international journal of smart electrical engineering-8

International Journal of Smart Electrical Engineering, Vol.2, No.4, Fall 2013 ISSN: 2251-9246 EISSN: 2345-6221

187

Unit Commitment in Presence of Wind Power Plants and

Energy Storage

Reza Khanzadeh1, Mahmoud Reza Haghifam2

1 Electrical Engineering Department, South Tehran Branch, Islamic Azad University, Tehran, Iran. Email: [email protected] 2 Electrical Engineering Department, South Tehran Branch, Islamic Azad University, Tehran, Iran. Email: [email protected]

Abstract

As renewable energy increasingly penetrates into power grid systems, new challenges arise for system operators to keep the

systems reliable under uncertain circumstances, while ensuring high utilization of renewable energy. This paper presents unit

commitment (UC) which takes into account the volatility of wind power generation. The UC problem is solved with the

forecasted intermittent wind power generation and possible scenarios are simulated for representing the wind power

volatility. The iterative process between the commitment problem and the economic dispatch(ED) problem will continue

until we find the optimum mode of committing the units. Furthermore we have considered a hydro pump storage (HPS) unit

to be a part of operating system in order to mitigating wind power forecasting errors and peak shaving. Numerical

simulations indicate the effectiveness of the proposed UC for managing the security of power system operation by taking into

account the intermittency and volatility of wind power generation.

Keywords: Unit Commitment, Economic Dispatch, Wind Power, Hydro Pump Storage Unit, Mont Carlo Simulation.

2013 IAUCTB-IJSEE Science. All rights reserved

Nomenclatures

Index

NG Number of thermal units

NW Number of wind units

NH Number of pump storage units

NS Number of scenarios

i Power generation unit index

s Scenario index

t Time index

T Time horizon

Binary

Variables

i,tI Commitment status of unit i at time t

i,tY If unit i starts up at time t , is equal to 1

i,tS If unit i shuts down at time t , is equal to 1

tIGH HPS generating mode decision variable

tIPH HPS pumping mode decision variable

tIIH HPS inactive mode decision variable

Parameters

i,tSU Startup cost of unit i at time t

i,tSD Shutdown cost of unit i at time t

i,minP Lower limit of real power generation of unit i

i,maxP Upper limit of real power generation of unit i

iRU Ramp up rate of unit i

iRD Ramp down rate of unit i

beginZ Initial water reserve inventory of HPS

lastZ Target water reserve inventory of HPS

h,maxV Upper limit of reservoir volume in HPS

h,minV Lower limit of reservoir volume in HPS

1A Efficiency of pumping cycle of HPS

2A Efficiency of generating cycle of HPS in

h,tL Lower limit of consumed power by HPS

inh,tU

Upper limit of consumed power by HPS out

h,tL Lower limit of generated power by HPS

outh,tU

Upper limit of generated power by HPS

h,tSGC Srarting to generate , cost of HPS

h,tSPC Starting to pump , cost of HPS

pp.187:193


188

oniT

Minimum up time of unit i off

iT Minimum down time of unit i

Variables

i,tP Generation of unit i at time t

w,i,tsP Generation of wind unit i at time t in scenario s

h,i,tinP Absorbed power by HPS i at time t

h,i,toutP Generated power by HPS i at time t s

tD Demand at time t in scenario s

s,i,tR Spining reserve prepeared by unit i at time t

s,tR System spinning reserve requirement at time t

tZ Water reserve inventory at time t

h,tV Water volume in HPS at time t

i,tonX Operating duration of unit i at time t

i,toffX Shutdown duration at time t

Function

i,tFC Generating cost of thermal unit i at time t

1. Introduction Wind energy has become increasingly popular

across the globe. It is reported by the Global Wind

Energy Council (GWEC) that global wind energy

installations rose by 11 531 MW in 2005, which

represent an annual increase of 40.5% [1]. Such

figures demonstrate the prosperous future of wind

power development. However, the intermittent and

volatile nature of wind power generation may impact

power system characteristics such as voltages,

frequency and generation adequacy which can

potentially increase the vulnerability of power

systems. Intermittency refers to the unavailability of

wind for an extended period and volatility refers to

the smaller and hourly fluctuations of wind within its

intermittent characteristics. The cumulative wind

power (representing several wind farms) in a power

system might not be intermittent. However, the

power output of a single wind farm could be

intermittent within a 24-h period. The intermittency

of individual wind farms is considered in the

proposed UC in order to ensure that prevailing

constraints are satisfied.

There are several techniques for predicting the

quantity of intermittent wind power [2], [3].Wind

forecasting is conducted by simulation, statistical

method, or a combination of the two. The simulation

method is based on a large number of wind scenarios

and starts by a numerical weather prediction (NWP)

followed by local wind pattern predictions using

analytical methods. The statistical method also starts

from NWP followed by statistical, artificial neural

network, or fuzzy logic methods instead of analytical

methods for calculating the hourly quantity of

intermittent wind power, in which large data sets are

needed and spikes in wind data are hard to predict

[4]. Although wind power is predictable to a limited

extent, it cannot be forecasted with 100% accuracy

for dispatching purposes. Hence, it is possible that

the actual wind power would be different from its

forecasted value. The uncertainty is characterized in

this paper by considering the volatility in multiple

scenarios.

The wind power forecasting and associated

forecasting accuracy issues are important in

analyzing the impact of wind power on power system

operation. However, a complete discussion on wind

speed, wind forecast, and wind power data analyses

is beyond the scope of this paper, and deserves

another full paper. Furthermore, the modeling of load

forecasting error (load profile) is also performed in

this work . Other uncertainties such as the modeling

of generation and transmission outages are also

important subjects for power system operation which

are beyond the scope of this paper.

Wind farms could be managed by utility

companies in which the real-time on/off status of

non-wind units would be decided based on the hourly

load behavior and the availability of intermittent

wind power. However, in certain parts of the United

States, the intermittency of wind could amount to

several hundred megawatts in a matter of hours.

Likewise, the volatility of wind power could have a

tremendous impact on power system operations

which poses new challenges for the electricity market

management. Control room operators and ISOs in

competitive electricity markets apply optimization

methods for managing the security of the system

while utilizing the merits of wind power generation

[5][7]. In [8] the impact of intermittent wind generation

on the operations of the Tennessee Valley Authority

(TVA) power system is investigated and the

operations of the TVA power system are outlined. In

reference [9] authors have presented a new method

for solving efficiently a large scale optimal unit

commitment problem that was included three types

of units (i) usual thermal units (ii) fuel constrained

thermal units and (iii) pumped storage hydro units .

The solution method in this paper uses a lagrangian

relaxation. A new simulation method that could fully

assess the impacts of large-scale wind power on

system operations from cost, reliability, and

environmental perspectives was introduced in [10].

For coordinating the wind and thermal generation

scheduling problem a hybrid approach of combining

branch and bound algorithm with a dynamic

programming algorithm was developed in [11]. In

[12] a security constrained unit commitment (SCUC)

algorithm which takes into account the volatility of

wind power generation is proposed. A stochastic cost

model and a solution technique for optimal

scheduling of the generators in a wind integrated

power system considering the demand and wind

generation uncertainties is presented in [13]. In [14]

the effects of stochastic wind and load on the unit

commitment and dispatch of power systems with


189

high levels of wind power is examined and showed

that stochastic optimization results in less production

cost and better performing schedules than

deterministic optimization. A computational

framework for integrating the state-of-the-art

numerical weather prediction (NWP) model in

stochastic UC/ED formulations is proposed in [15]

that accounts for wind power uncertainty. In [16]

authors have presented an efficient formulation of the

stochastic unit commitment problem that is designed

for use in scheduling simulations of single-bus power

systems. A robust optimization approach for

accommodating wind output uncertainty is proposed

in [17] that aims to providing a robust unit

commitment schedule for the thermal generators in

the day ahead market that minimizes the total cost

under the worst wind power scenario. In [18] a

stochastic dynamic programming approach to unit

commitment and dispatch has proposed that

minimizes the operating cost by making optimal unit

commitment, dispatch and storage decisions in the

face of uncertain wind generation. A novel approach

to the security constrained unit commitment with

uncertain wind power generation is presented in [19]

that its goal is to solve the problem considering multiple stochastic wind power scenarios but while

significantly reducing the computational burden

associated with the calculation of the reserve

deployment for each scenario. In [20] a robust

optimization approach is developed to derive an

optimal unit commitment decision for the reliability

unit commitment runs by ISOs/RTOs with the

objective of maximizing total social welfare under

the joint worst-case wind power output and demand

response scenario.

