1. Introduction
The role of optimal generation scheduling of a thermal-renewable power generation system aiming economic and environmental benefits is vital in the current scenario of increasing power demand, escalating the fuel price and high pollution rate. The optimal generation scheduling of a hydro-thermal-wind system aims to distribute the power demand among the generating plants in such a way that the net energy cost and emission of pollutants are minimised, while satisfying the various constraints of power plants. Earlier, the combined operation of hydro-thermal plants was successfully scheduled to reduce the fuel cost, as well as the emission of pollutants. Now, with the development and readiness of new and cost-effective technologies, the penetration of wind power plants in the energy sector has become significant, necessitating its inclusion in the scheduling process. But, research publications describing the optimal generation scheduling of such hybrid energy sources are scanty.
The HTWGS is a non-linear optimization problem with two conflicting objectives. The first approach for solving the HTWGS problem is adopting calculus-based solution techniques (conventional method) such as linear programming [
1], dynamic non-linear programming [
2], lagrangian relaxation [
3], and differential dynamic programming [
4], etc. These methods are iterative techniques, containing composite mathematical expressions and long computational steps. Also, they have a limited space to address discrete and non-differentiable problems. The second approach is to adopt heuristic optimization methods which mimic the natural behaviour of certain things or physical phenomena of certain items. This approach gained wide popularity due to the easiness of implementation, adaptability in searching for the best solution, and ability to address non-linear optimization problems. It can be broadly classified as, evolutionary algorithms, swarm intelligence based methods, and algorithms based on the principle of other natural phenomena.
A comprehensive review of short-term hydrothermal scheduling (STHTS) using different classes of heuristic algorithms has been elaborated in [
5]. The Genetic Algorithm is applied to generate optimal generation scheduling of a short-term hydrothermal system in [
6,
7]. An evolutionary algorithm is employed for addressing STHTS in [
8,
9]. Following evolutionary algorithms, swarm intelligence-based optimisation algorithms received wide acceptance due to the lower number of computational steps and control variables involved. One popular algorithm in this category is particle swarm optimization (PSO). In [
10], the PSO method was proposed for STHTS of the multi-reservoir cascaded system. The predator prey optimisation (PPO) technique, which is an extended version of PSO, was suggested in [
11] as a solution for STHTS. There are several reports on the use of heuristic algorithms to optimize wind-thermal plant scheduling. An artificial bee colony optimization algorithm was employed for emission and economic dispatch on a wind–thermal system in [
12]. A modified particle swarm optimization algorithm influenced by the gravitational search method was adopted to effect emission level reduction in [
13]. Many other heuristic algorithms based on natural phenomena and the random optimization process such as Harmony search (HS) and JAYA algorithms have been reported. The HS algorithm [
14] mimics the improvisation procedure of an orchestra. A solution to STHTS using the HS algorithm was also proposed in [
15]. In [
16], a newly introduced population-based heuristic algorithm (JAYA algorithm) was applied for optimal power flow solution. The optimal generation scheduling of a hybrid system consisting of hydro-thermal-wind plants has not often been reported in the literature. The intermittent behavior of wind power is the main hurdle in the massive incorporation of wind plants into the hydrothermal system. Many researchers have addressed the unpredictable nature of wind power using fuzzy logic, neural network, and time series analysis, etc. The Weibull distribution function [
17] is suitable for modelling wind speed characteristics with minimum parameters. An incomplete gamma function term is used in [
18] to illustrate the wind power impact. In [
19], the stochastic wind power was considered as a constraint. The fluctuating nature of wind power can be considerably mitigated by the wind-hydro joint operation since hydropower can be altered rapidly. Some of the associated works on HTWGS were described in [
20,
21,
22,
23,
24]. Security constrained hydrothermal generation scheduling accounting for the discontinuity and uncertainty of wind power is addressed in [
20]. But, in this model, the emission of pollutants from the thermal plant is not accounted for. In [
21], the NSGA-III technique is used for computing the optimal allocation load among the hydro-thermal-wind power units. In this paper, the thermal power is modelled as a quadratic polynomial where only limited generator constraints are considered. Reference [
22] presents a bee colony optimization method for finding short-term economic/environmental HTWGS, incorporating wind power uncertainty, along with non-linear generator constraints, into the approach. A distributionally robust optimization method is proposed in [
23] for solving the hydro-thermal-wind economic dispatch problem. In this paper, the S-lemma method is used to incorporate the wind power uncertainty within a confined set. In [
24], the spinning reserve was considered and allocated between the hydro-thermal units to mitigate the challenges that occurred due to the uncertain nature of wind power during HTWGS.
