Multi-objective glowworm swarm optimization for solving the optimal scheduling of thermal-wind power system

Abstract

This paper proposes the solution to an optimal scheduling problem of a thermal-wind power system. Here, the power output from a wind energy generator (WEG) is assumed to be schedulable, therefore the wind power penetration limits can be determined by the system operator (SO). The intermittent nature of wind power and speed is modeled using the Weibull density function. Here, 3 objective functions i.e., total operating cost, voltage stability enhancement index and system losses are selected. The total generation cost minimization objective has the cost of power generated from thermal and WEGs, under and over estimation costs of wind power. In the present paper, a multi-objective optimal power flow (MO-OPF) problems are framed by considering different objective functions simultaneously, and they are solved using the multi-objective Glowworm Swarm Optimization (MO-GSO) technique. The proposed optimization problem is solved on a modified IEEE 30 bus test system with two wind farms situated at two different buses in the system. The obtained simulation results show the suitability of proposed MO-OPF method for large scale power systems.

Keywords

Generation cost wind energy multi-objective optimization evolutionary algorithms transmission losses uncertainty voltage stability

Nomenclature

a_i, b_i, c_i	Fuel cost coefficients of i^th thermal unit.
N_obj	Number of objectives.
f_{p (p)}	Weibull probability distribution function.
C_p,wj	Penalty cost for j^th WEG.
n _t	Neighborhood threshold.
N_L	Number of demands
.r_s	Radial range of Luciferin sensor.
r _d	Local decision domain range.
C_r,wj	Reserve cost function for j^th WEG.
G _ij	Conductance of a line.
N _line	Number of lines in the system.
k_r,j	Reserve cost coefficient for j^th WEG.
k_p,j	Penalty cost coefficient for j^th WEG.
p_r	Rated wind power output.
P_Wj,avg	Available wind power (random variable).
ρ	Luciferin decay constant between 0 and 1.
γ	Luciferin enhancement constant.
β	Constant parameter.
δ _i	Voltage phase angle at i^th bus.
n_t	Threshold parameter.
v	Wind velocity (random variable).
x	Vector of decision variables.
J_j(t)	Objective function value at j^th agent location at time t.
t	Time (or step) index
d_ij(t)	Euclidian distance.
λ_j(t)	Luciferin level.
rⁱ_d(t)	Variable local-decision range.
r_s	Radial range of luciferin sensor.

1 Introduction

The electrical power grid is undergoing a transformation from a number of different perspectives. There is an increasing interest around the world in integrating higher levels of variable renewable energy resources (RERs) like wind and solar photovoltaic (PV) into electrical system. Wind power is uncertain and fluctuating. The power output from RERs is often uncertain, intermittent and uncontrollable. The ultra-high levels of RERs have a major impact on operation and planning of utility grids. However, the variable and uncertain nature of RERs will require grid operators to think about the best ways to provide operating reserves and in the case where there are organized market, they will need to understand the impacts of providing economic incentives for a variety of market products. The cost of power generation from RERs has decreased significantly and the large-scale penetration of renewable energy generators is introducing additional sources of uncertainty and variability into supply side of power system. In the modern power system, wind power can reach significant penetration and in the deregulated environment, the wind power introduces uncertainties in the transaction. Though, the electricity price is influenced by generation cost, but as wind power is environmental friendly that require to be used in optimum. However, the variability of wind power significantly affects the operation of power system [1].

Generally, the Optimal Power Flow (OPF) is a constrained and non-linear optimization problem. Aim of this OPF is to optimize an objective function (i.e., generation cost), and maintain the power outputs of generating units, shunt capacitors, bus voltages and tap settings of transformer within their secure limits. There have been excellent advancements made to the conventional optimization approaches like linear programming (LP), dynamic programming (DP) and nonlinear programming (NLP). However, they have the disadvantages of extremely limited capability to solve real-world power system optimization problems. To overcome the disadvantages of conventional optimization methods, meta-heuristic optimization techniques have been developed [2]. Regardless of the objective function, an OPF problem must be solved by ensuring the entire sets of power constraints are present and satisfied at the solution. OPF problem has many literature reviews from the past six decades. The classical OPF techniques include Gradient approach, Interior point (IP), Quadratic programming (QP), Newton and linear programming methods. The mathematical programming approaches, like QP [3], LP [4] and NLP [5] are used for the solution of OPF problem.