The rest of this paper is organized as follows.

Section 2 presents the uncertainty modeling

technique. Section 3 proposes the formulation of the

problem and the solution methodology. One case

study is studied in section 4. Section 5 concludes the

discussion.

2. Uncertainty Modeling Technique In order to taking into account wind power and

demand forecasting uncertainty, we use an

uncertainty modeling technique that is based on

scenario generation for uncertain parameter. In this

approach we use monte carlo simulation technique to

generate a large number of scenarios subject to a

normal distribution of forecasting errors that have

engendered in the past predictions. Since the number

of scenarios is very large, using all of those scenarios

in solving progress increases the computational

burden of our problem. Therefore, reducing the

number of scenarios is one of the necessities. So,

scenario generation and reduction methods are as

follows:

2.1. Scenario Generation

For scenario generation, first we have to

calculate the forecasting errors that have occurred in

the past wind power and demand predictions and

assume that they are subject to a normal or other

statistical distribution with an expected value () and a percentage of as its volatility (). Then, using monte carlo sampling technique, monte carlo paths

will create by sampling from this distribution and

juxtapose them. Now for constructing possible

scenarios we must add the obtained samples to the

predictions for next 24 hour, each scenario is

assigned an occurrence probability.

2.2. Scenario Reduction

The scenario reduction technique is employed

to decrease the number of obtained scenarios.

Scenario reduction will remove scenarios that have

low occurrence probability and conjunct those

scenarios that are the same as each other, in one

scenario [21], [22]. By reducing the number of

scenarios consequently the computational burden and

time will decrease remarkably.

3. Problem Formulation And Solution

Methodology

3.1. Stochastic Programming

In many situations there is a need to make, an

optimal, decision under conditions of uncertainty.

There is a disagreement, however, with how to deal

with such situations. Uncertainty can come in many

different forms, and hence there are various ways

how it can be modeled. In a mathematical approach

one formulates an objective function f: Rn R which should be minimized subject to specified

constraints. That is, one formulates a mathematical

programming problem:

Min f (x),

xX

(1)

Where the feasible set X Rn is typically defined by a (finite or even infinite) number of

constraints, say X := {x Rn : gi(x) 0, i I} (the notation := means equal by definition). Inevitably the objective and constraint functions

depend on parameters, which we denote by vector

Rd. That is, f (x, ) and gi(x, ), i I, can be viewed as functions of the decision vector x Rn and parameter vector Rd.

3.2. Problem Formulation

We formulate the UC problem in presence of

wind power plants and hydro pump storage unit in

(2) (17) as a stochastic optimization problem. The objective function (2) consists of generator operating


190

cost, start up and shutdown costs of thermal power

plants and the cost of staring to generate or absorb

power by pump storage unit. The constraints in our

UC problem including system constraints, thermal

power plant constraints, wind power plant constraints

and pump storage unit constraints are as follows.

System power balance constraint (3), system

spinning reserve requirements (4), unit generation

limits (5), unit minimum up time (6), unit minimum

down time (7), unit ramping up limits (8), unit

ramping down limits (9), unit initial state (10), hydro

water inventory constraints (11), constraints (12) and

(13) describe the upper and lower bounds of

electricity absorbed and generated by the pumped-

storage unit , constraints (14) and (15) give the initial

and target water inventory level for the pumped-

storage unit , water reservoir volume limit in pump

storage unit constraint (16) and finally constraint (17)

that ensures that the pumped-storage unit cannot

absorb and generate electricity simultaneously within

any specific time period .

, , , , , , ,

1 1 1 1

min ( )*NG T NH T

c i i t i t i t i t h t h t

i t h t

F P I SU SD SGC SPC

(2)

, , , , , , , ,

1 1 1

*NG NW NH

s out in s

i t i t w i t h i t h i t t

i i h

P I P P P D

(3)

, , , ,

1

*NG

S i t i t S t

i

R I R

(4)

,min , , ,max ,*i i t i t i i tP I P P I (5)

,( 1) ,( 1) ,* 0on on

i t i i t i tX T I I (6)

,( 1) , ,( 1)* 0off off

i t i i t i tX T I I (7)

, ,( 1) ,( 1) , ,min ,( 1) ,(2 ) (1 )i t i t i t i t i i t i t iP P I I P I I RU (8)

,( 1) , ,( 1) , ,min ,( 1) ,(2 ) (1 )i t i t i t i t i i t i t iP P I I P I I RD (9)

, , , ,( 1)i t i t i t i tY S I I (10)

,

( 1) 1 ,

2

out

h tin

t t h t

PZ Z A P

A

(11)

, , ,

in in in

h t h t h tL P U (12)

, , ,

out out out

h t h t h tL P U (13)

0 beginZ Z (14)

T lastZ Z (15)

,min , ,maxh h t hV V V (16)

, , , 1h t h t h tIGH IPH IIH (17)

4. Case Study

4.1. Applying on a Sample System

As a case study we have considered a single

bus system that includes 6 thermal units , 2 wind

units and one hydro pump storage unit that have

shown in fig.1 . The characteristics of thermal units

and pump storage unit are presented in table.1 and

table.2 respectively.

Fig.1. Studied system

Table.1

Generator data

(a)

Unit

St

Mbtu

Fuel

Price

($/Mbtu)

af

(Mbtu/MW2h)

bf

(Mbtu/MWh)

cf

(Mbtu/h)

G1 300 1 0.0109 8.6 70

G2 250 1 0.01059 8.3391 64.16

G3 100 1 0.003 10.76 32.96

G4 440 1 0.01088 12.8875 6.78

G5 100 1 0.01088 12.8875 6.78

G6 50 1 0.0128 17.82 10.15

(b)

Each of the wind farms and required demand

also has a predicted value and some scenarios for

modelling uncertainty of the forecasted quantities.

Figures 2, 3 and 4 show characteristics of wind unit

1, wind unit 2 and demand, respectively.

Unit

Pmin

(MW)

Pmax

(MW)

Min

ON

(H)

Min

Off

(H)

Ramp

Up

(MW/H)

Ramp

Down

(MW/H)

IniT

(H)

G1 100 580 10 10 250 250 10

G2 100 450 8 8 210 210 8

G3 100 380 6 6 175 175 6

G4 100 330 6 6 150 150 6

G5 100 300 5 5 150 150 5

G6 25 100 3 3 50 50 3


191

Table.2

Pump storage unit data

Unit

Pump

Cycle

Eff

Gen

Cycle

Eff

Max

Gen

Lim

(MW)

Min

Gen

Lim

(MW)

Max

Abs

Lim

(MW)

Min

Abs

Lim

(MW)

Min

ON

(H)

Min

OFF

(H)

1 0.8 0.8 40 5 40 5 1 1

Uphill Reservoir Downhill Reservoir

Unit

Ini

Vol (Hm3)

Tgt

Vol (Hm3)

Up

Lim

Vol (Hm3)

Low

Lim

Vol (Hm3)

Ini

Vol (Hm3)

Up

Lim

Vol (Hm3)

Low

Lim

Vol (Hm3)

Gen

And

pump

St Cost($)

1 180 60 250 50 380 600 200 75

Fig.2. Forecasted power and scenarios for wind1

Fig.3. Forecasted power and scenarios for wind2

Fig.4. Forecasted demand and scenarios

By means of the introduced system we solve the

UC and ED problem. In this paper we have used

GAMS24.1.3 and its CPLEX solver to solve the

Mixed Integer Program (MIP) that proposed in

section 3.2.