This paper investigates the capability of selected algorithms representing different heuristic groups for searching for the optimal solution for HTWGS considering economic and environmental factors. Here, well accepted and suitable mathematical functions were chosen for addressing the generation cost, emission of pollutants, and water discharge. An improvement is proposed to conventional PSO, named as modified particle swarm optimization (MPSO), and employed to obtain hour-by-hour optimal generation scheduling of integrated hydro-thermal-wind power plants. An optimum solution was searched for using the proposed method (MPSO) and four other algorithms (BGA, PSO, IHS, and JAYA) with a trial system consisting of a multi-reservoir cascaded system with four hydro, three thermal, and two wind power plants. The two objective functions dealing within the problem, namely economic and emission, are of a conflicting nature. Therefore, a balanced optimal operating point was searched for by combining the two objectives and treating it as one function by means of a penalty factor. This approach reduced the computational burden and provided a better compromised solution. A comparison of the results obtained from the various algorithms used has been presented. Among the algorithms employed, the MPSO method exhibited a better performance and capability for searching for a more optimal solution in the test case.
The paper is organized as follows:
Section 2 presents the modeling of hydro-thermal-wind generation scheduling considering economic and emission factors.
Section 3 illustrates the outline of GA PSO, HS, and JAYA algorithms.
Section 4 gives a short description of the MPSO method.
Section 5 presents the computational steps of MPSO. The application of the proposed method in a test system and its results are discussed in
Section 6.
Section 7 summarizes the conclusions.
2. Hydro-Thermal-Wind Generation Scheduling Considering Economic and Environmental Factors
HTWGS deals with the optimal distribution of power demand among existing generation plants so as to reduce the overall generation cost and pollutant emission during the specified period, satisfying the power limit of plants and water constraint of hydro plants of the integrated generation system.
The total cost of generation comprises the coal cost of the thermal plant and rate of wind power only, since the hydro power cost is independent of generation output. Hence, the objective function to be minimized involves the generation cost of thermal and wind power plants and the emission of pollutants. This problem is basically a nonlinear constrained multi-objective optimization problem. The overall objective function is given by:
where
CT is the overall cost of the generation of thermal-wind plants,
FT is the total fuel cost of thermal plants,
WT is the wind power generation cost, and
ET is the net pollutant emission from thermal plants.
Subject to a number of equality and inequality constraints as follows:
System active power balance:
where
NT,
NH, and
NW are number of thermal, hydro, and wind power plants, respectively;
Pgj,τ is the power output of the
jth thermal power plant;
Phm,τ is the power output of the
mth hydro power plant;
wl,τ is the power output of the
lth wind power plant in the sub-interval
τ;
PD,τ is the load demand during the sub-interval
τ;
PLoss,τ is the transmission loss in the sub-interval
τ; and
T is the scheduling period.
The dynamic water balance in the reservoir:
where
Vhm,τ and
Qhm,τ are the storage volume and water discharge rate of the
mth hydro plant in the sub-interval
τ, respectively;
Ihm,τ and
Shm,τ are the inflow rate and spillage of the
mth hydro power plant in the sub-interval
τ, respectively;
Rum is the number of upstream hydro plants directly above the
mth hydro power plant; and
tlm is the water transport delay from reservoir
l to
m.
Initial and final reservoir storage volume:
where
Vhm,begin and
Vhm,end are the initial and final storage volume of
mth hydro plant, respectively.
Thermal power plant generation limit:
where
and
are the minimum and maximum power output of the
jth thermal power plant, respectively.
Hydro power plant generation limit:
where
and
are the minimum and maximum power output of the
mth hydro power plant, respectively.
Wind power plant generation limit:
where
wrl is the rated power output of the
lth wind power plant.
Reservoir storage volume and discharge limit:
where
,
and
,
are the minimum and maximum reservoir volume and water discharge of the
mth hydro plant, respectively.
The hydro units power output is expressed as a function of reservoir volume and head [
25] given by:
where
C1m,
C2m,
C3m,
C4m,
C5m, and
C6m, are the generation coefficients of the
mth hydro plant in the sub-interval
τ.
In the present work, the multi-objective HTWGS considering economic and emission factors is modified into a single objective optimization problem using a penalty factor [
26]. The penalty factor converts the emission to the indirect cost of emission and hence allows treating fuel costs and emission together. Thus, the total cost of the thermal system is the sum of the fuel cost and the indirect cost of emission. The penalty factor
hj is given by the equation:
Thus, the objective function (1) can be modified as:
The fuel cost function of the thermal plant is expressed as a quadratic function of the real power output [
27]. The valve-point effects are taken into account by incorporating a sinusoidal term in the cost function [
28]. Consider a grid system with
NH hydro,
NT thermal, and
NW wind power plants. The objective of the problem is to reduce the energy cost of the hydro-thermal-wind system through optimal generation scheduling considering economic and emission factors. The fuel cost function of thermal power plant is denoted by:
where
Fj(
Pgj,τ) is the fuel cost function of the
jth thermal power plant in the sub-interval
τ in
$/h.