Reference [6] presents an economic dispatch (ED) for the power system with thermal, WEGs and the probability of intermittent wind power is considered as a constraint. An optimization algorithm to minimize the expected cost and emissions of the unit comment schedule for the set of scenarios is proposed in [7]. A method for the operation and planning of an energy storage system for a WEG is described in Reference [8]. Reference [9] proposes an OPF approach to dispatch the dispatchable power generation at optimum cost subjected to operational constraints on transmission and generation assets. Reference [10] formulates an AC unit commitment (UC) problem as a min-max-min optimization problem to represent the stochastic behavior of wind power outputs in terms of bounded intervals. A two-stage day-ahead UC approach and a rolling look-ahead ED method with the integration of concentrated solar plants and WEGs is presented in [11]. A stochastic ED model to assess the uncertainties in forecasts of wind and solar power outputs by considering a finite number of realizations obtained from a stochastic approach is proposed in [12]. Reference [13] presents an AC-OPF method to optimize system-level performance objectives while coping with uncertainty in wind, solar power outputs and demands. A joint scheduling approach of thermal, wind power outputs with energy storage system based on bi-level programming method is presented in References [14, 15].

An optimum scheduled power output of a WEG with forecasting and storage system is presented in Reference [16]. The objective of Reference [17] is to operate the wind-thermal power system with minimum cost while maintaining a voltage limits and reducing the system losses. Reference [18] proposes an optimum hourly schedule model of a system thermal, hydro and WEGs using the particle swarm optimization (PSO) algorithm. Reference [19] proposes an OPF approach for a thermal-wind generation system using an Evolutionary PSO approach. A general stochastic optimization and modeling framework for solving the wind integrated smart energy hub scheduling problem is proposed in [20]. A novel meta-heuristic hybrid optimization technique has been proposed in [21] for an environmental OPF problem considering wind and thermal generation. Reference [22] proposes an optimum generation scheduling problem for an electrical grid by including wind, thermal and solar photovoltaic modules.

To the best of the author’s knowledge, the MO-OPF considering wind power forecast uncertainties have not been proposed so far in the literature. The motivation of this paper is to bridge this existing gap. In this paper, the MO-OPF is performed by considering three conflicting objectives, i.e., total operating cost, system losses and voltage stability enhancement index, in the presence of wind forecast uncertainty. The reactive power capability of doubly fed induction generators (DFIGs) and synchronous generators are also considered in the proposed MO-OPF problem. Here, the OPF is framed by considering the factors involved due to the wind power forecast uncertainty, and swarm intelligence based Glowworm Swarm Optimization (GSO) algorithm is used for solving this proposed optimization problem. The determination of WEG power output is based on probability distribution function; where Weibull distribution is widely adopted for stochastic wind speed distribution. Here, the WEGs are represented by the doubly fed induction generators (DFIGs) and reactive power capability curve is derived for calculating the reactive power availability from the WEG. The OPF solution will also indicate the minimum real power requirement based on the wind variability at a particular location and the amount of reactive power to be installed in wind power plant to maintain the specified voltage profile in the system. The suitability of proposed MO-OPF problem is examined on a modified IEEE 30 bus - 6 generator system with WEGs situated at different buses in the system. This paper has a significant role in determining the best compromise solution between total operating cost, transmission losses and VSEI objectives for the electricitygeneration.

The remainder of this paper is organized as follows: Section 2 describes of proposed OPF problem formulation. Section 3 presents the description of Multi-Objective Glowworm Swarm Optimization (MO-GSO) algorithm for solving the MO-OPF Problem. Results and discussions are described in Section 4. Finally, the conclusions are presented inSection 5.