After solving the problem we must choose the

most optimum schedule of committing units in order

to minimize the production cost. The output results

of the program showed that the cost of the power

production in the considered day is 353834.056$ and

allocated power to each unit can be showed like

table.3.

Table.3 Commitment and dispatch of thermal and pump storage units

Hour G1

(MW)

G2

(MW)

G3

(MW)

G4

(MW)

G5

(MW)

G6

(MW)

H

OUT

(MW)

H

IN

(MW)

1 460.4 0 0 0 0 25 0 0

2 246.6 0 0 0 0 25 0 0

3 113.6 0 0 0 0 25 0 0

4 100 0 0 0 0 0 0 8.4

5 118.5 0 0 0 0 0 0 0

6 172.4 0 0 0 0 0 0 0

7 422.4 165.8 0 0 0 0 0 0

8 562.524 100 175 0 0 0 21.376 0

9 580 139 350 0 0 0 0 0

10 580 181.6 380 0 100 0 0 0

11 580 233.5 380 0 100 0 0 0

12 580 271.7 380 0 100 25 0 0

13 580 255.5 380 0 100 25 40 0

14 580 323.3 380 0 100 25 0 0

15 580 242.9 380 0 100 0 0 0

16 580 293.9 380 0 100 0 0 0

17 580 276.7 380 0 100 0 0 0

18 580 293 380 0 100 0 0 0

19 580 340.2 380 100 100 0 0 0

20 580 450 350 100 106.8 0 0 0

21 580 450 175 100 136.5 0 0 0

22 500 450 0 100 133.9 0 0 0

23 250 405.49 0 100 0 0 40 0

24 0 364.5 0 100 0 0 0 0

In order to ensure system reliability and

security we have allocated spinning reserve for each

hour. The spinning reserve is provided by thermal

units by means of committing those thermal units in

each hour that the sum of their maximum production

capacity is greater than or equal by 1.15*Demand in that hour.

In table.3 we can see that the pump storage

unit , in the period of times that demand is low and

wind production is high , treats like a load and

absorbs power to pump water from downhill

reservoir to uphill reservoir that . This work not only

causes an increase in energy storage but also

decreases the need to wind curtailment. Moreover

this unit has used for peak shaving in times that a


192

transition peak load has occurred and thus prevents

from more start up in thermal units.

It is obvious that if the capacity of the pump

storage unit be different from the value that we have

used, its commitment will be different too. Figures 5

and 6 show the change in uphill and downhill water

volume that have engendered because of the

generating and absorbing power during the day. It is

clear that the water volume had not transgressed

from its limits both in downhill and uphill reservoirs

and the volume in uphill reservoir have reached to

the predefined value at the end of the day.

Fig.5. Uphill reserve inventory changes

Fig.6. Downhill reserve inventory changes

4.2. Sensitivity Analysis

In order to evaluate the effect of some

parameters on the problem we have done sensitivity

analysis. By this goal we solved the problem without

wind power plants and pump storage unit. Results

show that this change increases the production cost

to 394171.138$ that is equal to 11.4% increase and

moreover the aggregate operating hour of units was

increased too.

We have also examined the effect of changes in

pump storage unit capacity and initial water volume

in uphill reservoir on the production cost. Figures 7

and 8 show the change in production cost by varying

the pump storage maximum capacity and initial

water volume in the uphill reservoir, respectively.

It is sensible in fig.7 that the production cost at

first decreases gradually as the maximum capacity of

pump storage unit increases, but after a specific

capacity the change in cost is more remarkable. It is

why by increasing the capability of pump storage

unit to take part in supplying the load and saving

energy, the efficiency of it, is also increased.

Besides, Fig.8 shows that if the volume of the

water that exists in the uphill reservoir at the

beginning of the day be more than we considered, the

production cost will also less than we obtained from

the implemented volume as the initial water volume

for the uphill reservoir of the hydro pump storage

unit.

Fig.7. Cost change by varying maximum capacity of the pump storage unit

Fig.8. Cost change by varying initial water level in uphill reservoir

5. Conclusion

In this paper, we proposed an approach that

includes applying optimization concepts and

incorporating pumped-storage units to hedge wind

power output uncertainty and peak shaving. We

provided scenarios that can capture the wind power

unpredicted changes and our proposed approach can

provide an optimal solution that minimizes the total

cost under the wind power fluctuations that can occur

in the system, while ensuring the higher penetration

of wind power. Meanwhile, this solution is feasible

with a high probability under wind power output

uncertainty. In addition, by incorporating pumped

storage hydro units in the real time, our optimization

model contains discrete decision variables in

problems. Finally, our computational results verify

the effectiveness of the presence of wind units and


193

pump storage unit for the system and power

production cost.

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195

Maximum Power Point Tracking of Wind Energy Conversion System using Fuzzy- Cuckoo Optimization Algorithm

Strategy

Mohammad Sarvi1, Mohammad Parpaei2

1 Electrical Engineering Department, Imam Khomeini International University, Qazvin, Iran. Email: [email protected] 2 Electrical Engineering Department, Imam Khomeini International University, Qazvin, Iran. Email: [email protected]

Abstract

Nowadays the position of the renewable energy is so important because of the environment pollution and the limitation of

fossil fuels in the world. Energy can be generated more and more by the renewable sources, but the fossil fuels are non-

renewable. One of the most important renewable sources is the wind energy. The wind energy is an appropriate alternative

source of fossil fuel. The replacement rate of renewable energy to fossil fuels is rising, although the production cost is higher

than fossil fuels. To further reduce cost of wind production, many methods have been proposed. One of the suitable

approaches is the maximum power point tracking strategy. In this paper, a new intelligent maximum power point tracker

called Fuzzy- Cuckoo strategy for small- scale wind energy conversion systems is proposed. The maximum power point

tracker proposed uses measured wind speed to detect the maximum output power and its respective optimal rotational speed.

The main contribution of the proposed approach is to exactly track the maximum power point, so the output power

fluctuations captured by wind turbine are less than conventional approaches. The simulations are performed in

MATAL/SIMULINK software. The superiority of the proposed approach is validated in two situations, low and rapid

changes in wind speed. The maximum power point of wind energy conversion systems can be tracked by the proposed

approach in any situation. The higher accuracy of the Fuzzy- cuckoo strategy than the conventional trackers is another

advantage of the proposed approach.

Keywords: Intelligent controller, Metaheuristic optimization approach, Wind energy conversion systems.


1. Introduction Fossil fuel reserves reduction causes that the

whole countries, especially the countries that have

not enough fossil fuel sources, pay special attention

to the renewable energy as a second energy source.

China and USA are two countries that concentrate on

the wind energy conversion systems (WECS) than

other countries. WECSs are used to change the wind

energy to electrical energy by electrical machines

such as the permanent magnet synchronies

generators (PMSGs). The small- Scale WECSs are

suitable alternative sources for urban regions or

remote places that connection to power grid is

impossible [1]. The main disadvantage of the

renewable energy is that the electricity production

costs from various renewable energy sources are

higher than fossil fuel. To improve this problem,

maximum power point tracking (MPPT) is a matter

that is expressed. The maximum power point trackers

control the WECSs at the optimal output power.

