some of the methods that have been proposed to solve the ED with the above constraints are neural networks [24], Lagrange relaxation [25] and mixed integer programming [26]. Also, a new stochastic optimization approach using hybrid Bacterial Foraging (BF) technique was suggested in [27] to solve ED problem including ramp rate limits and POZs. As mentioned before, appearance of WT in the new systems necessitates the reassessment of the available strategies for solving the ED problem. This issue is investigated in this paper. The most challenging issue regarding the use of WTs is the high uncertainty injected to the system. In this way, we suggest a new stochastic framework based on point estimate method (PEM) and krill herd algorithm (KH). PEM is categorized in the class approximate methods that replaces the probability density function (PDF) of each uncertain parameter with a few sample points [28]. In comparison with other methods such as Monte Carlo method, PEM requires less computational burden. Regarding the optimization tool, KH algorithm with a new modification based on Levy flight [29] is employed. KH algorithm is inspired from the social life of krill animals to search for food and can be a successful optimizer for the problem [30]. The feasibility and satisfying performance of the proposed problem are investigated using several standard test systems.

2 Problem formulation

This section describes the structure of the ED problem including the objective function and the relevant practical constraints.

2.1 Cost Objective function

ED target is to minimize the total cost of power generation for supplying the total load as follows: $\begin{matrix} Minimize & F_{T} = \sum_{i = 1}^{n} F_{i} (P_{i}) \end{matrix}$ (1)

The fuel cost for a specific fuel type is calculated using a second order polynomial function as follows: $F_{i} (P_{i}) = a_{i} + b_{i} P_{i} + c_{i} P_{i}^{2}$ (2)

Considering the valve loading effect, a sinusoidal term is added to the above cost function as follows: $F_{i} (P_{i}) = a_{i} + b_{i} P_{i} + c_{i} P_{i}^{2} + | e_{i} sin (f_{i} (P_{i}^{min} - P_{i})) |$ (3) where a _i, b _i, c _i, e _i and f _i are the cost coefficients of the ith power unit.

Considering the use of different fuel types, the cost function is changed to the below form:

$\begin{matrix} F_{i} (P_{i}) \\ = {\begin{matrix} \begin{matrix} a_{i 1} + b_{i 1} P_{i} + c_{i 1} P_{i}^{2} & fuel 1, P_{i}^{min} \leq P_{i} \leq P_{i 1}^{max} \end{matrix} \\ \begin{matrix} a_{i 2} + b_{i 2} P_{i} + c_{i 2} P_{i}^{2} & fuel 2, P_{i 2}^{min} \leq P_{i} \leq P_{i 2}^{max} \end{matrix} \\ ⋮ \\ \begin{matrix} a_{ij} + b_{ij} P_{i} + c_{ij} P_{i}^{2} & fuelj, P_{ij}^{min} \leq P_{i} \leq P_{i}^{max} \end{matrix} \end{matrix} \end{matrix}$ (4)

It is worth to note it that $P_{ij - 1}^{max} = P_{ij}^{min}$ . If the valve-loading is considered, the above formulation is changed as follows: $\begin{matrix} \begin{matrix} F_{i} (P_{i}) = a_{ij} + b_{ij} P_{i} + c_{ij} P_{i}^{2} \\ + | e_{ij} sin (f_{ij} (P_{ij}^{min} - P_{i})) | \\ if fuelj, P_{ij}^{min} \leq P_{ij} \leq P_{ij}^{max} \end{matrix} \\ \begin{matrix} \forall_{i}, i = 1, \dots, n & and & \forall_{j}, j = 1, \dots, N_{fuel} \end{matrix} \end{matrix}$ (5)

2.2 Constraints

Two of the significant constraints (valve-loading effect and multi fuel option) were discussed andincorporated in the cost function part. The other constraints as follows:

–Demand and power balance: The total power generation should equal the total load and power losses as follows $\sum_{i = 1}^{n} P_{i} = P_{D} + P_{Loss}$ (6)

Since ED does not consider the network topology, the below formulation is used for estimating the amount of power losses: $P_{Loss} = \sum_{i = 1}^{n} \sum_{j = 1}^{n} P_{i} B_{ij} P_{j} + \sum_{i = 1}^{n} B_{0 i} P_{i} + B_{00}$ (7)

–Ramp rate limits: Ramp rate limits are reflected based on the power of units in the preceding hourP _0i. ${\begin{matrix} \begin{matrix} P_{i} - P_{0 i} \leq {RUP}_{i} & if & generation & increases \end{matrix} \\ \begin{matrix} P_{0 i} - P_{i} \leq {RDN}_{i} & if & generation & decreases \end{matrix} \end{matrix}$ (8)