2 MO-OPF: Problem formulation

Here, three objective functions, i.e., total operating cost, system losses and voltage stability enhancement index are considered.

2.1 Objective 1: Generation/operating cost minimization

This objective is formulated as minimization of both operating cost of thermal and WEGs along with factor involved for over and/or under estimation of wind power (P_w). This objective function is formulated as [23, 24],

Minimize, J₁, i.e., $\begin{matrix} \sum_{i}^{N_{g}} C_{i} (P_{gi}) + \sum_{j}^{N_{w}} [C_{wj} (P_{wj}) + C_{p, wj} (P_{wj, avg} - \\ P_{wj}) + C_{r, wj} (P_{wj} - P_{wj, avg})] \end{matrix}$ (1)

The first term in Equation (1) is operating/generation cost of thermal generating units. Second term in Equation (1) is cost for the power drawn from WEG. This cost depends upon the ownership of wind farm. For example, if a wind farm is owned by SO itself, then this cost doesn’t exist in the Equation (1). Suppose, if the wind plant is owned by independent power producer (IPP), then SO has to pay for the power output from the wind farms. Third term associate with not using all the wind power that is available in the system. The fourth term is cost for the over-estimation of windpower.

Here, the quadratic cost function is used for determining the generation cost function of thermal generators, and it is represented by, $C_{Gi} (P_{Gi}) = c_{i} P_{Gi}^{2} + b_{i} P_{Gi} + a_{i}$ (2)

As mentioned earlier, there is no cost is involved to the SO, if the WEG is owned by itself. However, in a IPP owned WEG, the cost of wind power have to be paid according to the special contractual agreements. Suppose, the SO purchase wind power from a IPP, then he has to pay a fixed tariff to owner of wind power plant for the amount of wind power that is scheduled (P_wj). Here, a direct cost function is assumed for the P_wj, and it is expressed as [29], $C_{wj} (P_{wj}) = d_{j} P_{wj}$ (3)

In this paper, the two cost functions that are proposed in [25] are used for accounting the uncertainty of wind power. Cost for the under estimation concept of wind power is termed as penalty cost. This penalty cost function can also be related with the variance of the probability distribution, and it is normally produced above the scheduled value. This penalty cost is useful to find excess amount of power than the scheduled wind power. $\begin{matrix} C_{p, wj} (P_{wj, avg} - P_{wj}) = K_{P, j} (P_{wj, avg} - P_{wj}) = \\ K_{p, j} \int_{P_{wj}}^{P_{r, j}} (p - P_{wj}) f_{p} (p) dp \end{matrix}$ (4)

The second cost function is the cost due to amount of wind power that is less than scheduled one. It is termed as over-estimation cost. This function is used to find the deficit power that might produce from probability distribution function (PDF). It is expressed as, $\begin{matrix} C_{r, wj} (P_{wj} - P_{wj, avg}) = K_{r, j} (P_{wj} - P_{wj, avg}) = \\ K_{r, j} \int_{0}^{P_{wj}} (P_{wj} - p) f_{p} (p) dp \end{matrix}$ (5)

2.2 Objective 2: Active power/transmission losses minimization

Here, the minimization of total system losses is considered as an objective to be optimized for the reactive power optimization in the system. This objective function is expressed as [26],

Minimize, $\begin{matrix} J_{2} = \sum_{i = 1}^{N_{line}} {Loss}_{i} = \frac{1}{2} \sum_{i = 1}^{N} \sum_{\begin{matrix} j = 1 \\ j \neq i \end{matrix}}^{N} G_{ij} [V_{i}^{2} + V_{j}^{2} - \\ 2 V_{i} V_{j} \cos (δ_{i} - δ_{j})] \end{matrix}$ (6)

