2020 2nd International Conference on Smart Power & Internet Energy Systems

Multi-Objective Optimal Reactive Power Dispatch

Considering Probabilistic Load Demand Along with
Wind and Solar Power Integration
M.U. Keerio1, Aamir Ali1*, Muhammad Saleem1, Noor Hussain1,
Riaz Hussain1,
1Department of Electrical Engineering, Quaid-E-Awam University of Engineering Science and Technology, Nawabshah Sindh, Pakistan

usmankeerio@quest.edu.pk, aamirali.bhatti@quest.edu.pk, just_salim72@quest.edu.pk, noorhussain@quest.edu.pk,

r.hmemon@quest.edu.pk,

Abstract—Fast growing of uncertain renewable energy swarm optimization based multi-objective (IPSOMO) [10].
sources (RES) in power system results difficult to control The perspectives of considering multiple uncertainties on the
reactive power. The purpose of optimal reactive power dispatch MO-ORPD issue have been explored in recent research. MO-
(ORPD) is to find the appropriate values of PV bus voltages, ORPD considering the integration of uncertain wind power
transformer tapings and shunt var compensation. For the has been examined in various papers. Enhanced firefly
solution to ORPD two conflicting objective functions, active algorithm (EFA) proposed in [11] to optimize both active and
power loss and voltage deviation are minimized simultaneously. reactive power, a scenario-based approach considered in [12-
Because of the stochastic behavior of wind and solar power 14], chance-constrained programming technique considering
generation, appropriate probability distribution functions are
uncertain nodal power injection and branch outages proposed
considered to model them with Monte-Carlo simulation
technique. Solution to multi-objective ORPD (MO-ORPD)
in [15]. Single objective optimization and weighted sum
problem, NSGA-II along with constraint technique is proposed. multi-objective optimization considering uncertain solar-wind
Furthermore, IEEE standard 30-bus system is adopted to find and load demand has been proposed in [16]. In summary, the
the superiority and effectiveness of NSGA-II. Two study cases conventional and probabilistic MO-ORPD regarding the
such as deterministic and stochastic (scenario-based) are integration of wind energy has been well examined in current
considered to analyze the simulation results. The obtained studies. On the other hand, solar energy is also plentiful and
simulation results show that the proposed algorithm has the almost everywhere, nowadays the solar PV has become an
ability to find the global optimal solution in all the scenarios. essential part of the smart grid. The solution of MO-ORPD
problem, which combines together wind and solar power,
Keywords—Optimal reactive power dispatch, Wind and solar would therefore, be important. As an alternative, the existence
power, NSGA-II, constraint handling techniques of several renewable energy sources enhances the difficulty of
the optimal power dispatch. Furthermore, with rising the
I. INTRODUCTION integration of RESs, reactive power management problem is
Reactive power plays a vital role for losses reduction and complex and causes excessive power loss and voltage failure
enhance voltages at all the buses in the transmission line. in the system [17].
ORPD is the most effective measure for both loss reduction
Therefore, this article mainly focuses solution of MO-
and voltage deviation (VD) of PQ buses. In recent years due
ORPD in view of the impact of both uncertain wind and solar
to the integration of renewable resources, ORPD requires
power integration along with uncertain load demand. For the
special attention. For example, an output power of the wind
generation of uncertain power and load demand, various
and solar photovoltaic (PV) sources is uncertain/probabilistic
probability distribution functions (PDFs) are utilized. Weibull
in nature may produce negative impacts such as excessive
and lognormal PDFs are considered to model the uncertain
power loss and unwanted voltage drops, if the PV bus
wind speed and solar irradiance respectively, whereas,
voltages, tap ratio of transformer, and reactive power of shunt
stochastic nature of the load is modeled by normal PDF [16].
capacitors are not within desirable limit. Single objective
For all of these uncertainties, 800 scenarios are formed by
optimization of ORPD considering conventional thermal
conducting the Monte Carlo simulation. The scenario
generators was extensively used in the literature. Numerous
reduction strategy proposed in [18] is applied to choose a
optimization methods such differential evolution (DE) in [1],
particular scenarios. In all the scenarios minimization of active
quasi oppositional DE (QODE) in [2], moth-flame
power loss and voltage deviations are considered the objective
optimization (MFO) [3], gravitational search algorithm (GSA)
functions. MO-ORPD problem is proposed to solve by using
[4] have been popularly used for the solution of single
NSGA-II [19] that is an outstanding multi-objective
objective deterministic ORPD problems. Moreover, the
evolutionary algorithm (MOEA). MO-ORPD is the
hybrid algorithms in [5, 6] show competitive output results as
constrained type problem, such as during optimization of
compared to individual optimization algorithms.
various constraints such as bus voltages, generator’s MVAr
Multi-objective ORPD (MO-ORPD) is other concern that and shunt VAr compensators, transmission line flow,
has attracted the consideration of authors in recent decades. transformer tap ratio must be with their allowable limit and
The most commonly used multi-objective functions are the balance active and reactive power. So, in order to increase the
combination of power loss, VD and enhancement of L-index superiority of NSGA-II method for finding the feasible
for better voltage stability. MO-ORPD considering solution, representative constraint handling techniques (CHT)
conventional thermal generators are discussed in NSGA-II must be integrated with NSGA-II. Moreover, most of the
[7], multi-objective GSA [8], hybrid fuzzy multi-objective research work considered the integration of the penalty
evolutionary algorithm (HFMOEA) [9], improved particle function approach with the MOEA for finding feasible