Pgi,τ is the power output of the
jth thermal power plant in the sub-interval
τ in MW.
aj,
bj,
cj are the fuel cost coefficients and
hj,
ej are the coefficients of the valve point effect of the
jth power thermal plant.
is the minimum power output of the
jth thermal plant.
The pollutant emission from a coal-based power plant depends on the power output of that plant.
The total emission of pollutant
E can be expressed [
22] as:
where
αj,
βj,
γj,
ηj, and
δj are the coefficients of emission of the
jth thermal plant.
The total operating cost of a wind-powered generator consists of three components: (a) direct cost, (b) cost for not utilizing existing wind power (underestimation), and (c) overestimation cost [
13]. The cost function of a wind generator is formulated as:
where
Cd,l is the direct cost function of the wind power plant
l.
wl,τ is the scheduled wind power output of plant
l in the sub-interval
τ in MW.
Cu,l is the penalty cost function for underestimation and
Co,l is the penalty cost function for overestimation of the
lth wind power plant.
Wl,avl is the available power of the
lth wind power plant.
Direct cost is involved when the utility is purchasing the power from the wind farm, which is expressed as a linear cost function of actual power usage.
where
dl is the coefficient of direct cost of the
lth wind plant.
The underestimation and overestimation of wind power are mainly due to the uncertainty involved in the available wind power. The power output of a wind turbine depends on the blowing strength of the wind, which relies on many environmental parameters. Hence a reliable and accurate prediction of wind energy is difficult. In this paper, the uncertain nature of the wind generation is accounted for by a probability distribution function. The wind speed frequency distribution can supply a clear-cut picture about the wind speed pattern of a given location. Then, a proper statistical function can be fitted to express the wind speed distribution mathematically.
The penalty cost due to the underestimation of wind energy occurs when the available wind power is more than the predicted power (or actual wind power used), and the system operator should then pay a reasonable amount to the utility to compensate for the wastage of available wind power. Conversely, if the available wind power is less than the expected power (or actual wind power needed), then the system operator should purchase power from alternative sources or the load must be shut down.
The expression for penalty cost corresponding to underestimation and overestimation of wind power presented in [
17] was used in this work. The penalty cost function for underestimation of the wind power plant
l in the sub-interval
τ is expressed as a linear relation showing the difference between available wind power and actual wind power, and is given by:
where
ku,l is the cost coefficient of underestimation of wind power plant
l.
wr,l,τ is the rated wind power output of the unit
l in the sub-interval
τ.
fW(
w) is the probability density function (PDF) of wind power.
The penalty cost function for overestimation of the
lth wind power plant in the sub-interval
τ is given by:
where
ko,l is the cost coefficient of overestimation of the wind power plant
l.
Modelling of Wind Speed and Power
The numerical value of the underestimation and overestimation cost is obtained only by assuming a proper statistical function for the wind power output. Weibull distribution is the most popular distribution function, which closely follows the wind speed profile [
17,
19]. The Weibull probability density function is expressed as:
where
υ is the wind speed of the given location.
κ and
c are the shape parameter and scale parameter, respectively.
The wind turbine power output can be mathematically expressed [
22] as:
where
w is the power output of the wind turbine (kW or MW);
wr is the rated wind power output; and
υin,
υr, and
υo are the cut-in, rated, and cut-out wind speed, respectively.
Thus, the wind turbine power output is a combination of discrete and continuous random variables, ie, wind turbine power output is a discrete random variable between υr and υo and also a continuous random variable between υin and υr.
The Weibull probability distribution function can be obtained for three portions of wind power output, described in Equation (23).
The Weibull PDF for the continuous range of wind power output equation is expressed as:
where
and
.
6. Simulation Results
In this work, the two conflicting objectives are treated together using the penalty factor. The maximum penalty factor approach has been chosen for combining the fuel cost and emission; it offers an acceptable solution for the problem of emission and fuel cost.
The parameter setting is counted as the main limitation of any heuristic algorithm. Once the parameters are suitably chosen, the algorithm follows the logical pattern and converges to an optimal solution. In this study, the following values are assigned to the control parameters of each algorithm. The range of these parameter values is considered by observing similar published case studies, and the fine turning is done by a trial-and-error process.