2.3 Objective 3: Minimization of voltage stability enhancement index (VSEI)

In this paper, to monitor the system voltage stability, the L-index [27] of demand buses is utilized. VSEI uses information from load flow solution, and is in the range of 0 (i.e., no load) to 1 (i.e., voltages collapse). VSEI of the system is formulated as sum of squared L-indices for a particular system operating condition, and it is formulated as [28],

Minimize, $J_{3} = VSEI = \sum_{j = N_{G} + 1}^{N} L_{j}^{2}$ (7)where $L_{j} = | 1 - \sum_{i = 1}^{N_{G}} F_{ji} \frac{V_{i}}{V_{j}} | j = N_{G} + 1, \dots, N$ (8)

The L-indices are calculated at all demand buses in the system, and it provides the proximity of system to voltage collapse. L-indices at each demand bus is calculated by using, $L_{j} = | 1 - \sum_{i = 1}^{N_{G}} \bar{F_{ji}} \frac{E_{i}}{E_{j}} |$ (9)where j = N_G+1,…,N. The values of $\bar{F_{ji}}$ are calculated by using admittance (Y) matrix, and they are expressed as, $[\begin{matrix} \bar{I^{G}} \\ \bar{I^{L}} \end{matrix}] = [\begin{matrix} \bar{Y^{GG}} & \bar{Y^{GL}} \\ \bar{Y^{LG}} & \bar{Y^{LL}} \end{matrix}] [\begin{matrix} E_{G} \\ E_{L} \end{matrix}]$ (10)

In the above equation, $\bar{I^{G}}$ , $\bar{I^{L}}$ , E_G, E_L are current and voltage at generator and demand buses, respectively. By rearranging Equation (10), we can obtain, $[\begin{matrix} E_{L} \\ \bar{I^{G}} \end{matrix}] = [\begin{matrix} \bar{Z^{LL}} & \bar{F^{LG}} \\ \bar{K^{GL}} & \bar{Y^{GG}} \end{matrix}] [\begin{matrix} \bar{I^{L}} \\ E_{G} \end{matrix}]$ (11)where $\bar{F^{LG}} = - {[\bar{Y^{LL}}]}^{- 1} [\bar{Y^{LG}}] .$

Both discrete and continuous control variables are considered in this paper. The continuous control variables include generator voltage magnitudes, generator real powers except slack bus power, whereas, the discrete controls include the switchable shunt devices and transformer tap settings.

2.4 Problem constraints

These include the equality and inequality constraints.

2.4.1 Equality constraints

They are the typical nodal power balance equations, and they are expressed as, $P_{Gi} - P_{Di} - V_{i} \sum_{j = 1}^{N} V_{j} (G_{ij} \cos δ_{ij} + B_{ij} \sin δ_{ij}) = 0$ (12) $Q_{Gi} - Q_{Di} - V_{i} \sum_{j = 1}^{N} V_{j} (G_{ij} \sin δ_{ij} + B_{ij} \cos δ_{ij}) = 0$ (13)

In (12) and (13), i = 1,2,3,…,N. In this paper, the solution of Newton-Raphson (NR) load flow is utilized for solving these equality constraints.

2.4.2 Inequality constraints

These are system operating constraints.

Generator Constraints: These are the generator voltage magnitudes (V_Gi), thermal generator’s active power outputs (P_Gi), active power outputs of WEGs (P_Wj) and generator reactive power outputs (Q_Gi) are limited by their minimum and maximum limits, and they are expressed as, $V_{Gi}^{\min} \leq V_{Gi} \leq V_{Gi}^{\max} i = 1, 2, \dots, N_{G}$ (14) $P_{Gi}^{\min} \leq P_{Gi} \leq P_{Gi}^{\max} i = 1, 2, \dots, N_{G}$ (15) $P_{Wj}^{\min} \leq P_{Wj} \leq P_{Wj}^{\max} i = 1, 2, \dots, N_{W}$ (16) $Q_{Gi}^{\min} \leq Q_{Gi} \leq Q_{Gi}^{\max} i = 1, 2, \dots, N_{G}$ (17)