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solutions. In such an approach objective function of infeasible considered, in which 800 scenarios are reduced to 15
solutions are penalized and the penalty of penalizing objective representative scenarios.
function relies on constant parameter and that is chosen by the
time-consuming trial and error process. Limitation of penalty TABLE I. PARAMETERS AND PDFS OF UNCERTAIN LOAD, WIND
SPEED, AND SOLAR IRRADIANCE
function approach is that it either delaying the optimization
process during over explore the infeasible solution when small Uncertainty PDF Parameters
penalty co-efficient is added with the infeasible objective Load Normal Mean (µL=90); SD (σL=10)
function or may not explore the infeasible region when large Wind speed Weibull Shape (aw=9); Scale (bw=2)
penalty co-efficient is added [20]. The author in [21] utilized Solar irradiance Lognormal Mean (µGir=5.5); SD (σs=0.5)
several representative CHTs that include feasibility rule, In each scenario the % loading is multiplied with the nominal
penalty less and adaptive trade-off model (ATM) for finding load of the 30-bus system. However, actual wind speed (vw)
the feasible solution. And in this paper, the proposed MO- and solar irradiance (Gir) are used to calculate the actual power
ORPD problem is solved by using NSGA-II with the of wind turbine (Pw) and solar PV (PS) using Eq. (4) and (5).
integration of ATM representative CHT in order to explore
and exploit the feasible search space [21]. Furthermore, two 0, 𝑖𝑓𝑣𝑤 ≤ 𝑣𝑤,𝑖𝑛 𝑎𝑛𝑑 𝑣𝑤 ≥ 𝑣𝑤,𝑜𝑢𝑡
study cases without integration of wind and solar generation 𝑣𝑤 −𝑣𝑤,𝑖𝑛
considering standard IEEE 30-bus. 𝑃𝑤 (𝑉𝑤 ) = {𝑃𝑤,𝑟 (𝑣 ) , 𝑖𝑓 𝑣𝑤,𝑖𝑛 ≤ 𝑣𝑤 < 𝑣𝑤,𝑟 ()
𝑤,𝑟 −𝑣𝑤,𝑖𝑛