There are many approaches to track the maximum

power point, but all approaches are based on three

main classifications. The first strategy is the methods

based on iteratively search, the second strategy is the

methods based on the static parameters of the wind

turbine and wind speed, and the third strategy is the

methods based on hill- climb searching (HCS)

pp.195:200


196

control. The many approaches have been proposed to

implement the maximum power point trackers in the

WECSs. Reference [2] presents a relationship

between DC electrical variables and mechanical

variables of turbine. Reference [3] proposes an

alternative approach using mechanical model of the

permanent magnet synchronous generator (PMSG) in

order to estimate both position and speed of the wind

turbine. Reference [4] develops a new approach

using perturb and observe (P&O) and optimum

relationship based (ORB). Reference [5] presents a modified MPPT controller using Fuzzy logic

controller (FLC). Reference [6] proposes a new

MPPT approach using neural networks (NNs) for

modeling of wind turbine systems and NNs describe

relationship between input and output. Reference [7]

introduces a novel approach of radial basis function

neural network (RBFNN) and particle swarm

optimization (PSO). Reference [8] presents a two

stages MPPT approach. In the first stage, the large

iterative steps and in the second stage, the

conventional P&O have been used to exactly track

the maximum power point. Reference [9] describes a

direct control using output observation and directly

adjusting duty cycle of boost converter. Reference

[10] develops a control strategy for the generator side

converter with output maximization of a PMSG.

Reference [11] present a new maximum power point

tracking based on adaptive neuro- fuzzy interface

system (ANFIS). Reference [12] presents a new

MPPT strategy based on Wilcoxon radial basis

function network (WRBFN) with hill-climb

searching (HCS). Reference [13] proposes a new

sensorless control strategy based on a model

reference adaptive system (MRAS) observer for

estimating the rotational speed. Reference [14]

proposes a new adaptive intelligent optimization

algorithm that uses a new advance P&O to detect the

maximum power point. Reference [15] uses a new

approach to estimate position and speed of the wind

turbine in order to track the maximum power point.

In reference [16], the rotor position is estimated

based on the flux linkage.

In this paper, a novel strategy based on an

intelligent optimization algorithm, namely Cuckoo

optimization algorithm (COA), and the fuzzy logic

controller is proposed. The proposed approach is

based on the optimum relationship- based (ORB).

The main contribution of the proposed approach is to

exactly track the maximum power point, so the

output power fluctuations captured by wind turbine

are less than the conventional approaches such as

PSO and fuzzy logic trackers. The higher accuracy of

the Fuzzy- Cuckoo strategy than the conventional

trackers is another advantage of the proposed

approach.

This paper has been organized as follows:

section 2 describes the wind energy conversion

system. Section 3 introduces the Cuckoo

optimization algorithm. Section 4 presents the

proposed maximum power point tracker. The

simulation results in several case studies are given in

Section 5, and finally section 6 states the conclusion.

2. Description of Wind Energy Conversion

System

The proposed approach will be applied to the

following WECS as Figure 1. As shown in Fig.1,

PMSG is coupled to a wind turbine. The mechanical

output power is transferred to the electrical power by

PMSG. The AC electrical power produced by PMSG

will be rectified by a three-phase diode bridge

rectifier, and the rectified power is fed to the boost

converter.

PMSGdcC o

C R

DL

Fig.1. Wind energy conversion system

2.1. Wind Energy Conversion System Characteristics

The output power derived by the wind turbine

blades is expressed as [16]:

P =1

2 A Cp V

3 (18)

Where P is the output power (in watt), is the air density (in kg/m3), Cp is the power coefficient,

and V is the wind speed (in m/s). A is area swept by the blades (in m2) determined as:

A = R2 (19)

Where R is radius of blades (in m).

Power coefficient Cp is a nonlinear function of

the pitch angle and tip speed ratio as follows:

= 0.5176 (116 (1

) 0.4 5)

exp (21 (1

)) + 0.0068

(20)

1

i=

1

+ 0.08

0.035

3 + 1 (21)

=RtV

(22)

Where is the tip speed ratio (TSR), is the pitch angle (in degree), and t is the rotational speed of turbine (in rps).


197

Fig.2 shows the output power captured by wind

turbine versus rotational speed. According to this

figure, the output power function has a one global

optimum point, and it is a non-linear function of the

rotational speed.

Fig.2. Output power versus rotational speed

3. Cuckoo Optimization Algorithm

Cuckoo optimization algorithm (COA) is a new

intelligent evolutionary algorithm that is proper for

continuous non- linear problems. It is inspired by the

special kind of bird called Cuckoo. The superiority of

COA has been proven than particle swarm

optimization (PSO) and genetic algorithm (GA) by

[17]. The main advantage of the COA is that this

optimization algorithm is robust to dynamic changes.

The special lifestyle of Cuckoo has been formulated

as an optimization algorithm in order to fine the

maximum/minimum value of objective functions.

The following section meticulously describes the

COA.

To start the optimization process, it is necessary

to initialize the starting points as an array. Each of

the evolutionary optimization algorithms specifies a

special name for this array. For instant, in PSO, DE,

GA this array is called Particle Position, Stochastic Population, and Chromosome, respectively. In cuckoo optimization algorithm, this

array is also called Habitat. In order to solve a N- dimensional optimization

problem, we need to form an array of 1N as

follows:

= (1

, 2 , ,

) (23)

Where d is dimension of the objective function, i is ith habitat, and t is tth generation. Therefore, a candidate habitat matrix of size NP Nd (NP is the number of habitats) is randomly generated.

In nature, each cuckoo lays 5 to 20 eggs, so the

number of eggs is randomly determined from 5 to 20

for each cuckoo at each generation of the

optimization process. Each cuckoo lays in the special

distance from its habitat called egg laying radius

(ELR) as follows:

=

( ) (24)

Where K is an integer number, xu , xl are upper and lower of variable limits, respectively.

Cuckoos lay their eggs in other host birds nests according to their ELR. Some of the eggs in host

birds nests will be recognized and thrown out by

host bird, so p% of all eggs will be killed. When the cuckoos eggs hatch, they throw the host birds eggs out from nests. Cuckoos chicks grow in the host birds nests. When they become mature, they immigrate to new area with more food and more

similarity of eggs to host birds eggs. To immigrate, the best value experienced will be determined as a

new area that other cuckoos immigrate to there.

When they want to immigrate to a new area, they do

not fly all way to the destination habitat. They only

fly a part of whole way with a deviation like Figure

3.

Current

Cuckoo's location

New area

Cuckoo's location

for next generation

W

hole le

ngth

A pa

rt of w

hole

length

Fig.3. Immigration of a cuckoo to a new location for next

generation

4. Fuzzy logic controller

The five main steps of fuzzy controller are input

and output variables, fuzzy rule, fuzzification,

inference, and defuzzification. The maximum power

point Pm,optimal and its respective rotational speed

optimal are calculated by COA. These values are compared with the actual values of the output power

Pm,actual and of the rotational speed actual, respectively. As seen in Figure 4, we have two input

variables for fuzzy logic controller; the first input is

the difference between the maximum power point

and the actual power point as Equation (8). The

second input is the difference between the optimal

rotational speed that belongs to the maximum power

point and the actual rotational speed as Equation (9).

0 5 10 150

0.5

1

1.5

2

2.5

3x 10

4

The locus of

maximum output

power

Ou

tput

po

wer

(W

)

Rotational speed (rad/s)


198

Fuzzifier

Fuzzy Rule

Fuzzy Interface Defuzzifier

Gain

Gain

m,optimalP Cuckoo Optimization

optimal

m,actualP

actual

Gain

m,actualP

Fig.4. Configuration of the proposed MPP tracker

= , , (25)

= optimal actual (26)

A mamdani inference is used as fuzzy inference

system. Fuzzy controller based on these inputs and

fuzzy rules change the duty cycle of boost converter.

The input and output membership functions are

shown in Figures 5 and 6. Fuzzy rules are shown in

Table 1.

