i UNIVERSITY OF NAIROBI COLLEGE OF ARCHTECTURE AND ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING DYNAMIC ECONOMIC DISPATCH (DED) USING ANALYTICAL HIERACHIAL PROCESS (AHP) PROJECT INDEX: 54 SUBMITTED BY: MAISIBA ZACHARY CHOROKE F17/1354/2010 SUPERVISOR: PROF: N. O ABUNGU EXAMINER: Dr.WEKESA PROJECT REPORT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT OF THE AWARD OF THE DEGREE OF BACHELOR OF SCIENCE IN ELECTRICAL AND INFORMATION ENGINEERING OF THE UNIVERSITY OF NAIROBI 2015 SUBMITTED ON: 24/4/2015
Transcript
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UNIVERSITY OF NAIROBI COLLEGE OF ARCHTECTURE AND ENGINEERING
DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING
DYNAMIC ECONOMIC DISPATCH (DED) USING ANALYTICAL HIERACHIAL PROCESS (AHP) PROJECT INDEX: 54
SUBMITTED BY:
MAISIBA ZACHARY CHOROKE
F17/1354/2010
SUPERVISOR: PROF: N. O ABUNGU
EXAMINER: Dr.WEKESA
PROJECT REPORT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT OF THE AWARD OF THE DEGREE OF BACHELOR OF SCIENCE IN ELECTRICAL AND INFORMATION ENGINEERING OF THE UNIVERSITY OF NAIROBI 2015
SUBMITTED ON: 24/4/2015
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DECLARATION OF ORGINALITY.
NAME OF STUDENT: MAISIBA ZACHARY CHOROKE
REGISTRATION NUMBER: F17/1354/2010
COLLEGE: Architecture and engineering.
SCHOOL: Engineering.
DEPARTMENT: Electrical and electronic engineering.
COURSE NAME: BSc. electrical and electronic
engineering.
TITLE OF WORK: DYNAMIC ECONOMIC
DISPATCH (DED) USING
ANALYTICAL HIERACHIAL
PROCESS (AHP).
1) I understand what plagiarism is and I am aware of the university policy in this regard.
2) I declare that this final year project report is my original work and has not been submitted
elsewhere for examination, award of a degree or publication. Where other people’s work or my
own work has been used, this has properly been acknowledged and referenced in accordance with
the University of Nairobi’s requirements.
3) I have not sought or used the services of any professional agencies to produce this work
4) I have not allowed, and shall not allow anyone to copy my work with the intention of passing
it off as his/her own work.
5) I understand that any false claim in respect of this work shall result in disciplinary action, in
accordance with University anti-plagiarism policy.
Signature:
…………………………………………
Date:
…………………………………………..
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CERTIFICATION: This report has been submitted to the Department of Electrical and Information Eng. University of
Nairobi with my approval as supervisor:
Prof. Nicodemus Abungu Odero
Signature ………………………………..
Date ………………………………………..
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DEDICATION. To Samuel Zachary, Kwamboka Mokua ,Daniel Karasi and my family.
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ACKNOWLEDGEMENTS. I thank God for the far He has brought me, my family for their unending support and my supervisor
Mr. Musau for his motivation, criticism and insight.
I also thank my supervisor Prof. Nicodemus Abungu for his motivation.
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LIST OF FIGURES. Figure1. 1Input output characteristics of a generating unit ....................................................................... 19
Figure1. 2 A simple model of a fossil power plant ...................................................................................... 20
Figure1. 3 Cost function with five valves .................................................................................................... 21
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LIST OF TABLES.
Table 1. 1 Analogy of Annealing process in Solids ........................................................................................ 8
TABLE OF CONTENTS DECLARATION OF ORGINALITY. ..................................................................................................................... i
CERTIFICATION: ............................................................................................................................................ iii
DEDICATION. ................................................................................................................................................ iv
ACKNOWLEDGEMENTS. ................................................................................................................................ v
List of figures. ............................................................................................................................................... vi
List of tables. ............................................................................................................................................... vii
List of abbreviations ................................................................................................................................... viii
Abstract. ....................................................................................................................................................... xi
Literature review ..................................................................................................................................... 18
2.1 literature review on dynamic economic dispatch ............................................................................ 18
2.2 literature review on analytical hierarchical process ......................................................................... 22
Results and analysis ................................................................................................................................ 33
4.1 Case study: IEEE bus system ....................................................................................................... 33
Program listing. ....................................................................................................................................... 45
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ABSTRACT. Power systems are built and maintained to achieve reliable and economic power supply.