The remainder of the paper comprised as follows. Section 𝑃𝑤,𝑟 , 𝑖𝑓 𝑣𝑤,𝑟 < 𝑣𝑤 ≤ 𝑣𝑤,𝑜𝑢𝑡
II contains the problem formulation of MO-ORPD problem In this paper, the parameters of Enercon E82-E4 wind turbine
that includes modeling of uncertain RES and demand, model are considered. Furthermore, a wind farm consists of
generation and selection of representative scenarios and 25 turbines, rated power (Pw,r) of each turbine is 3MW,
formulation of objective functions and constraints. Whereas, whereas, vw,in, vw,out and vw,r respectively are the cut-in, cut-out,
Section III deliberates the NSGA-II algorithm and ATM and rated wind speed.
constraint handling technique. Section 4. provides the
investigation of simulation results. Section 5 gives the 𝐺2
𝑖𝑟
𝑃𝑠,𝑟 ( ) 𝑓𝑜𝑟 0 < 𝐺𝑖𝑟 < 𝐺𝑖𝑟,𝑐
conclusion of this work. 𝑃𝑆 (𝐺𝑖𝑟 ) = {
𝐺𝑖𝑟,𝑠𝑡𝑑 ×𝐺𝑖𝑟,𝑐
()
𝐺
II. PROBLEM FORMULATION 𝑃𝑠,𝑟 ( 𝑖𝑟 ) 𝑓𝑜𝑟 𝐺𝑖𝑟 ≥ 𝐺𝑖𝑟,𝑐
𝐺𝑖𝑟,𝑠𝑡𝑑
In this section, uncertain renewable generation (wind and The rated power (Ps,r) of solar PV unit is 50 MW, whereas, the
solar PV) and load are modeled first than formulation of multi- other parameters Gir,std and Gir,c measured in W/m2
objective ORPD (MO-ORPD) and constraints are discussed. respectively are the solar irradiance in the standard and certain
A. Modeling of Uncertain Demand and Generation environment.
In the field of an electric power system, load demand is B. Formulation of MO-ORPD Problem
always uncertain in nature and hence the generation to meet Mathematically, the constrained MO-ORPD problem
the uncertain load. Further complexity of power system expressed as:
planning is increased with the integration of uncertain solar
PV and wind. Therefore, in this paper, appropriate PDFs are 𝑚𝑖𝑛 𝐹(𝑥) = (𝑓1 (𝑥), 𝑓2 (𝑥), 𝑓𝑚 (𝑥))𝑇
used to model the demand and output power of renewable 𝑔(𝑥) ≤ 0, ℎ(𝑥) = 0, 𝑥 = (𝑥1 , 𝑥2 , . . 𝑥𝐷 )𝑇 ∈ 𝑆 ()
generation. Typically, normal [14], Weibull and lognormal
[16] PDFs have been used for the load, wind speed (vw) and Where 𝐹(𝑥) is comprised of m objective function, 𝑥 is the