Fig.5. Input and output membership function of Pm and

,

respectively

Fig.6. Input membership function of

Table.1.

Rules table of fuzzy logic applied to the WECS

++ ++

++ ++

++

5. Simulation results

In order to validate the performance of the

proposed approach, the proposed Fuzzy- Cuckoo

tracker is applied to the wind turbine with following

parameter values:

Table.2.

Parameter values of the simulated system

Air density () 1.2929 (/3)

Radius of blade () 7.4 (m)

Pitch angle () 0

Parameter values of PMDG

Stator phase resistance 0.98 ()

Stator phase resistance 2.83 (mH)

Pole pairs 3

Inertia 30 (kg.m2)

Parameter values of BOSST converter

Inductance 400e-6 (H)

Capacitance 800e-6 (F)

Load

Load 30 ()

The simulations are performed in MATLAB

environment. The schematic diagram of the proposed

MPP tracker is completely shown in Figure 7.

PMSG

Gain

Gain

m,optimalP

optimal

m,actualP

actual

Gain Cuckoo

OptimizationFuzzy logic

m,actualP

Gain

D

PID PWMD

dcC oC R

L D

wind speed

Fig.7. Completely schematic diagram of the proposed MPP tracker

A three- phase diode bridge rectifies the voltage

generated by PMSG, and the dc- link capacitor Cdc is used to smooth the voltage ripple of the dc voltage

generated by the three- phase diode bridge rectifier.

Finally, the boost converter is used to control wind

turbine in the maximum power point by adjusting the

duty cycle.

To confirm the ability of the proposed MPP

tracker, in the first case, wind speed pattern will be

changed slowly as Fig.

Fig.8. Assumed wind speed profile

0 1 2 3 4 5 6 7 8 9 107

8

9

10

11

12

13

Win

d s

pee

d (

m/s

)

Time (sec)


199

Fig.9- a shows the output power of wind turbine

tracked by the proposed Fuzzy- Cuckoo controller,

and Figure 9- b shows the optimal rotational speed

calculated by cuckoo optimization according to the

wind speed pattern (as Fig.8).

Fig.9. Output maximum power captured by the proposed MPP

tracker and rotational speed calculated by cuckoo optimization

algorithm while the wind speed is slowly changed

As seen in Fig.9, the proposed fuzzy- cuckoo

tracker is capable of tracking the maximum power

point in normal wind changes. To prove the

superiority of the proposed fuzzy- cuckoo approach

than conventional MPP trackers such as PSO and

fuzzy trackers in normal wind changes, a comparison

with these trackers is provided in following figures.

Figure 10 shows the new assumed pattern of wind

speed.

Fig.10. Assumed wind pattern for comparison purpose

Fig.11 shows the maximum output power

tracked by fuzzy- cuckoo, PSO, and fuzzy trackers.

Fig.11. Maximum output power captured by the fuzzy- cuckoo,

PSO, and fuzzy MPP trackers

As it is clearly observed, the proposed fuzzy-

cuckoo outperforms PSO and fuzzy MPP trackers.

Figure 11 shows that fluctuations of the proposed

approach are also less than PSO and fuzzy, as well as

higher mean value of the output power.

One of the difficulties in MPPT trackers is the

fast variations of the wind speed. In this case, for

further demonstration, the fast variations of the wind

speed are applied to the MPP trackers. Figure 12

shows the fast wind variations, and Figure 13 shows

the power tracked by fuzzy- cuckoo, PSO, and fuzzy

trackers.

Fig.12. Fast wind variations profile.

Fig.13. Simulation results in fast variations of the wind speed

As it is shown in Fig.13, the proposed fuzzy-

cuckoo controller outperforms the conventional MPP

trackers such as PSO and fuzzy trackers.

6. Conclusion

In this paper, an accurate MPP tracker called

Fuzzy-cuckoo tracker has been applied to the small-

scale WECS with a PMSG and a three-phase diode

bridge rectifier. The proposed MPP tracker uses

Cuckoo optimization algorithm to detect the

maximum power point of the mechanical power

curve and its respective rotational speed. The wind

speed is measured by anemometer as an input of the

Cuckoo optimization algorithm. The difference of the

optimum output power that is calculated by the

Cuckoo optimization algorithm and actual output

power has been fed the Fuzzy as the first input, and

the difference of the rotational speed that belongs to

the optimum output power and actual rotational

speed has also been fed the Fuzzy as the second

input. Finally, the output of Fuzzy logic has been

used as a set- point. The implementation of the

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8x 10

4

0 1 2 3 4 5 6 7 8 9 102

4

6

8

10

12

14

0 1 2 3 4 5 6 7 8 9 107

8

9

10

11

12

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8x 10

4

COA- Fuzzy

Fuzzy

PSO

0.9 1 1.1

3.8

3.85

3.9x 10

4

2.9 3 3.14.45

4.5

4.55

4.6x 10

4

4.9 5 5.1

6.9

7

7.1

x 104

6.9 7 7.1

5.2

5.25

5.3

5.35x 10

4

8.9 9 9.13.8

3.85

3.9x 10

4

0 1 2 3 4 5 6 7 8 9 106

7

8

9

10

11

12

13

14

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12x 10

4

COA- Fuzzy

Fuzzy

PSO

2.2 2.4 2.66

6.5

7x 10

4

6.5 6.6 6.78

9

10x 10

4

4.2 4.255

6

7x 10

4

9 9.2

4.6

4.8

5x 10

4

Time (sec)

Ou

tput

po

wer

(W

) R

ota

tio

nal

spee

d (

rad

/s)

Time (sec)

Win

d s

pee

d (

m/s

)

Time (sec)

Time (sec)

Time (sec)

Win

d s

pee

d (

m/s

)

Time (s)

Ou

tput

po

wer

(W

)

Ou

tput

po

wer

(W

)


200

proposed MPP tracker is very simple because only

the measurement of the wind speed and the rotational

speed are required. In comparison with other

conventional MPP trackers such as PSO, and Fuzzy

trackers, the proposed MPP trackers are more

accurate and robustness (Figures 11 and 13).

References

[1] Z. M. Dalal, Z. U. Zahid, W. Yu, and J. S. Lai, Design and Analysis of an MPPT Technique for Smal- Scale Wind

Energy Conversion System, IEEE Transactions on Energy Conversion, Vol.28, No.3, pp.756-767, 2013.

[2] A. Urtasun, P. Sanchis, I. S. Martn, J. Lpez and L. Marroyo,

Modeling of samll wind turbines based on PMSG with diode bridge for sensorless maximum power tracking, Renewable energy, Vol.55, pp.138-149, 2013.

[3] C. Ming, C. H. Chen and C. H. Tu, Maximum power point tracking- based control algorithm for PMSG wind generation

system without mechanical sensors, Energy conversion and management, Vol.69, pp.58-67, 2013.

[4] Y. Xia, K. H. Ahmed and B. W. Williams, A New Maximum Power Point Tracking Technique for Permanent Magnet

Synchronous Generation Based Wind Energy Conversion System, IEEE Transactions on Power Electronics, Vol.26, No.12, pp.3609-3620, 2011.

[5] A. M. Eltamaly and H. M. Farh, Maximum power extraction from wind energy system based on fuzzy logic control, Electrical power systems research, Vol.97, pp.144-150, 2013.

[6] M. Narayana, G. A. Putrus, M. Jovanovic, P. S. Leung and S. McDonald, Generic maximum power point tracking controller for small-scale wind turbines, Renewable Energy, Vol.44, pp.72-79, 2012.

[7] C. Y. Lee, P. H. Chen and Y. X. Shen, Maximum power point tracking (MPPT) system of small wind power generator

using RBFNN approach, Expert Systems with Applications, Vol.38, pp.12058-12065, 2011.