To achieve the above objectives, analysis and computations have to be done to determine the most
efficient generators to load (unit commitment) problem and how much of the load each generator
will carry for minimal fuel cost (economic dispatch).
The dynamic economic dispatch problem has been formulated as a mathematical optimization
problem and its objective is to minimize the fuel cost as the load demand increases by calculating
the amount of load each generator should carry for maximum efficiency.
In solving the above problem, various mathematical and engineering algorithms have been
developed.
In this project, the analytical hierarchical process is used to allocate generating levels to various
generators to obtain the optimum cost function while considering the equality constraint, inequality
constraint and ramp rate constraints.
The DED equation formulated as;
Fi(Pi)=(ai+bi Pi+ci Pi2)+(eisinxi) ………(8)
Where Xi=fi(Pi min- Pi) ……….(9)
The above cost function is used to form a hierarchy tree form which the AHP method is used to
find the rank of each generator for each load demand.
AHP is a multi-objective decision making process which can be used both qualitatively and
quantitatively.
AHP algorithm is tested on a 30 bus IEEE network. The algorithm is run to solve a classical
economic dispatch problem without constraints, CED with minimal and maximum constraints and
DED with ramp rate limits and valve point effects.
The algorithm was tested for the following load demands;
Power Demand (MW) =[137 283.4 374.3 479.3 ]; …(a)
The costs obtained for the above power demands were as follows;
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Total cost($)=[454.3845 886.5009 1084.8 1209.9]…(b)
The results obtained were compared to each other and from the comparison, the cost reduces as
the algorithm is improved from classic economic dispatch to dynamic dispatch where the cost is
the least.
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CHAPTER 1
INTRODUCTION
DYNAMIC ECONOMIC DISPATCH
1.1.1 Economic dispatch
Economic operation is very important for a power system to return profit to the capital invested
which in turn reduces the cost of a kilowatt hour to the consumer [1] [2].
Operational economics involving power generation and delivery can be subdivided into two parts;
A. Minimum cost of power production(economic dispatch)
B. Minimum loss delivery of the generated power to the load
Both the economic dispatch problem and minimum loss problem can be solved by various methods
and power flow algorithms.The process of allocating generating levels to the generating units so
that the system load is fully supplied in the most economical way. The process of allocating
generating levels to the generating units so that the system load is served entirely and most
economically while considering operating limitations of the generating units. Traditional
economic dispatch problem attempts to minimize the cost of supplying energy subject to
constraints on static behavior of the generating units. It is assumed that the amount of power to be
supplied to be supplied to the generating units is constant for a given interval of time. Classic
economic dispatch can also be defined as the solution of the economic dispatch problem subject
to static constraints (behavior) of the generating units.
To avoid shortening of the life of their equipment, plant operators, try to keep thermal gradients
inside the turbines within safe limits. This mechanical constraint is usually translated into a limit
on the rate of increase of the electrical output. Such ramp rates distinguish the DED problem from
the static economic dispatch.
The DED problem serves to schedule the generator outputs with the predicted load demands over
a certain period of time so as to operate an electric power system most economically.
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1.1.2 Dynamic economic dispatch
The problem of determining minimum system cost of dispatch generators, taking into
consideration the constraints imposed in system operation by generator ramping rate limitations.
Power output of fossil plants is increased sequentially by opening a set of valves to its steam
turbine at the inlet. The throttling losses are large when a valve is slightly opened and small when
the valve is fully opened [3].
The fuel cost function with valve points in the generating units is accurate model of the DED
problem.
1.2 SURVEY OF EARLIER METHODS
There are a number of traditional methods to solve the economic dispatch problem. The methods
can be classified into the following categories [1];
Mathematical programming based methods/heuristically based methods.
a) Lambda iterative method.
b) Gradient projection method.
c) Lagrange relaxation method.
d) Linear/non-linear.
e) Interior point methods.
f) Dynamic programming.
Methods based on artificial intelligence.
a) Artificial neural networks.
b) Stochastic optimization methods.
c) GA, SA, EP, DE, PSO
Hybrid methods. A combination of two or more methods.
a) EP-SQP
b) PSO-SQP
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c) HNN-QP
d) EP-PSO-SQP
In reality, due to valve point effect, the cost fuel function is non-smooth and non-monotonically
increasing and traditional methods fail.
1.2.1 Methods based on artificial intelligence.
In order to solve this problem, many stochastic methods such as;
a. Genetic algorithm.
b. Particle swarm optimization.
c. Simulated annealing.
d. Differential evolution.
Stochastic methods may prove to be very effective in solving the non-convex ED problems without
any restriction on the shape of the cost curve.