–Generation capacity constraint: Each unit can produce power in the limited range as follows:

$\begin{matrix} max (P_{i}^{min}, P_{0 i} - {RDN}_{i}) \leq P_{i} \\ \leq min (P_{i}^{max}, P_{0 i} + {RUP}_{i}) \end{matrix}$ (9)

–Prohibited Operating Zones: Due to mechanical limitations, the operating area for each power unit can be limited: $\begin{matrix} {\begin{matrix} \begin{matrix} P_{i}^{min} \leq P_{i} \leq P_{i, j}^{LB} & j = 1 \end{matrix} \\ \begin{matrix} P_{i, j - 1}^{LB} \leq P_{i} \leq P_{i, j}^{LB} & j = 2, 3, \dots, {NP}_{i} \end{matrix} \\ \begin{matrix} P_{i, j}^{UB} \leq P_{i} \leq P_{i}^{max} & j = {NP}_{i} \end{matrix} \end{matrix} \\ i = 1, \dots, N_{GP} \end{matrix}$ (10)

It is clear that a generating unit with NP _i POZ will and a convex set will have NP _i+1 separate operating regions [3].

–Spinning reserves capacity: Spinning reserve is also another constraint of the problem: $\begin{matrix} {SR}_{i} & = & {\begin{matrix} \begin{matrix} min (P_{i}^{max} - P_{i}, {SR}_{i}^{max}) & \forall i \in (Ω - Θ) \end{matrix} \\ \begin{matrix} 0 & \forall i \in Θ \end{matrix} \end{matrix} \\ {SR}_{r} & = & \sum_{i = 1}^{n} {SR}_{i} \end{matrix}$ (11) ${SR}_{r} \geq {SR}_{R}$

2.3 WT modeling

WT as a popular renewable source has experienced rapid growth in the last years. WT makes use of the wind speed energy to convert to the electrical energy. Nevertheless, the volatile nature of the wind speed is a challenging issue in front of its popularity yet. In fact, the nonlinearity and complexity of the wind speed behavior is very high such that the most accurate forecasting methods will have some error [31]. This will result in much uncertainty in the system and directly to the ED with the presence of WT. PEM is the tool that will be described in the next section for modeling these uncertainties. The relationship between the wind speed and wind turbine output power can be estimated using the below equation:

$\begin{matrix} P_{w, f} = P_{r, w, f} \\ {\begin{matrix} \begin{matrix} s_{1, w, f} v_{w, f} - v_{ci, w, f} & v_{ci, w, f} \leq v_{w, f} \leq v_{1, w, f} \end{matrix} \\ \begin{matrix} s_{1, w, f} (v_{w, f} - v_{ci, w, f}) + s_{2, w, f} (v_{w, f} - v_{1, w, f}) \\ v_{1, w, f} \leq v_{w, f} \leq v_{2, w, f} \end{matrix} \\ \begin{matrix} s_{1, w, f} (v_{w, f} - v_{ci, w, f}) + s_{2, w, f} (v_{w, f} - v_{1, w, f}) \\ + s_{3, w, f} (v_{w, f} - v_{2, w, f}) v_{2, w, f} \leq v_{w, f} \leq v_{3, w, f} \end{matrix} \\ \begin{matrix} 1 & v_{r, w, f} \leq v_{w, f} \leq v_{co, w, f} \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{matrix} \\ w = 1, \dots \dots, N_{w} f = 1, \dots, N_{farm} \end{matrix}$ (12)

According to the above equation, there is no power produced below the cut-in speed or above the cut-out speed.

3 Levy Krill Herd algorithm

This section describes the KH algorithm and its Levy version for solving the proposed practical ED.

3.1 Original KH

KH algorithm was first proposed by Gandomi et al. [30] to mimic the foraging behavior of krill for food in their society. Each krill X is a promising solution for the problem in hand. The main characteristics of this algorithm can be named as simple concept, usable for both continuous and discrete optimization problems, using powerful crossover and mutation operators and easy implementation. According to the researches, each krill movement toward the food is affected by three parameters: 1) surrounding effect by other krill called induction movement V _ind, 2) last experience of the krill itself about food position called foraging movement V _frg and 3) a random movement with uniform distribution called diffusion V _diff. After generation of the initial population of the krill randomly, the fitness function is calculated and the best krill X _Gbest is stored. Now the position of each krill is updated using the below combinatorial equation: $\begin{matrix} X_{i}^{m + 1} = X_{i}^{m} + V_{r, i}^{m} κ \sum_{j = 1}^{N_{v}} (u_{j} - l_{j}) \\ V_{r, i}^{m} = V_{ind, i}^{m} + V_{frg, i}^{m} + V_{diff, i}^{m} \end{matrix}$ (13)