[8] V. Agarwal, R. K. Aggarwal, P. Patidar and C. Patki, C, A Novel Scheme for Rapid Tracking of Maximum Power Point

in Wind Energy Generation Systems, IEEE Transactions on Energy Conversion, Vol.25, No.1, pp.228-236, 2010.

[9] E. Koutroulis, and K. Kalaitzakis, Design of a Maximum Power Tracking System for Wind- Energy- Conversion Applications, IEEE Transactions on Industrial Electronics, Vol.53, No.2, pp.486-494, 2006.

[10] M. E. Haque, M. Negnevitsky and K. M. Muttaqi, A Novel Control Strategy for a Variable-Speed Wind Turbine With a

Permanent-Magnet Synchronous Generator, IEEE Transactions on Industry Applications, Vol.46, pp.331-339, 2010.

[11] A. Meharra, M. Tioursi, M. Hatti and S. A. Boudghene, A variable speed wind generator maximum power tracking based on adaptative neuro-fuzzy inference system, Expert Systems with Applications, Vol.38, pp.7659-7664, 2011.

[12] W. M. Lin and C. M. Hong, Intelligent approach to maximum power point tracking control strategy for variable-

speed wind turbine generation system, Energy, Vol.35, pp.2440-2447, 2010.

[3] C. H. Chen, C. M. Hong and F. S. Cheng, Intelligent speed sensorless maximum power point tracking control for wind

generation system, Electrical Power and Energy Systems, Vol.42, pp.399-407, 2012.

[14] I. Kortabarria, J. Andreu, I. M. Alegria and J. Jimenez, A novel adaptative maximum power point tracking algorithm for small wind turbines, Renewable Energy, Vol.63, pp.785-796, 2014.

[15] T. Senjyu, Y. Ochi, Y. Kikunaga, M. Tokudome and A. Yona,

Sensor-less maximum power point tracking control for wind generation system with squirrel cage induction generator, Renewable Energy, Vol. 34, pp.994-999, 2009.

[16] T. Senjyu, S. Tamaki, E. Muhando, N. Urasaki, H. Kinijo, T.

Funabashi, F. Hideki and H. Sekine, Wind velocity and rotor position sensorless maximum power point tracking control for

wind generation system, Renewable Energy,Vol.31, pp.1764-1775, 2006.

[17] R. Rajabioum, Cuckoo Optimization Algorithm, Applied Soft Computing, Vol.11, pp.5508-5518, 2011.


201

Solving the Economic Load Dispatch Problem Considering

Units with Different Fuels Using Evolutionary Algorithms

Mostafa Ramzanpour1, Hamdi Abdi2

1 Electric Engineering Department, Science and Research branch, Islamic Azad University Kermanshah, Iran

[email protected] 2 Electric Engineering Department, Faculty of Engineering, Razi University, Kermanshah, Iran

Abstract

Nowadays, economic load dispatch between generation units with least cost involved is one of the most important issues in

utilizing power systems. In this paper, a new method i.e. Water Cycle Algorithm (WCA) which is similar to other intelligent

algorithm and is based on swarm, is employed in order to solve the economic load dispatch problem between power plants.

In order to investigate the effectiveness of the proposed method in solving non-linear cost functions which is composed of

the constraint for input steam valve and units with different fuels, a system with 10 units is studied for more accordance with

literatures in two modes: one without considering the effect of steam valve and load of 2400, 2500, 2600 and 2700 MW and

the other one with considering the effect of steam valve and load of 2700 MW. The results of the paper comparing to the

results of the other valid papers show that the proposed algorithm can be used to solve in any kind of economic dispatch

problems with proper results.

Keywords: Economic load dispatch, water cycle algorithm, valve- point effect.


1. Introduction Most of the optimization problems in power

systems such as economic load dispatch have

complicated and non-linear features with equality

and inequality constraints. This causes it hard to be

mathematically solved [1]. Economic load dispatch is

an important issue in the field of management and

utilization of power system where the aim is to

determine the production value of each power plant

in a way the load of system is supplied with least cost

while all the constraints are fulfilled.

Obtaining the optimal solution is sometimes

difficult in solving the economic load dispatch

problem due to complication of fuel cost function of

power plants and also due to some limitation. One of

these complications is related to the actual form of

cost function. It has to be noted that in practice cost

function of a power plant has no smooth form. In

other words, this cost function has local maximum

and minimum and usually it is not derivable in these

points [2-3].

Different optimization methods and techniques

have been used to solve the problem of economic

load dispatch. Some of these methods are based on

mathematical methods such as linear and quadratic

programming [4-5]. Linear programming methods

are generally quick and use linear and step

approximation of fuel cost which in turn reduces the

accuracy of the problem. To overcome this problem,

non-linear programming methods is used. However,

non-linear programming method has difficulties in

convergence and also with an increase in the number

of units the required time and memory to solve the

problem would considerably increase.

To overcome such problems evolutionary

algorithms are proposed. These algorithms have

pp.201:208


202

several distinct advantages such as using probable

search instead of explicit methods and also

effectively finding the general optimal points instead

of local optimal points. For instance, Artificial

Neural Network [6], Genetic Algorithm [7],

Simulated Annealing [8], Ant Colony Optimization

[9] are some of these algorithms. In ref [10], a new

algorithm named Artificial Life is proposed to

minimize a non-convex combinatorial function.

Interactive Honey Bee Mating Optimization [11] and

Artificial Immune Systems [12-13] are used to solve

economic load dispatch problem.

In this paper, a novel evolutionary algorithm

which is based on the water cycle algorithm and is

inspired from water cycle in the nature is proposed to

solve the economic load dispatch (ELD) between

power plants considering actual model of non-

smooth cost functions on a test system with 10 units

with and without the effect of steam valve. The rest

of the paper is organized as follow. Problem

description is presented in section 2. In section 3, a

comprehensive definition of the water cycle

algorithm (WCA) is expressed and the application of

WCA in solving the ELD is presented in section 4.

Simulation and conclusion are presented in section 5

and 6, respectively.

2. Economic Load Dispatch Problem As mentioned before, ELD is defined as a

process of allocating the levels of generation for each

power plant in combinatorial form in a way the

demand of system is supplied completely and

economically [14].

In order to reach to optimal production for each

power plant, curve of fuel cost has to be modeled as

a mathematical relation. In classic case, this function

is modeled as quadratic function (Figure 1) but in

practical and developed cases; this model is modeled

as non-linear and discontinues form due to several

constraints.

Fig.1. The curve of fuel cost for generators in smooth and

continues condition

The first relation in optimal load dispatch

problem is the law of conservation of energy. The

sum of generated power by power plants has to be

equal to the load demanded from grid.

=

=1

(27)

Where PD is the demand power, Pgi power

generated (output power) by ith generator, ng number

of generators in the system.

In more complex load dispatch problems, the

loss of transmission network (PL) has to be added to

the equation (1).

=

=1

+ (28)

Equation (2) expresses that the sum of

generated power is equal to sum of the consuming

power including power consumed in loads and power

wasted in transmission line. This equation is a form

of the law of conservation of energy. The value of PL

is calculated by equation (3).

=

=1

=1

+0 + 00

=1

(29)

Where Bij, B0i and B00 are factors of loss

function for transmission network. Fuel cost of each

power plant is calculated from the following relation.

() = 2 +

+ (30)

Where F is fuel cost and ci, bi, ai are factors for

fuel cost function of ith unit.

In Figure 1, Pgimin is the minimum loading

range that below this range it would not be

economical (or technically impossible) for the unit

and Pgimax is output maximum range for unit.

Therefore, output power of generator has to be within

minimum and maximum ranges.

(31)

In steam plants, several steam valves are used

in turbine for controlling the output power of the

generators. Opening the steam valve would lead to a

sudden increase in loss and causes ripples in input-

output curve and consequently causes cost function

non-smooth. If the effect of steam valve is

considered in power plants, cost function of their

generation would take a non-smooth form due to

related mechanical effects.