Transformer Tap Constraints: Transformer tap settings have minimum and maximum limits, and they are expressed as, $T_{i}^{\min} \leq T_{i} \leq T_{i}^{\max} i = 1, 2, \dots, N_{T}$ (18)

Constraints on VAR sources: They are expressed as, $Q_{Ci}^{\min} \leq Q_{Ci} \leq Q_{Ci}^{\max} i = 1, 2, \dots, N_{C}$ (19)

Security constraints: These are the constraints on flow limits in transmission lines and voltage magnitude limits on load buses, and they are expressed as, $S_{Li} \leq S_{Li}^{\max} i = 1, 2, \dots, N_{line}$ (20) $V_{Li}^{\min} \leq V_{Li} \leq V_{Li}^{\max} i = 1, 2, \dots, N_{L}$ (21)

As mentioned earlier, in this paper the single objective OPF problems are solved by using evolutionary based Glowworm Swarm Optimization (GSO), and the proposed multi-objective optimization problems are solved using the MO-GSO. For the uncertainty modeling of wind speed and power, the reader may refer Reference [23] for the detailed explanation.

3 Multi-objective glowworm swarm optimization (MO-GSO) for OPF problem

There are three important aspects that are to be noted in the context of multi-objective optimization (MOO) problem. First, a MOO problem has two or more objectives that are need to be optimized simultaneously. Second, there may be constraints imposed on the objectives. Third, generally the objectives in MOO problem are conflict with each other; otherwise, a single solution may exist which may be obtained by optimizing the objectives in a sequential order. Suppose, a problem involving N variables (x₁, x₂, x₃,….., x_N), in a search space X⊂R^N, and M objectives (f₁(.),….., f_M(.)) in the objective function space Y⊂R^M, that are to be optimized, then the MOO problem is defined as,

Minimize, $f (x) = [f_{1} (x_{1}, \dots, x_{N}),, \dots . ., f_{M} (x_{1}, \dots, x_{N})]$ (22)

Subjected to, $h_{k} (x) \leq 0 k = 1, 2, 3, \dots, N_{ineq}$ (23) $g_{i} (x) = 0 j = 1, 2, 3, \dots, M_{eq}$ (24)

Equation (22) refers the objective function. Equations (23) and (24) refer the set of inequality and equality constraints. Let x, y∈X be the two vector inputs. We say x dominates y, if they satisfy the following: $(x, y) \in X, \forall i \in {1, 2, \dots {. M} | f}_{i} {(x) = f}_{i} (y)$ (25) $(x, y) \in X, \exists j \in {1, 2, \dots {. M} | f}_{j} {(x) = f}_{j} (y)$ (26)

A solution x is assumed as a non-dominated solution, if there is no other solution which satisfies the above conditions. The Pareto optimal set (P) projection in M dimensional objective space (Y) is termed as Pareto optimal front (F). $F = {(f_{1} (x), f_{2} (x), \dots . ., f_{M} (x) | x \in P}$ (27)

3.1 Performance measurements for MOO

To test the performance of different MOO techniques, various measures for quantitative comparison of results are required. A variety of performance measures have been proposed in Reference [29]. The goals of a good MOO algorithm are [30]:

The size of the non-dominated front should be maximized.

The non-dominated front found by the MOO technique should be as close as possible to the true optimal front.

The solutions should be as uniformly distributed as possible.

As mentioned earlier, here a Glowworm Swarm Optimization (GSO) algorithm is used for solving single objective optimization problem, and it is described next:

3.2 Description of GSO algorithm

GSO [31] is a novel swarm intelligent based optimization technique, and it is developed by D. Ghose and K.N. Krishnanad. The main motivation of GSO is derived from natural activities of glowworm’s during night. Usually, glowworms exercise in group, their inter-attraction and interaction among them by one’s luciferin. If a glowworm emits luciferin more light, and it can attract more glowworms and they move towards it [32]. GSO technique starts randomly by positioning the glowworms in workspace, to make them well dispersed. At begining, all glowworms have an equal quantity of luciferin. Every generation includes a movement and update phases based on a transition principle. These phases are described next:

Luciferin Update Phase: This phase is based on functional value available at the glowworm position. During this phase, every glowworm is added to its last luciferin level, and this luciferin quantity proportional to the measured value of a sensed profile at that specific point. This principle is expressed as, $ℓ_{j} (t + 1) = max {0, (1 - ρ) ℓ_{j} (t) + γ J_{j} (t + 1)}$ (28)

Movement Phase: In this phase, every glowworm follows a probabilistic approach to move towards a neighbor that has more luciferin value than its previous one. It means that they are attracted to the neighbors that glow much brighter. For the i^th glowworm, the probability of moving towards a j^th neighbor is expressed as, $Pj (t) = \frac{l_{j} (t)}{Σ_{k \in N_{i} (t)} l_{k} (t)}$ (29)where $k \in N_{i} (t) . N_{i} (t) = {j : d_{ij} {(t) < r}_{d}^{i} (t); ℓ_{i} (t) < ℓ_{j} (t)}$ . Suppose, the glowworm i select a glowworm jÎN_i(t) with Pj(t) expressed by Equation (29), then the discrete time model for the movements of glowworm can be expressed as, $X_{i} (t + 1) = X_{i} (t) + s (\frac{x_{j} (t) - x_{i} (t)}{| | x_{i} (t) - x_{i} (t) | |})$ (30)

The step size (s) is given by, $s = [\begin{matrix} δ & {ifd}_{ij} (t) \geq δ \\ d_{ij} (t) & else \end{matrix}$ (31)

Updating the local-decision range: When glowworms depend only on local information for deciding their movements, then it can be expected that the number of peaks captured would be a strong function of radial sensor range. This is because, the prior data about objective function is not available. To detect the multiple peaks, the sensor range must be made by using a varying parameter. For this, we associate each agent i, with a local-decision domain whose radial range (rⁱ_d) is dynamic in nature (0 < rⁱ_d≤rⁱ_s). The local decision range of i^th glowworm is expressed using, $r_{d}^{i} (t + 1) =$ (32) $min [r_{s}, max [0, r_{d}^{i} (t) + β (n_{t} - | N_{i} (t) |)]]$

The flow chart of MO-GSO for solving MO-OPF is depicted in Fig. 1.

Fig. 1

Flowchart of MO-GSO technique for solving MO-OPF.

The pseudo code for the proposed MO-GSO algorithm is presented in detailed manner in Reference [32].

4 Simulation results and discussion

The MO-GSO technique proposed in this paper is used to solve the MO-OPF approach and its application is demonstrated on modified IEEE 30 bus, 41 branch test system [33]. IEEE 30 bus system has 6 generating units, among them 4 units are considered to be thermal generating units situated at bus numbers 1, 2, 5 and 8; and the remaining 2 are considered to be WEGs situated at bus numbers 11 and 13. In this paper, 4 transformer tap settings are considered between the transmission lines (6–10), (6–9), (4–12) and (28-27); 9 bus shunt compensations are considered at the bus numbers 10, 12, 15, 17, 20, 21, 23, 24 and 29. Here, a total of 24 control variables are considered, and they are 3 thermal generating units active power outputs, 2 WEGs real power outputs, six generator bus voltages, four transformer tap settings and nine bus shunt susceptances. The objective here is to find the set of control variables, which optimize all the objective functions considered. The solution of OPF problem is depends on different factors like Weibull PDF shape, scale factors, penalty costs for under-estimation of the wind power and reserve costs for over-estimating. In the present paper, reserve cost coefficient (k_rj) is selected as 1$/MW and the penalty cost coefficient (k_pj) is considered as 5$/MW. The generator data considered for modified IEEE 30 bus test system is presented in Table 1.