The problem with stochastic search techniques is that they cannot guarantee an optimal or near
optimal solution in a single run. Even after fixing up the parameters initially cannot guarantee near
optimal solution because of the randomness involved in the solution technique.
Therefore, the solution obtained is not unique in each trial having a fixed number of iterations.
1.2.1.1 Genetic algorithms
Genetic algorithms are adaptive heuristic search algorithms based on the evolutionary ideas of
natural selection and genetics. GA algorithms are designed to simulate processes in a natural
system necessary for evolution, specifically those that the principles laid down by Charles Darwin
of survival of the fittest [4] [5].
1.2.1.2 Differential evolution algorithm.
A population based algorithm that uses three operators; mutation, crossover and selection to evolve
from randomly generated initial population to find individual solution.
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The basic idea behind differential evolution is that it starts with an initial population of feasible
target vectors (parents) and new solutions (offsprings) are generated (by mutation, crossover and
selection operations) until optimal solution is reached.
For the mutation operation, three different vectors are selected randomly from the population and
a mutant vector is created by perturbing one vector with the difference of the other two vectors.
For the crossover operation, a new trial vector (offspring) is created by replacing certain
parameters of the target vector by the corresponding parameters of the mutant vector on the bases
of a probability distribution. For the DED the competition between parents and offspring is one to
one. Individuals with the best fitness will remain until the next generation.
The iteration process then continues until a user-specific stopping criterion is met.
Differential evolution has three control parameters, which are differentiation (mutation) factor f,
crossover constant CR, and the size of the population Np. According to Storn and Price the basic
strategy of DE for m-dimensional optimization problem can be described as follows:
(1) Initialization: generate a population of Np initial feasible target vectors (parents)
Xi = {X1i , X2i , … ,Xmi},
i=1,2,…,Np randomly as
Xji=Xjmin +S1.(Xj
max-Xjmin), j=1,2,…,m, i=1,2, … ,Np
….(1)
Where
S1 is uniform random number in [0,1]; Xjmin and Xj
max are lower and upper bounds of the jth
components of the target vector.
(2) Mutation: let XiG
={X1iG
, X2iG, … ,Xmi
G} be the individual at i at the current generation
G. A mutant vector ViG+1=(V1i
G+1, V2iG+1,..., Vmi
G+1) is generated according to the
following;
ViG+1= Xrj
G+F. (Xr2G- Xr3
G)…(2)
Where r1≠ r2≠ r3≠I i=1,2, …, Np.
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with randomly chosen integer indexes r1,r2, r3 ε{1,2,3,…,Np}
(3) Crossover: According to the target vector XiG and the mutant vector Vi
G+1 a new triall vector
(offspring)
UiG+1={ U1i
G+1, U2iG+1,…, Umi
G+1}…(3)
is created with
UjiG+1={ vji
G+1 if (rand (j)≤CR) or j=rnb(i)
XjiG otherwise
Where j= 1,2, …, m, i= 1,2 ,…., NP and rand(j) is the jth evaluation of a
Uniform random number generator between [0,1] CR is the crossover
Constant between [0, 1] which has to be determined by the user. rnb(i)
Is a randomly chosen index from 1,2, 3,…, m which ensures that UiG+1
Gets at least one parameter from ViG+1 .
(4) Selection: determines which vector is to be chosen fro the next generation by implementing
one-to – one competition between the new generated trial vectors and their corresponding
parents. The selection operation can be expressed as follows ;
XiG+1={ Ui
G+1 if f(UiG+1)≤f(Xi
G)…(4)
XiG otherwise
Where i=1,2,…,Np and f is the objective function to be minimized.the value of f of each trial
vector UiG+1 is compared to that of the its parent vector Xi
G. if the value of f , of the target vector
XiG, is lower then that of the trial vector , the target vector is allowed to advance to the next
generation . hence all the individuals of the next generation are as good as or than their counter
parts in the current generation. The above steps of reproduction and selection are repeated
generation after generation untill some stopping criteria are satisfied. The evaluating function for
evaluating the fitness of each individual in the population in the DE algorithm as follows ;
F=Ct+λ∑i=1T(∑i=1
N pit – (Dt+losst))2…(5)
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Where
λ=penality value
The objective is to find fmin, the minimum evaluation of all the individuals in all the iterations.The
penality term reflects the violation of the equality constraint . Once the minimum f is reached ,the
equality constraint is satisfied.
Generation power output of each unit at a time t should be adjusted to satisfy the following
constraints.