203

Fig.2. Fuel cost curve for generators with 4 steam valves

This influence is usually modeled by adding a

sinusoidal term in cost function of power plants.

Therefore, equation (4) considering the effect of

input steam valve is expressed as equation (6) [15].

() = 2 +

+ + | ( ( ))| (32)

Where egi and fgi are factors for the effect of

valve point on ith generator.

When generation units perform with different

kinds of fuels, cost function of each unit is expressed

by some quadratic equations where each one of them

corresponds to one kind of fuel. This is formulated

mathematically as in equation (7) [16-17].

() =

{

1

2 + 1 + 1 , 1 , 1

22 + 2 + 2 , 1 2 , 2

::

2 + + , ,1

,

(33)

, , express the factors for fuel cost function of ith unit.

3. Water Cycle Algorithm

In this section, a new algorithm inspired by

water cycle in nature is presented which is not

employed for optimization on any power systems

[18]. Similar to other heuristic swarm algorithms, the

proposed method is started with an initial population

named rain drops (Npop). A matrix is produced as rain

drops with a dimension of Npop*Nvar for initializing

the optimization. Rows and columns of this matrix

are composed of population (Npop) and number of

design variables (Nvar) or generation units,

respectively.

Population of raindrops =

[ 123

]

=

[ 1

1 21 3

1

12 2

2 32

1

2

1 2

3

]

(34)

In a random matrix with certain dimension, the

values of every variable X1, X2 , X3, , XN can be real or complex. Cost function or cost of the problem is

defined as below:

= (1 , 2

, , ) , = 1,2,3, , (35)

After producing the initial matrix, NSR number

of them is considered as sea and rivers. Besides, a

drop with the best answer is chosen as sea. The other

members as rain drops would flow to the rivers or

directly to the sea. In fact, NSR is the sum of river

(user parameter) and sea.

= + 1

(36)

It would be understood which drop will go to

which river based on the stream intensity of each

river:

= {|

=1

| }

= 1,2, ,

(37)

After flowing the drop to the river, it has to be

known that how rivers are flown down to seas.

Movement of each stream to a river is known by a

line that joins them. This distance is calculated

randomly.

(, C ) , C > 1 (38)

Fig.3. Schematic of flowing a stream to a river

Where C is a value between 1 and 2 (near to 2).

The current distance between stream and river is

shown by parameter d. X in equation (14) is a

random value between 0 and C*d. The values of C

greater than 1 enables stream to flow to the river in

different directions. Therefore, the best value for C

would be 2. This concept can be used in flowing

rivers to the sea. Hence, new positions for stream and

river can be defined as follow [19].

+1 =

+ (

) (39)

+1 =

+ (

) (40)

Where rand is a random number with uniform

distribution between 0 and 1. After updating the


204

position of each drop and investigating the objective

function, if the proposed solution by stream is better

than a river connected to it, position of stream and

river are changed (i.e. stream would be river and vice

versa). This displacement could be happened for

river and sea.

The evaporation criterion is investigated after

the abovementioned stages. When the position of a

stream or river is completely corresponding with the

position of a sea, it means that it has flown to sea. In

this condition, evaporation is done and new streams

and drops are again flown to the mountains by rain

and the above procedure is repeated.

|

| < , = 1,2,3, , 1 (41)

Where dmax is a small number near to zero and

controls the depths of search close to sea. Greater

values of dmax increase the search space and small

values if depth. dmax is decreased in any repetition.

+1 =

(42)

If the above condition is set, rain will come and

all the above procedure would be repeated. Equation

(19) is used for specifying the new position of newly

produced streams.

= + ( ) (43)

Where LB and UB are the lower and upper

range of the problem, respectively. Besides, there

may some streams that would flow directly to the sea

without flowing to a river.

=

+ (1, ) (44)

Where is the search domain near to sea. randn is a random number of normal distribution. Great

values for increase the possibility of going out of the possible area. Besides, small values for causes the algorithm to search in a smaller area near to sea.

The best value for is selected 0.1. Stop criterion in heuristic algorithm is usually

the best calculated answer in which stop criterion

would be defined as maximum number of iteration,

processing time or a non-negative small value as tolerance between two previous results. Performance

of WCA is agreeable to maximum iteration as a

convergence criterion.

4. Application of WCA on ELD problem

In this paper, a new algorithm called Water

Cycle is used to optimize the overall fuel cost of

power plants. At first, the initial values of WCA like

Npop, Nsr and data related to generation units such

as factors of fuel cost function of generators, output

limitations of generators and the demanded load are

summoned by the system. After this, population

matrix of drops is produced in a random manner. At

third stage, constraints of the ELD problem are

investigated.

In order to investigate the power balance

condition, the value of is calculated for each population like according to the following equation:

= ( + )

=1

(45)

If =0, it means that the inequality constraint is met; otherwise, the calculated is added to a unit randomly. Inequality constraint is then checked for

that unit. If the power is more than maximum power

of that unit, power is set to the maximum value.

Since can also be negative, so if the power is less than the minimum power of that unit, then the power

is set to minimum value. Therefore, we have:

= { >

<

(46)

Now we go back to the second stage and repeat

it until = 0. For solving the ELD problem when there are

different kinds of fuels, the considered production

range for each unit is divided to some sub range

(usually 3 sub ranges) and if the production power is

put in a sub range, the fuel kind of that range is

introduced as the optimum fuel for that unit. In fact,

this constraint makes a piecewise quadratic cost

function and consequently difficult optimization.

There are a few studies in this field considering the

simultaneous effect of input steam valve.

At fourth stage, cost of one drop and at fifth

stage the intensity of stream for river and sea re

calculated. At sixth stage, flowing the streams to the

rivers and rivers to the seas are investigated. After

updating the position of each drop and investigating

the objective function, if the proposed solution by a

stream is better than a river connected to it, position

of stream and river are changed (seventh stage). This

stage could be happened for river and sea (eighth

stage). Evaporation condition which has an important

role in preventing the algorithm to be trapped in the

local minima is investigated. At next stage, dmax is

reduced. When the distance between river and sea is

less than dmax, it means that river has reached to the

sea. In other words, if the evaporation condition is

met, rain will occur based on equations (19) and (20).

And finally stop criterion is checked. Performance of

WCA is usually agreeable to maximum iteration as a

convergence criterion. If the stop criterion is met,

algorithm will stop; otherwise it goes back to the

third stage.


205

5. Simulation

In this kind of problem some different

consuming fuels are considered for different

production ranges of generators which make a

quadratic cost function.

5.1. System with 10 units without considering the effect of steam valve

To investigate this kind of ELD problem, a

system with 10 units with loads of 2400, 2500, 2600

and 2700 MW without considering the effect of input

steam valve is taken into account as the first test

system. Parameters of the algorithm in this test is set

as follow

Npop = 100 , Nsr = 30 , dmax = 0,1 , C = 2 , U = 0.1

Figure 4 and Tables 1 and 2 show the

convergence diagram, optimum solutions and

comparison with other methods, respectively.