MPSO and PSO parameters:
Swarm size (population) = 10
Learning factors, c1, c2 = 2.05
Maximum iterations = 500
wtmin = 0.4, wtmax = 0.9
Binary Coded GA parameters:
Size of Population = 60
Probability of crossover = 0.7
Probability of mutation = 0.1
Probability of elitism = 0.15
Maximum iterations = 500
Harmony Search parameters:
Harmony Memory Size (HMS) =10
Harmony Memory Consideration Rate (HMCR) = 0.85
Pitch Adjustment Rate (PAR): PARmin = 0.2, PARmax = 2
Bandwidth (bw): bwmin = 0.45, bwmax = 0.9
JAYA Algorithm parameters:
Size of Population = 10
Maximum iterations = 500
In this work, a test system consisting of a multi-stream cascaded hydro system with four hydro plants, three thermal plants, and two wind plants has been considered for investigating the feasibility and performance of the solution techniques. The schematic diagram of the hydro-thermal-wind test system is shown in
Figure 2. The HTWGS considering economic and emission factors has been conducted by implementing the algorithm based on conventional PSO, MPSO, Binary Coded GA, IHS, and JAYA algorithms. The simulations were executed in MATLAB 2015a platform. The program was run 30 times for the test case and the results were analyzed on the basis of the best, average, and worst case with standard deviation. The proposed MPSO shows competency and effectiveness in terms of solution quality and consistency of results.
Thermal system coefficients and constraints are taken from [
31]. The hydro system data is taken from [
25]. The scheduling period is taken as one-day, which is split into 24 numbers of a 1-h time span.
Figure 3 shows the system power demand curve. The wind system parameters are taken from [
18,
22]. All the necessary data of the hydro-thermal-wind system are shown in
Table A1,
Table A2,
Table A3,
Table A4 and
Table A5 in
Appendix A.
The optimal allocation of demand among the hydro-thermal-wind system and corresponding economic and emission values obtained from the best run are tabulated in the following tables.
Table 1 shows the optimal hydro-thermal-wind generation scheduling of the test system accounting for economic and emission factors obtained from the MPSO method.
Table 2 presents hourly water discharge and reservoir storage volume values. The storage volume satisfied the end conditions of each reservoir by adjusting the water discharge from each reservoir.
Table 3 shows the total fuel cost, emission, and wind penalty cost of the optimal generation schedule of the test system. Statistical analysis and comparison of performance of the proposed method (MPSO) with other heuristic algorithms (conventional PSO, BCGA, IHS, and JAYA algorithm) in terms of total fuel cost and emission are presented in
Table 4. The simulation results obtained using PSO, BCGA, IHS and JAYA methods are shown in
Tables S1–S12 in the Supplementary Materials.
The comparison of total fuel cost and emission shown in
Table 4 indicates that the MPSO method is capable of providing the optimal generation schedule. Also, the MPSO solution maintains the lower value of standard deviation, representing the consistency in the results compared with conventional PSO, BCGA, IHS, and newly introduced JAYA algorithms. To show a quantitative measure, here the MPSO solution is compared with the next best performing algorithm (conventional PSO). The total fuel cost and emission of pollutants by the MPSO algorithm are
$66,083.6629 and 63.4564 lb, respectively, whereas the PSO-based algorithm shows total fuel cost and emission values of
$68,646.8010 and 65.7942 lb, respectively. In other words, over the specified time schedule and demands, the proposed MPSO-based method attains an average reduction of 109.7974
$/h in generation cost and 0.0974 lb/h in emission of pollutant compared with the PSO-based algorithm. This quantitative comparison exhibits the efficiency of the MPSO algorithm for providing the optimal generation schedule accounting for economic and emission factors, without being trapped in the local minima.
Figure 4 shows the optimal load allocation among hydro, thermal, and wind plants of the test system over the 24-h time span. The thermal generation shows dominancy from 8.0 h to 20.0 h, because of the increased power demand on the system.
Figure 5 and
Figure 6 show the fuel cost and emission release of thermal plants over the scheduling period obtained by MPSO, PSO, BCGA, IHS, and JAYA algorithms. MPSO maintains a lower fuel cost and emission over the scheduling period.
Figure 7 and
Figure 8 show the hourly water discharge from the hydro plant and storage volume of reservoirs, respectively. The convergence characteristics of MPSO, conventional PSO, BCGA, IHS, and JAYA algorithms in terms of total fuel cost are shown in
Figure 9. The JAYA method exhibits an almost constant fuel cost in the beginning stage. The MPSO, conventional PSO, BCGA, and IHS methods exhibit a similar curve, but MPSO shows the lowest position.