Each of the induction, foraging and diffusion movements are described at below:

–Induction movement of ith krill in mth iteration $V_{ind, i}^{m}$ is calculated as follows: $V_{ind, i}^{m} = α_{ind, i} V_{ind, i}^{max} + ω_{ind} V_{ind, i}^{m - 1}$ (14)

$\begin{matrix} α_{ind, i} & = & \sum_{j = 1}^{N_{s}} [\frac{f_{i} - f_{j}}{f_{w} - f_{b}} \times \frac{X_{i} - X_{j}}{| X_{i} - X_{j} | + ɛ}] \\ + 2 [rand (.) + \frac{i}{{Iter}_{max}}] f_{i}^{b} X_{i}^{b} \end{matrix}$ (15)

The radius of the surrounding area of each krill X _j is evaluated as below: $R_{Vicinity} = \frac{1}{5 N_{p}} \sum_{j = 1}^{N_{p}} | X_{i} - X_{j} |$ (16)

–Foraging movement of ith krill in ith iteration $V_{frg, i}^{m}$ is calculated as follows:

Each krill will update its foraging velocity according to its present position and former food position as follows:

$\begin{matrix} V_{frg, i}^{m} & = & 0.02 [2 (1 - \frac{i}{{Iter}_{max}}) f_{i} \frac{\sum_{i = 1}^{N_{s}} \frac{X_{i}}{f_{i}}}{\sum_{i = 1}^{N_{s}} \frac{1}{f_{i}}} + f_{i}^{b} X_{i}^{b}] \\ + ω_{frg} V_{frg, i}^{m - 1} \end{matrix}$ (17)

–Diffusion movement of ith krill in mth iteration $V_{diff, i}^{m}$ is calculated as follows: $V_{diff, i}^{m} = υ \times ω_{diff}$ (18) wherein υ is distributed uniformly between −1and 1.

As mentioned before, KH algorithm is equipped with crossover and mutation operators from GA that more descriptions can be found in [30].

3.2 Modified Levy KH (MLKH)

KH is a new intelligent evolutionary algorithm that has shown superior performance than a number of other algorithms in the area [30]. Nevertheless, we propose a new modification method to improve its performance in the ED problem. The first part of the proposed modification is constructed based on the Levy Flight [29]. Levy Flight is a random movement around each krill to provide a deep local search. Therefore, in each iteration, the position of each krill is updated using the Levy flight as follows: $X_{i}^{m} = X_{i}^{m - 1} + ϕ_{1} \oplus L e' υ y (ρ)$ (19) $L e' υ y (ρ) \sim τ = m^{- ρ}; (1 < ρ \leq 3)$ (20)

The second part of the modification method is used to increase the variety of the krill population. In this way, for each krill X _i, three krill X _k1, X _k2 and X _k3 are chosen from population such that i≠k ₁ ≠ k ₂ ≠ k ₃. By the use of mutation and crossover operators, the following two test solutions are generated: $X_{Test 1} = X_{k 1} + σ_{1} \times (X_{k 2} - X_{k 3})$ (21) $\begin{array}{l} x T e s t 2, j = \{\begin{array}{c} x_{i}^{b} & if Σ_{3} \leq Σ_{2} \\ x_{i} & otherwise \end{array} \\ X_{i} = [x_{i, 1}, x_{i, 2}, \dots, x_{i, d}] \\ X_{G b e s t} = [x_{1}^{b}, x_{2}^{b}, \dots, x_{d}^{b}] \end{array}$ (22)

In the above formulation, Equation (21) simulates the mutation operator from GA. Also, Equation (22) shows the crossover mutation between the best krill X ^b and X _i. This procedure will enable the solutions to search in the local area of the best krill. Finally, the solution with higher fitness function between X _Test1 and X _Test2 is compared with X _i and replaces it if it is morefit.