Table.1

Results of WCA for system with 10 units considering the

effect of input steam valve

Unit No 2400(MW) 2500(MW) 2600(MW) 2700(MW)

Output F Output F Output F Output F

1 189.73 1 206.52 2 216.58 2 218.27 2

2 202.35 1 206.45 1 210.88 1 211.67 1

3 253.89 1 265.74 1 278.51 1 280.72 1

4 233.05 3 235.95 3 239.07 3 239.63 3

5 241.85 1 258.01 1 275.53 1 278.49 1

6 233.04 3 235.95 3 239.18 3 239.63 3

7 253.26 1 268.87 1 285.70 1 288.60 1

8 233.04 3 235.95 3 239.13 3 239.63 3

9 320.38 1 331.49 1 343.52 3 428.47 3

10 239.40 1 255.05 1 271.97 1 274.87 1

Ftotal (S/h) 481.7216 526.2279 574.3791 623.798

Table.2

Comparison of WCA results by different methods without

considering the effect of input steam valve

Method Fuel cost $/h

2400(MW) 2500(MW) 2600(MW) 2700(MW)

HNUM[20] 488.50 526.70 574.03 625.18

HNN[21] 487.87 526.13 574.26 626.12

AHNN[22] 481.72 526.23 574.37 626.24

ELANN[23] 481.74 526.27 574.41 623.88

IEP[24] 481.779 526.304 574.473 623.851

DE[25] 481.723 526.239 574.381 623.809

MPSO[26] 481.723 526.239 574.381 623.809

RCGA[27] 481.723 526.239 574.396 623.809

AIS[28] 481.723 526.240 574.381 623.809

HICDEDP[29] 481.723 526.239 574.381 623.809

EALHN[30] 481.723 526.239 574.381 623.809

WCA 481.7216 526.2279 574.3791 623.7980

Fig.4. Convergence diagram of WCA for system with 10 units and

loads of 2400 to 2700 MW

As illustrated in the results of system with 10

units without considering the effect of steam valve,

this system has the best cost in loads of 2400 to 2700

MW in comparison to the other methods. Besides,

Figure 4 shows that this system without considering

the effect of steam valve with load of 2400 MW

converges quicker than the same system with load of

2700 MW.

5.2. System with 10 units considering the effect of input steam valve

There are no too many researches to combine

the effect of input steam valve and the constraint of

different consuming fuels in the ELD problem. For

this purpose, the second test system is considered

with adding the effect of steam valve to previous

system and load of 2700 MW. Parameters of

algorithm for this system are similar to the previous

system. Figure 5 and Table 3 present convergence

diagram and comparison of the results with other

methods, respectively.

Table.3

Comparison of WCA results for system with 10 units considering

the effect of input steam valve and load of 2700 MW

CGA_MU[31] IGA_MU[31] WCA

Unit No Output F Output F Output F

1(MW) 222.0108 2 219.1261 2 220.0550 2

2(MW) 211.6352 1 211.1645 1 210.9169 1

3(MW) 283.9455 1 280.6572 1 279.6702 1

4(MW) 237.8052 3 238.4770 3 238.7458 3

5(MW) 280.4480 1 276.4179 1 280.1485 1

6(MW) 236.0330 3 240.4672 3 239.6610 3

7(MW) 292.0499 1 287.7399 1 287.7073 1

8(MW) 241.9708 3 240.7614 3 239.6864 3

9(MW) 424.2011 3 429.3370 3 427.4088 3

10(MW) 269.9005 1 275.8518 1 275.9998 1

Ftotal (S/h) 624.7193 624.5178 623.8509


206

Fig.5. Convergence diagram of WCA for system with 10 units

considering the effect of input steam valve and load of 2700 MW

There are a few researches to simultaneously

study the effect of steam valve and units with

different fuels in system with 10 units with

considering the effect of steam valve due to

complications. Therefore, there was just a system

surveyed once with load of 2700 MW. Fuel cost of

this system is 623.8509 which are improved slightly

comparing to the other two algorithms. Figure 5

illustrates a comparison of system with 10 units and

load of 2700 MW with and without the effect of

steam valve. The effect of steam valve makes

convergence a little difficult and increases the

probability of converging to the local minimums. In

the first mode, convergence was achieved at about 10

iterations but the effect of steam valve delayed it to

25 iterations.

5.3. Sensitivity analysis of algorithm C

As mentioned before, this parameter controls

the way of flowing from streams to rivers and seas

and has a value between 0 and 2. Values higher than

1 causes the streams to be able to flow in two

directions. Table 4 and Figure 6 present the

comparison of results for 4 different values of this

test system.

Table.4

Results of WCA for system with 10 units and load of 2700 MW and different values of C

C =0.1 C =0.8 C=1.4 C =2

Unit No Output F Output F Output F Output F

1(MW) 219.5905 2 215.2694 2 212.0377 2 220.0550 2

2(MW) 207.1839 1 222.9899 1 215.1524 1 210.9169 1

3(MW) 297.7758 1 283.4461 1 279.4089 1 279.6702 1

4(MW) 242.3708 3 241.0252 3 240.6257 3 238.7458 3

5(MW) 311.3986 1 294.1461 1 279.5790 1 280.1485 1

6(MW) 245.5736 3 244.8194 3 240.0692 3 239.6610 3

7(MW) 302.0291 1 277.4023 1 285.1218 1 287.7073 1

8(MW) 244.5238 3 243.4481 3 236.9990 3 239.6864 3

9(MW) 350.3311 3 393.6467 3 424.3429 3 427.4088 3

10(MW) 279.1511 1 283.6891 1 286.5712 1 275.9998 1

Ftotal (S/h) 627.4213 626.3751 624.3440 623.8509

Fig.6. Sensitivity analysis diagram of WCA for system with 10

units and load of 2700 MW and parameter C

5.4. Sensitivity analysis of Parameter dmax

As mentioned in this section, this parameter

prevents algorithm from quick convergence and to be

trapped in local minima. The value of this parameter

is updated after each iteration. Results and

convergence diagram for three different values of

dmax are shown in Figure 7 and Table 5, respectively.

6. Conclusion

In this paper, a new algorithm is employed in order

to solve the economic load dispatch problem. This

algorithm uses fewer factors to reach to the

optimum solution comparing to the other algorithms

which causes this algorithm to solve the problems

quicker while maintaining the accuracy. In order to

show the ability of WCA in solving non-linear

problems, a system with 10 units in two modes is

investigated i.e. with and without considering the

effect of steam valve.

Table.5

Results of WCA for system with 10 units and load of 2700 MW

and different values of dmax

dmax =2 dmax =0.1 dmax =0.01

Unit No Output F Output F Output F

1(MW) 218.5807 2 220.0550 2 216.3999 2

2(MW) 214.1351 1 210.9169 1 211.4120 1

3(MW) 280.6462 1 279.6702 1 284.4471 1

4(MW) 239.2832 3 238.7458 3 240.7613 3

5(MW) 273.7120 1 280.1485 1 278.8212 1

6(MW) 239.1236 3 239.6610 3 237.2423 3

7(MW) 287.7891 1 287.7073 1 289.7176 1

8(MW) 239.9551 3 239.6864 3 240.8957 3

9(MW) 427.5427 3 427.4088 3 425.7371 3

10(MW) 279.1428 1 275.9998 1 274.5656 1

Ftotal (S/h) 623.9863 623.8509 623.9657

In this system when the effect of steam valve is

not considered, as observed from the related Table,

the cost of this system is obtained as 481.7216,

526.2279, 574.3791 and 623.7980 for loads of 2400


207

to 2700 MW. These results are less than all the other

methods and more near to optimum value. Although

this reduction is not very noticeable, it is very

valuable considering the type of problem and other

performed researches up to now in ELD problem

with different kinds of fuels. In this system when the

effect of steam valve is considered, this effect makes

convergence a little difficult and increases the

probability of converging to the local minimums. In

the first mode, convergence was achieved at about 10

iterations but the effect of steam valve delayed it to

25 iterations. At last, Sensitivity analysis of

Parameters C and dmax was carried out on a system

with 10 units and load of 2700 MW. As shown by

the results and section 3, the best value for parameter

C is 2 which give the best results. The results of

parameter dmax shows that higher values of this

algorithm increase discovery range and search span

of the algorithm but too high values of this

parameter worsens the quality of the solution.

Besides, too low values of this parameter would also

lead to solution with little quality but a proper

reduction in this parameter increases the

convergence.

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time of unit i variables

time t i

tp generation of unit

tsp generation of wind

time of unit i offit

ti commitment status

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tsd shutdown cost of