Solar irradiance (Gir) respectively, expressions of such PDFs decision vector, g(𝑥) and ℎ(𝑥) are the inequality and equality
are, constraints. From Eq. (6) decision vector of ORPD problem
(𝑃 −𝜇 )
[− 𝐿 𝐿 ]
2 consists of generator voltages (VG), tap ratio of transformers
1
Normal 𝛥𝐿 (𝑃𝐿 ) = 𝑒 2𝜎 2
𝐿 () (Tk) and reactive power of VAR compensating devices (Qc)
𝜎𝐿 √2𝜋
and given as:
𝑣 𝛽
𝑣𝑤 (𝑏𝑤 −1) [−( 𝑤 ) ]
Weibull 𝛥𝜈 (𝑣𝑤 ) = (
𝑏𝑤 𝑎𝑤
() 𝑥 = [𝑉 , ⋯ , 𝑉𝐺,𝑁𝐺 , ⏟ 𝑇𝑘,1 , 𝑇𝑘,𝑁𝑇 𝑇
𝑄𝑐,1 , 𝑄𝑐,𝑁𝐶 , ⏟ ()
𝑎𝑤
)(
𝑎𝑤
) 𝑒 [ ⏟𝐺,1 ]
𝑉𝐺 𝑄𝑐 𝑇𝑘
−( ln 𝐺𝑖𝑟 −𝜇𝐺𝑟 )2
1 [
2𝜎2
] The two objective functions power loss (PTloss) and VD [2] for
Lognormal 𝛥𝐺 (𝐺𝑖𝑟 ) = 𝑒 𝑠 ()
𝐺𝑠 𝜎𝑠 √2𝜋 the optimal solution calculated as
Where, parameters of normal, Weibull and lognormal PDFs 𝑃𝑇𝑙𝑜𝑠𝑠 = ∑𝑛𝑙 2 2
𝑘=1 𝐺𝑘(𝑖𝑗) [𝑉𝑖 + 𝑉𝑗 − 2𝑉𝑖 𝑉𝑗 cos (𝛿𝑖𝑗 )] ()
proposed in this paper are given in Table I. A very simple
approach of scenario generation implemented in [12, 13], in where 𝐺𝑘(𝑖𝑗) is the shunt conductance of kth line between bus i
which the typical forecasted load uncertainty distributed into and j.
three intervals (specific scenarios) and Weibull PDF has been
utilized for the uncertain wind speed generation of five 𝑉𝐷 = (∑𝑁𝐿
𝑝=1 | 𝑉𝐿𝑝 − 1|) ()
intervals. Therefore, total 15 scenarios were chosen with all
possible combinations of uncertain load and wind generation. Where 𝑉𝐿𝑝 is the PQ bus voltage and 𝑁𝐿 is the number of
If a similar technique is utilized in this work, total of 75 load buses. The equality constraints hj(x) in Eq. (6) can be
(3*5*5) scenarios for each wind, solar and uncertain load were defined as:
created and it is impractical to handle them. Therefore, in this 𝑃𝐺𝑖 − 𝑃𝐷𝑖 = 𝑉𝑖 ∑𝑁𝐵 (10)
𝑗=1 𝑉𝑗 [𝐺𝑖𝑗 cos (𝛿𝑖𝑗 ) − 𝐵𝑖𝑗 sin (𝛿𝑖𝑗 )]
work, first 800 scenarios of each renewable generation and
load are created by Monte-Carlo technique, a histogram of all 𝑄𝐺𝑖 − 𝑄𝐷𝑖 = 𝑉𝑖 ∑𝑁𝐵
𝑗=1 𝑉𝑗 [𝐺𝑖𝑗 sin (𝛿𝑖𝑗 ) − 𝐵𝑖𝑗 cos (𝛿𝑖𝑗 )] ()
scenarios considering 150 bins are shown in “Fig. 1”. After
that scenario reduction techniques same as in [16] is