Pit={ Pi
t,min if Pit< Pi
t,min…(6)
Pit if Pi
t,min≤ Pit ≤ Pi
t,max
Pit,max if Pi
t> Pit,max
Where
Pit,min={ Pi
t,min if t=1
Max(Pimin, Pi
t-1- DRi) others
Pit,max={ Pi
t,max if t=1
Max(Pimax, Pi
t-1- DRi) others
1.2.1.3Particle swarm optimization
A population based stochastic search algorithm which was first introduced by Kennedy and
Eberhart in 1995. The original objective of their research was to graphically model the social
behavior of bird flocks and fish schools. This original version can only handle the non-linear
continuous optimization problems. Advancements in the PSO algorithm can explore the global
solution of complex problems of engineering and sciences [6].Among various versions of PSO,
the most familiar version was proposed by Shi and Eberhart.
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The key attractive feature of PSO is its simplicity as it involves the following two equations;
Vik+1=k(Vi
k *w+C1*R1*(p-best(i) –Xik ) +C2*R2*(G-best – Xi
k)…(17)
Xik+1= Xi
k + Vik+1….(18)
K=2/abs(2-c-sqrt(c2-4*c) …..(19)
Where
Vik - velocity of particle I at iteration k
w-inertia weight factor.
C1,C2 - acceleration constants.
R1,R2 - uniformly distributed random number between 0 and 1.
Xik – position of particle I at k iteration.
P-best – best position of particle I until iteration k.
G-best - best position in the group until iteration k.
K – constant factor
In PSO , the coordinates of each particle represent a possible solution called particles associated
with position and velocity vector. At each iteration, the particle moves towards optimal solution
,through its present velocity, personal best solution obtained by each particle and global best
solution obtained by all the particles. In a physical d dimensional search space, the position and
velocity of the particle I are represented as vectors of Xi=[ Xi1, Xi2, … , Xid] and
Vi=[ Vi1, Vi2, … , Vid] in PSO algorithm. Let the pbest i=[Xi1pbest, Xi2pbest, … , Xidpbest] and Gbest=[
X1gbest , X2gbest, … , Xngbest] be the best position so far respectively.
The modified velocity and position of each particle can be calculated using the current velocity
and distance from Pbest I and Gbest by equations (a), (b), (c).
In this velocity updating process, the value of parameters such as w,C1, C2 ,k should be determined
in advance. The inertia weight w is linearly decreasing as iterations proceeds and obtained as;
W=wmax – (Wmax- Wmin)iter/(iter-max)…(20)
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Where
Wmax: final inertia weight
Wmin: initial inertia weight
Iter: current iteration number.
Iter_max: maximum iteration number.
1.2.1.4 simulated annealing
A stochastic optimization technique inspired by the natural process of crystallization (gradual
cooling of matter).
Annealing – a process involving heating and controlled cooling of a material to get perfect crystals
with minimum defects. There is a significant correlation between the terminology of
thermodynamic annealing process (the behavior of systems with many degrees of freedom in
thermal equilibrium at finite temperature) and combinatorial optimization (finding global
minimum of a function based on many parameters).
A detailed analogy of annealing process in solids provides a framework for optimization [7] [6].
Table 1. 1 Analogy of Annealing process in Solids
THERMODYNAMIC ANNEALING SIMULATED ANNEALING
System state Feasible solution
Energy Cost
Change of state Neighboring solution
Temperature Control parameter
Frozen state Heuristic solution
The main advantage of SA approach is that it does not require large computer memory. The main
purpose of optimization is to achieve fast convergence as well as better exploration capability.
Annealing process in thermodynamics
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Molecules of a metal become unstuck from their initial positions and wander randomly at high
temperatures.
By gradual cooling, thermal mobility is lost and atoms start to get arranged in the form of of a
crystal.
If the reduction in temperature is done at a very fast rate, a meta-stable state (i.e crystalline state
transforms to amorphous structure) is obtained which responds to a local minimum of the energy
level.
For a thermal equilibrium state of a system for a temperature T, afterwards the probability Pt(s)
with its pattern S depend on energy level of the corresponding pattern E(s), and is depending on
Boltzmann distribution ;
Pt(s)= e-E(s)/kt/ ∑w e-E(s)kt…(21)
Where, k is known as Boltzmann constant and the sum ∑w consists of all the promising states of
N.
Let the system have a configuration, g, which corresponds to energy E (g). If one one of the
molecules of the system is displaced from its initial position, then a new state σ corresponding to
energy E (g) occurs. If E (σ) ≤E (g), then a new state is accepted. If E (σ)>E (g), then a new state
is accepted with probability:
e-(E(σ)-E(g))/kt…(22)
1.2.1.4.1Critical temperatures of SA algorithm.