Table 1
Thermal and wind generating units data for the modified IEEE 30 bus test system

Bus No.	Type of Generator	$P_{Gi}^{\min}$ (MW)	$P_{Gi}^{\max}$ (MW)	$Q_{Gi}^{\min}$ (MVAR)	$Q_{Gi}^{\max}$ (MVAR)	b ($/MWh)	c ($/MW² h)
1	Thermal	50.0	200.0	− 20.0	200.0	2.00	0.0037
2	Thermal	20.0	80.0	− 20.0	100.0	2.75	0.0175
5	Thermal	15.0	50.0	− 15.0	80.0	1.00	0.0625
8	Thermal	10.0	35.0	− 15.0	60.0	3.25	0.0083
11	Wind	10.0	30.0	− 10.0	50.0	3.00	0.0
13	Wind	10.0	30.0	− 15.0	60.0	3.00	0.0

The selected parameters for GSO algorithm are: γ is 0.95, ρ is 0.95, β is 0.0005, n_t is 4, r_s is 0.005 and r_d is 0.0005. To implement the MO-GSO, the considered population size is 200 and the size of Pareto optimal front is 30. Here, the strong dominated solutions (i.e., 30) are obtained from the total set of population (i.e., 200). A fuzzy based approach (i.e., fuzzy min-max approach) is considered in this paper to select the best-compromised solution from obtained Pareto optimal set. The maximum number of generations/iterations selected is 100. The amount of wind power penetration into the power system is defined as percentage of power generated from the WEGs to the total generation dispatch at a certain period of time.

In this paper, the OPF problem with various objectives is considered as a MOO problem. The problem is first treated as two objective optimization problem with total operating cost and system losses; total operating cost and VSEI/L-index as competing objectives. The OPF problem is then treated as three objective optimization problem with the mentioned three objectives as competing objectives. The following case studies are performed in this paper:

Case 1. Total generation cost and loss minimizations using MO-GSO algorithm.

Case 2. Total generation cost and VSEI minimizations using MO-GSO algorithm.

Case 3. Total generation cost, loss and VSEI minimizations using MO-GSO algorithm.

There is no need to optimize, the transmission losses and VSEI minimizations. Because, optimizing the transmission losses/VSEI inherently improves the other objective value (i.e., VSEI/losses).

4.1 Case 1: Total operating cost and transmission losses minimization using MO-GSO

In this case, the total operating cost and transmission losses are considered to be optimized simultaneously using the MO-GSO algorithm. Figure 2 depicts the Pareto optimal set for total generation cost and transmission losses minimization objectives using MO-GSO algorithm. Optimum control variables settings and the best-compromised solution obtained from the Pareto optimal front for Case 1 are presented in Table 1. The obtained best-compromised solution has total operating cost of 1012.30$/hr, and system losses of 7.9871 MW.

Fig. 2

Pareto optimal front obtained for total generation cost and transmission losses objectives for Case 1 using MO-GSO algorithm.

4.2 Case 2: Total generation cost and VSEI minimization using MO-GSO

As mentioned earlier, in this Case, the two objectives, i.e., total operating cost and VSEI are optimized at a time using MO-GSO algorithm. The Pareto optimal front for total operating cost and VSEI minimization objective functions is depicted in Fig. 3. Table 2 also shows optimum control variables settings and optimum values of objective functions for Case 2. The best-compromised solution obtained has total operating cost of 998.62$/hr and VSEI of 0.1634$/hr.

Fig. 3

Pareto optimal front of total generation cost and L-index for Case 2 using MO-GSO algorithm.