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Where 𝐵𝑖𝑗 is the susceptance and 𝑃𝐷𝑖 and 𝑄𝐷𝑖 are the real and • Generator constraints:
reactive power demand. The number of inequality constraints
gj(x) given as:

Fig. 1. Uncertain distribution of percentage of load, wind speed and solar irradiance

𝑉𝐺min
𝑖
≤ 𝑉𝐺𝑖 ≤ 𝑉𝐺max
𝑖
∀𝑖 ∈ 𝑁𝐺 () Whereas, the overall degree of constrained violation
calculated as:
𝑄𝐺min ≤ 𝑄𝐺 ≤ 𝑄𝐺max ∀𝑖 ∈ 𝑁𝐺 () 𝑗
𝑖 𝑖 𝑖
𝐺(𝑥) = ∑𝑘=1 𝐶𝑗 (𝑥) ()
• Transformer constraints:
A solution x is said to be feasible if 𝐺(𝑥) = 0, else it is an
𝑇𝑗min ≤ 𝑇𝑗 ≤ 𝑇𝑗max ∀𝑗 ∈ 𝑁𝑇 () infeasible solution. ATM divides the current population into
three situations such as all the solutions are infeasible, all
• Shunt compensator constraints: feasible solutions and the combined infeasible and feasible
𝑄𝑐min ≤ 𝑄𝐶 ≤ 𝑄𝐶max ∀𝑘 ∈ 𝑁𝐶 () solutions. During each situation, ATM handles the constraint
𝑘 𝑘 𝑘 violation with a different technique.
• Security constraints: • All the solutions are infeasible (ATM1):
𝑉𝐿min
𝑝
≤ 𝑉𝐿𝑝 ≤ 𝑉𝐿max
𝑝
∀𝑝 ∈ 𝑁𝐿 In this situation, ATM converts the constraint violation
additional m+1th unconstrained objective function then non-
𝑆𝑙𝑞 ≤ 𝑆𝜄max ∀𝑞 ∈ 𝑛𝑙
𝑞 dominated sorting [19] is applied. Later on, select the first half
In this work IEEE 30-bus system is incorporated into two of the individual with a less constrained violation in the first
study cases such as deterministic (single solution of ORPD layer and deleted them from the population. The same process
called a base case) and stochastic (multiple solutions of is applied and continued on the remaining population until the
various scenarios) are adopted. In deterministic study case desired population number is achieved.
continuous rating of transformer tap ratio (Tk) and shunt VAR • All the solutions are feasible (ATM2):
compensator (QC), however, in probabilistic case study
discrete values of Tk and QC are selected. In this situation G(x)=0, non-dominated sorting [19] is applied
for all the population.
III. NSGA-II AND CONSTRAINT HANDLING TECHNIQUE
• Both feasible and infeasible solutions (ATM3):
In MOEAs, the comparison of each individual fitness is
explicitly based on Pareto dominance. Pareto dominance[22] In this condition current population is divided into two groups
can be defined by consider f1(x), f2(x), …., fm(x) m objective such as feasible (Za) and infeasible (Zb). In this situation
functions and assume that two solutions xa and xb in which xa transformed the objective function and penalize the objectives
is non-dominated by xb (𝑥𝑎 ≺ 𝑥𝑏 ) if the following both the of infeasible solutions according to feasibility proportion
conditions are true: (𝜑 = 𝑍𝑎 /𝑁𝑝 ) as:
𝑓𝑖 (𝑥𝑎 ) ≤ 𝑓𝑖 (𝑥𝑏 ), ∀ 𝑖 ∈ {1,2, … , 𝑚} 𝑓𝑖 (𝑥𝑗 ), 𝑖𝑓 𝑥𝑗 ∈ 𝑍𝑎 ;
𝑓𝑖′ (𝑥𝑗 ) = { ()
𝑓𝑗 (𝑥𝑎 ) < 𝑓𝑗 (𝑥𝑏 ), ∃ 𝑎𝑡𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑗 ∈ {1,2, … , 𝑚} max[𝜑 × 𝐴 + (1 − 𝜑) × 𝐵, 𝑓𝑖 (𝑥𝑗 )] , 𝑖𝑓 𝑥𝑗 ∈ 𝑍𝑏

All the solutions in the search space satisfying both the above Where, 𝐴 = 𝑚𝑖𝑛 𝑓𝑖 (𝑥𝑗 ) and 𝐵 = 𝑚𝑎𝑥 𝑓𝑖 (𝑥𝑗 ).
𝑥𝑗 ∈𝑍𝑎 𝑥𝑗 ∈𝑍𝑎
conditions are called Pareto set and objective functions
considering Pareto set are called Pareto front (PF). In the The final fitness in ATM is obtained by addition of
following sub-sections first the constraint technique then normalized the objective functions ( 𝑓𝑖̅ ) and constrained
NSGA-II is described. violation (𝐺̅𝑖 ) according to Eq. (19):
A. Adaptive trade-off Model (ATM) Constraint Technique 𝐹𝑖 (𝑥𝑗 ) = 𝑓𝑖̅ (𝑥𝑗 ) + 𝐺̅𝑖 (𝑥𝑗 ) ()
The global optimal solutions of MO-ORPD problem given
B. NSGA-II
in Eq. (6), it is desirable to handle the feasible and infeasible
solutions efficiently. Typically, the jth constrained violation is In this work, one of the most efficient MOEA based on
calculated as: Pareto dominance, NSGA-II [19] along with an adaptive
trade-off model (ATM) [21] constraint technique is proposed.
max (𝑔𝑗 (𝑥), 0) Steps of NSGA-II algorithm with the integration of ATM for
𝐶𝑗 (𝑥) = { ()
max (|ℎ𝑗 (𝑥)| − 𝛿, 0) finding the MO-ORPD is as shown in “Fig. 2”. There are three
steps are used in any generation of NSGA-II.
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Fig. 2. Proposed Methodolog