4 2m scheme point estimate method

The stochastic method is formed based on 2m PEM to undertake the uncertainty of the WT output power. Here, m shows the number of random variables of the problem. The main reason for choosing PEM is that it requires less computational effort to model the uncertainties with appropriate accuracy. Regarding the 2m scheme, it requires little information regarding the first few statistical moments of the random variable including mean value, variance value, skewness value, and kurtosis value. Considering the input random variables vector as z and the stochastic problem as a simple nonlinear function as F (z), the below simple formulation is considered: $S = F (z)$ (23)

The 2m PEM will first replace each random variable z _l with its probability density function (PDF) f _zl. The stochastic problem with m random variables is solved 2m times as follows: $S = F (μ_{z 1}, μ_{z 2}, \dots z_{l, k}, \dots, μ_{zm}); k = 1, 2$ (24)

Now, two sample points z _l,1 and z _l,2 are taken from f _zl matching its mean, variance and skewness coefficient as follows: $z_{l, k} = μ_{z_{l}} + ξ_{l, k} . σ_{z_{l}}; k = 1, 2$ (25) $ξ_{l, k} = \frac{λ_{l, 3}}{2} + (- 1)^{3 - k} \sqrt{m - (λ_{l, 3}^{2} / 2)^{2}}, k = 1, 2$ (26)

Skewness coefficient (λ _l,3) as the third central moment of PDF is calculated as below: $λ_{l, 3} = \frac{E [{(z_{l} - μ_{z_{l}})}^{3}]}{{(σ_{z_{l}})}^{3}}$ (27)

Each concentration point z _l,1 and z _l,2 have specific weighting factor ω _l,1 and ω _l,2 that determines their influence in determining the final solution. The values of ω _l,1 and ω _l,2 are determined as follows: $ω_{l, k} = \frac{1}{2 m}$ (28)

The standard deviation and expected value of the output variable S _i is evaluated as follows:

$\begin{matrix} σ & = & \sqrt{var (S_{i})} = \sqrt{E (S_{i}^{2}) - {[E (S_{i})]}^{2}} \\ E (S_{i}^{j}) & = & \sum_{l = 1}^{m} \sum_{k = 1}^{2} \\ (ω_{l, k} \times S_{i}^{j} (μ_{z 1}, μ_{z 2}, \dots, z_{l, k}, \dots, μ_{zm})) \end{matrix}$ (29)

5 Analytical results

This section is paid to assess the performance of the proposed method for solving the ED problem. In this way, two standard test systems are employed in two different cases of considering WT and neglecting WT. The first test system is the 10-unit thermal unit [32]. This test system includes valve-loading effect and multi-fuel options. Regarding the WT, the characteristics of the power curve is shown in Table 1. About the MLKH algorithm, the size of population is 30 and the termination is 1000 experimentally. In the first part of the simulation, WTs are neglected in the system.

The simulation results for the first test system are shown in Table 2. For better comparison, the simulation results of a number of other well-known algorithms are shown comparatively. It is worth noting that all simulations are repeated for 40 times and the results of the best solution, worst solution and average value of cost function are shown. As it can be seen, the proposed MLKH algorithm has shown superior performance than the other algorithms. Also, the results of the best, average and worst solutions show the high stability of the algorithm.

The second test system consists of 40 units with valve-loading effects. The complete data are taken from [13]. This test system has many local optimal that makes the optimization problem very hard. The simulation results are shown in Table 3. According to these results, the proposed MLKH algorithm has found more optimal solution that is not found by the other methods.

Until now, the WTs were neglected in the test systems. From now on, WTs are considered and uncertainty of power generation is modeled using the proposed stochastic method. In the 10-unit system, WTs are installed as the first and second units. This change in the system will force other power units to increase their generation for supplying load. In order to see the effect of uncertainty, the simulations in both deterministic and stochastic frameworks are demonstrated comparatively. Table 4 shows the simulation results. According to these results, the use of WT has increased the total cost of the system. Of course, this result could be guessed before. Since the optimal output power of WT is low, the other thermal units have increased their power which has resulted in incremental cost value. However, considering uncertainty has resulted in the lower expected cost value. Finally, Table 5 shows the optimal output power of units for both considering and neglecting WTs in the system.

About the 40-unit test system, WTs are placed in units 27, 28 and 29. Similarly, the analyses are implemented for both deterministic and stochastic frameworks and the results are shown in Table 6. The use of WT in this test system has resulted in reducing the total cost of the system. This event roots in the fact that WTs have replaced power units with low output power and thus has forced other units to reduce their power. This issue can be inferred from Table 7 easily. According to these results, the proposed stochastic method could solve the problem suitably.

6 Conclusion

This paper proposed a new stochastic framework based on 2m PEM and MLKH algorithm to assess the effect of considering WT in the optimal ED problem. The problem is formulated in a practical structure considering all mechanical and operational limitations.The simulation results on two standard test systems show the high search ability and satisfying performance of the method. According to the simulation results, the use of WT in the ED problem can result either in reduction or increasing of the cost function depending on the location of WT. Also, it was seen that considering uncertainty can affect the cost function value and help the system to be scheduled in more reliable situation. From the operation view, the proposed MLKH algorithm could reach better results than a number of other well-known methods in the area.

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