Table 2

Optimum control variables settings and best-compromised solution for Cases 1, 2 and 3 using MO-GSO

Control Variables and Objective Function Values	Case 1: Generation Cost and Losses Minimization	Case 2: Generation Cost and VSEI Minimization	Case 3: Generation Cost, Loss and VSEI Minimization
P_G1 (in MW)	135.01	188.02	140.35
P_G2 (in MW)	69.11	51.52	68.60
P_G5 (in MW)	43.05	23.88	42.44
P_G8 (in MW)	34.76	28.76	34.93
P_W11 (in MW)	29.92	30.00	29.96
P_W13 (in MW)	29.93	30.00	30.00
V_G1 (in p.u.)	1.0994	1.0607	1.10
V_G2 (in p.u.)	1.0888	1.0453	1.0918
V_G5 (in p.u.)	1.0647	1.0159	1.0582
V_G8 (in p.u.)	1.0670	1.0389	1.0741
V_W11 (in p.u.)	1.0970	1.0206	1.0882
V_W13 (in p.u.)	1.0994	1.0629	1.0853
T_6,9 (in p.u.)	0.9625	0.95	1.05
T_6,10 (in p.u.)	0.9500	0.9	0.9
T_4,12 (in p.u.)	1.0375	0.9625	1.0125
T_28,27 (in p.u.)	1.0375	1.0375	1.0125
b_sh10 (in p.u.)	0.05	0.05	0.05
b_sh12 (in p.u.)	0.02	0.01	0.02
b_sh15 (in p.u.)	0.04	0.03	0.04
b_sh17 (in p.u.)	0.01	0.04	0.03
b_sh20 (in p.u.)	0.02	0.03	0.02
b_sh21(in p.u.)	0.03	0.00	0.04
b_sh23(in p.u.)	0.05	0.00	0.03
b_sh24 (in p.u.)	0.05	0.05	0.02
b_sh29 (in p.u.)	0.02	0.04	0.00
Generation Cost (in $/hr)	1012.30	998.62	1034.02
System Losses (in MW)	7.9871	11.3101	8.3252
VSEI	0.1845	0.1634	0.1737

4.3 Case 3: Total generation cost, transmission losses and VSEI minimization using MO-GSO

In this Case, all 3 objective functions, i.e., total generation cost, transmission losses and VSEI are optimized simultaneously. Table 2 shows the optimum values of objective function and the optimum values of control variables for Case 3. The Pareto optimal front for Case 3 is depicted in Fig. 4. In this case, the obtained best-compromised solution has the total generation cost of 1034.02$/hr, transmission losses of 8.3252 MW and the VSEI of 0.1737.

Fig. 4

Pareto optimal front of three objectives (i.e., total generation cost, system losses and VSEI) for Case 3 using MO-GSO algorithm.

5 Conclusions

The present paper solves the multi-objective optimal power flow (MO-OPF) in a hybrid (wind-thermal) system using the multi-objective glowworm swarm optimization (MO-GSO) algorithm. Here, a methodology is proposed to include the WEGs in an OPF problem. The intermittent behavior of wind speed and power are modeled using Weibull PDF. In addition to classical cost minimization, the factors which account for under and over estimation of wind power are selected in this paper. Here, the GSO algorithm is used to solve the proposed problem. The solution of the OPF problem presented is depends on several factors involved like Weibull shape, scale factors, reserve cost for over estimating the wind power, and penalty cost for under estimating the wind power. In this paper, the WEG is represented by doubly fed induction generator and reactive power capability is derived for estimating the reactive power availability from the WEG. The simulation results for multi-objective OPF problems are presented on modified IEEE 30 bus system. These results show the suitability of proposed MO-GSO algorithm. The role of renewable power generation for the reduction in carbon emission, to meet the green house gas target and to receive credits/subsidy for local pollutant emission reduction is a scope for future work.

Footnotes

Acknowledgments

This research work is based on the support of “2018 Woosong University Academic Research Funding”.

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[Online]. Available: https://www2.ee.washington.edu/research/pstca/

Multi-objective glowworm swarm optimization for solving the optimal scheduling of thermal-wind power system

Abstract

Keywords

Nomenclature

1 Introduction

2 MO-OPF: Problem formulation

2.1 Objective 1: Generation/operating cost minimization

2.4.1 Equality constraints

3.2 Description of GSO algorithm

Table 1 Thermal and wind generating units data for the modified IEEE 30 bus test system

Footnotes

Acknowledgments

References

Table 1
Thermal and wind generating units data for the modified IEEE 30 bus test system