• Coding: 1 𝑓𝑜𝑟 𝑓𝑚𝑘 ≤ 𝑓𝑚𝑚𝑖𝑛

𝑓𝑚𝑚𝑎𝑥 −𝑓𝑚
𝑘
A real coded scheme has been utilized, when dealing with 𝑘
𝜇𝑚 = 𝑚𝑎𝑥 𝑚𝑖𝑛 𝑓𝑜𝑟 𝑓𝑚𝑚𝑖𝑛 < 𝑓𝑚𝑘 < 𝑓𝑚𝑚𝑎𝑥 ()
𝑓𝑚 −𝑓𝑚
continuous search space along with large dimensions of
decision vector within the upper and lower bound. However, {0 𝑓𝑜𝑟 𝑓𝑚𝑘 ≥ 𝑓𝑚𝑚𝑎𝑥
for the mixed-integer problems, around operator is used. Where subscript m and superscript k respectively are the
• Genetic operators: indexes of objective functions and non-dominated solutions,
𝑘
𝜇𝑚 is membership function; 𝑓𝑚𝑘 is the objective function.
Crossover and mutation are the genetic operators, for the Normalized membership (𝜇 𝑘 ) function is computed by
real coding scheme, a crossover is done by simulated binary 𝑀 𝜇𝑘
𝛴𝑚=1
crossover whereas normally distribution mutation operator 𝜇𝑘 = 𝑁𝑑 𝑀
𝑚
()
𝑘
is employed for the mutation. 𝛴𝑘=1 𝛴𝑚=1 𝜇𝑚

• Selection Out of all non-dominated solutions (Nd), a larger value

of 𝜇 𝑘 is the best compromise solution. “Fig. 3” shows the
After the genetic operator combines the parent and final non-dominated solutions for the study case 1 of all the
offspring population and apply the non-dominated sorting six runs. For finding the best PF out of six PFs, an HV
principle according to feasibility proportion. Moreover, the indicator is used to find the single PF of 600 population size.
parameters of proposed NSGA-II algorithm for proposed After that fuzzy satisfying approach given in Eq 1 is utilized
cases are given in Table II. to find the best compromise solution. The NSGA-II method
TABLE II. PARAMETERS OF NSG-II converges to find the optimal PF over 100 generations.
Further, the decision vector and objective functions of the
Name Symbol Value
best compromise solution are shown in Table III. Table
Population size NP 600
Dimension of decision vector D 19 reveals that the lowest loss obtained in Case 1 is 5.1830
Maximum generation Max_gen 500 MW, whereas the voltage deviation value is 0.1543 p.u
Crossover probability Pc 0.9 subject to satisfy all the constraints. Also, the values of all
Mutation probability Pm 1/D the decision variables within their allowable limits.
Cross over distribution index dc 20 However, some of the values of reactive power near the max
Mutation distribution index dm 20
limit, therefore decision-maker must be carefully select the
optimal solutions. Also, the load bus voltage lies with the
IV. SIMULATION RESULTS AND ANALYSIS
short band of limits. The second objective function will not
In the following subsection simulation results of both lead to under or over-voltage issues, due to this system
cases are discussed and analyzed. IEEE standard 30-bus test voltage level force to near unity. Nevertheless, this may also
system is considered to implement the proposed NSGA-II affect on generators or VAR compensating devices to the
algorithm considering uncertain load and wind and solar degree that they reach the maximum limits of reactive
generation. The data of IEEE 30-bus test system is taken capacity. Furthermore, supply and compensation of reactive
from [16]. power in the power system depend on several other factors
A. Case 1: Deterministic MO-ORPD such as constant parameters of a transmission line, load bus
voltages type and location of load. However, in this work,
Case 1 is run for six times, six Pareto fronts are obtained NSGA-II with ATM constraint handling technique finds the
and the best PF is selected by using hypervolume (HV) solution within feasible search space where the supply of
performance indicator [23]. The final compromise solution reactive power must lie within desirable limit.
from the PF is selected by using the fuzzy set theory [24].
The membership function of the fuzzy set theory are:

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 units. Due to the stochastic nature of RES, the active and
reactive power capacity of a thermal generator of swing bus
is increased, because even at a smaller generation of wind
or solar PV, conventional generators can meet the demand.
Table IV gives results of all the 15 representative scenarios.
Table indicates that the odds of different scenarios are
practical, PV plant produces almost zero output power with
a 50 percent chance (scenario 14). Naturally, power
produced by wind and solar PV are a complement, both the
sources do not generate maximum or minimum power
simultaneously. In this work, expected power loss and
expected VD are computed for the selection of optimal
solutions for all the scenarios. The 𝐸𝑃𝐿 achieved as shown
in Table IV is 8.9726 MW and 𝐸𝑉𝐷 is 0.2163 p.u.
Fig. 3. PF of case 1 for the IEEE 30-bus systems
Moreover, the PFs, best compromise solutions and expected
TABLE III. SIMULATION RESULTS OF CASE1
values of all the scenarios are shown in “Fig. 4”. The
expected values of objective functions are near scenario 8
Parameters Min Max Case 1 and 15. Among all the scenarios the smallest values of both
V1 (p.u.) 0.95 1.1 1.0452 the objective functions appeared when the percentage of
V2 (p.u.) 0.95 1.1 1.0344 loading is minimal (scenario 5). Low loading means the
V5 (p.u.) 0.95 1.1 1.0088
V8 (p.u.) 0.95 1.1 1.0103
minimum load current (thus low loss) in the system. On the
V11 (p.u.) 0.95 1.1 1.0465 other hand, with the non-available solar power and a small
V13 (p.u.) 0.95 1.1 1.0236 amount of wind power at the critical loading condition, both
T11 (p.u.) 0.9 1.1 1.0784 the objective functions are the highest such as in scenarios
T12 (p.u.) 0.9 1.1 0.9097 14 and 15.
T15 (p.u.) 0.9 1.1 1.0183
T36 (p.u.) 0.9 1.1 0.9729 V. CONCLUSION
QC10 (MVAr) 0 5 4.1969
QC12 (MVAr) 0 5 1.5019 In this paper, an outstanding optimization algorithm
QC15 (MVAr) 0 5 4.9795 NSGA-II along with the ATM constraint technique is
QC17 (MVAr) 0 5 3.0783 proposed to solve MO-ORPD problem considered
QC20 (MVAr) 0 5 4.5764
QC21 (MVAr) 0 5 4.0903 deterministic (base case) and stochastic (scenario-based)
QC23 (MVAr) 0 5 4.3077 cases. In scenario-based MO-ORPD, realistic 800 scenarios
QC24 (MVAr) 0 5 4.9828 are created by applying the Monte Carlo technique
QC29 (MVAr) 0 5 2.1328 considering the appropriate PDFs for the uncertain load,
Ploss (MW) 5.1830 wind and solar PV. Moreover, scenario reduction technique
VD (p.u.) 0.1543
QG1 (MVAr) -20 150 2.5123
is applied to select the 15 most appropriate scenarios.
QG2 (MVAr) -20 60 15.332 Selected scenarios clearly show that these are the
QG5 (MVAr) -15 62.5 22.375 representation of entire scenarios and are close to them. In
QG8 (MVAr) -15 48.7 28.546 order to show the performance of proposed algorithm IEEE,
QG11 (MVAr) -10 40 23.844 30-bus system is considered to minimize both conflicting
QG13 (MVAr) -15 44.7 10.186
objective functions such as active power loss and VD.
B. Case 2: Probabilistic MO-ORPD Simulation results show that NSGA-II along with ATM
obtained well-distributed Pareto Front that give good trade-
In this case, the conventional thermal generators at bus off between power loss and VD.
5 and 8 are replaced with the stochastic wind and solar PV
TABLE IV. MO-ORPD CASE STUDIES WITH UNCERTAIN DEMAND AND RENEWABLE POWER
Scene # % of Load vw Gir Probability (ρsc) Pw PS Ploss VD
1 91.422 2.7065 893.167 0.00125 0 44.658 6.15055915608834 0.145013237154246
2 95.119 7.3743 472.410 0.00875 25.236 23.620 5.85981631813363 0.148897368089472
3 94.845 8.5426 1542.92 0.00125 31.976 50 4.04475662289920 0.123875625693577
4 90.676 8.4884 1332.64 0.00125 31.664 50 3.41584709580112 0.134604813532246
5 70.980 3.8557 763.470 0.00875 4.937 38.1735 2.80214533430777 0.0858537042744566
6 95.037 9.6897 531.763 0.01000 38.595 26.588 4.69170560905451 0.136095288737168
7 85.935 25.026 640.249 0.00250 0 32.01245 5.60879957742659 0.211580160576356
8 100.380 5.5907 176.900 0.12250 14.946 8.8450 9.08036732451477 0.174235993312822
9 97.759 8.1229 233.578 0.09125 29.555 11.678 6.89581714977921 0.132969244246257
10 96.904 9.4537 588.719 0.01500 37.233 29.435 4.97667263943906 0.134823070337390
11 89.165 2.0243 403.591 0.03250 0 20.179 6.99211073549432 0.208455581594288
12 92.444 6.1759 111.120 0.09625 18.322 5.144 7.07502552612887 0.180581540325209
13 92.788 3.1938 300.723 0.12000 1.1181 15.036 8.01518463005242 0.206966656040823
14 101.764 4.8502 0 0.48750 10.674 0 10.5053369526490 0.260699491065041
15 115.716 7.3945 680.931 0.00125 25.353 34.046 9.00606309753776 0.214567789522528
𝑁𝑆𝐶

Expected power loss 𝐸𝑃𝑇𝑙𝑜𝑠𝑠 ∑ 𝜌𝑆𝐶 × 𝑃𝑇𝐿𝑜𝑠𝑠,𝑆𝐶 8.9726 MW

𝑠𝑐=1
𝑁𝑆𝐶

Expected VD 𝐸𝑉𝐷 ∑ 𝜌𝑆𝐶 × 𝑉𝐷𝑆𝐶 0.2163 p.u

𝑠𝑐=1

506

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Fig. 4. Final PF of all the scenario

Furthermore, various extreme and critical load and multi-objective model at the presence of uncertain wind power
generation scenarios are considered for the analysis and generation," Iet Generation Transmission & Distribution, vol. 11,
no. 4, pp. 815-829, Mar 9 2017.
comparison purpose in order to select the single feasible
[13] S. M. Mohseni-Bonab, A. Rabiee, and B. Mohammadi-Ivatloo,
decision vector for all the scenarios which is selected by "Multi-objective Optimal Reactive Power Dispatch Considering
using expected values of objective functions. Uncertainties in the Wind Integrated Power Systems," in Reactive
Power Control in AC Power Systems: Fundamentals and Current
ACKNOWLEDGMENT Issues, N. Mahdavi Tabatabaei, A. Jafari Aghbolaghi, N. Bizon, and
F. Blaabjerg, Eds. Cham: Springer International Publishing, 2017,
This project is funded by Quaid-e-Awam University of pp. 475-513.
Engineering Science and Technology. [14] S. M. Mohseni-Bonab, A. Rabiee, and B. Mohammadi-Ivatioo,
"Voltage stability constrained multi-objective optimal reactive
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