For the successful application of the SA algorithm, the annealing schedule is important. We have
four control parameters that are associated with the convergence and efficiency of the simulated
annealing algorithm;
1. Initial temperature.
2. Final temperature.
3. Rate of temperature decrement.
4. Iteration at each temperature.
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1.2.1.4.2Initial temperature.
Must be at a higher value (to get more probability of acceptance for non-optimized solutions during
the first stages of the algorithm.
Too much higher initial temperature makes the algorithm and computationary inefficient.
Low initial temperature may not be capable of searching a minimum especially for a multimodel
function.
1.2.1.4.3Final temperature.
While working with SA algorithm generally the final temperature is set to zero degrees Celsius.
Temperature decrement.
As initial and final temperature have predefined values, it is essential to find the approach of
transition from starting to its final temperature as the success of the algorithm depends on it.
Decrement of temperature at a time t is;
T (t) =d/ (log (t))… (23)
Where d is a positive constant.
The temperature decrement can also be implemented by using
T (t+1) =aT (t)… (24)
Where a, is a constant close to 1. Its effective range is 0.8<a<0.99 iterations at each temperature.
To enhance efficiency of the algorithm, selection of proper number of iterations is another
important factor. Lundy et al (1985) suggests the realization of only one iteration for each
temperature and the fall in temperature should take place at a really slow rate which can be
expressed as;
T (t) =t/ (1+β.t)… (25)
Generally β has a very small value.
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1.2.1.4.4 Simulated annealing algorithm
1. For initialization choose temperature T, parameter α and maximum number of iterations ‘max
tries’ , to generate an initial feasible solution by random process and store it as current solution
Si. Then perform the DED in order to evaluate the total cost Fcost , while satisfying the equality,
inequality and ramp rate constraints.
2. Set the iteration counter to μ=1.
3. Create an adjacent solution Sj through the rand operator and compute the new cost, Fcost.
4. If the new solution is found to be better, accept it; otherwise find the deviation of cost ΔS=Sj-
Si and generate a random number Ω ϵ (0, 1) out of a uniform distribution using the following
logic. If e-Δs/t≥Ωϵ (0, 1), accept the new solution Sj to replace Si.
5. Reduce the temperature by a parameter α, until the stopping criterion is satisfied T (t) =α.T and
go back to step 2.
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1.2.1.5 SDOA
Consider a blind person who wants to reach an object, he will rely intuitively on his remaining
senses: hearing, tasting, touching, and smelling with various degrees of dependency. However the
three senses [taste, touch and smell] will only provide him with feedback of nearby obstacles [8].
The feedback of the remaining sense –hearing- could symbolize the slightly distant obstacles.
Utilizing these feedback will assist the blind person to his goal.
It is clear that the exploitation and exploration process are present in the previous assumption.
However, for the exploration process, the process is divided into 2 strategies;
1. Employ hearing in order to receive feedback from adjacent region of the search space of
the remaining senses.
2. Retires the worst- performing sensory-deprived person (population) and replaces them
by those who perform better in parallel (temporary) search process.
This parallel search dynamically diminishes when the number of iterations increases.
Each time the solution will be modified into sense, it follows this form;
1. For taste , touch , smell
Xijnew= Xij
old+u. (Xijold- Xkj)… (26)
2. For hearing
Xijnew= Xij
old+U. ( Xijold- Xkj)…(27)
Where
uϵ[-1.1]
Uϵ [-5, 5] and ϵ≠u
Each constraint of the DED problem has a degree of dominance affecting the algorithm
performance, and somehow directs the algorithm towards the optimal or quasi-optimal region.
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The proposed search-tactic utilizes those constraints “specifically the dominance power-balance
constraint” to accelerate the algorithm performance towards the optimal feasible region.
The following procedures describe the proposed constrained search-tactic for the scenario for a
one hour dispatch period.
On the other hand, only steps 2 and 3 are utilized when multiple time intervals are considered.
Proposed constrained-search tactic
1. The units’ output power are updated every dispatch hours as follows;
pI,mint=max (pI,min, pi
t-1-DRi)…(28)
pI,maxt=min(pI,max, pi
t-1+URi)…(29)
2. The objective function is altered “temporary” to minimizing the violation of the real power
balance equation. The main objective function is retained once the loop (cycle), (ɸ+1) starts.
3. The handling mechanism for the unit’s prohibited zones is as follows;
Consider a unit (i) in a solution vector (xi) operates at a time t within a prohibited operating
zone (j);
(3.1) divide the solution (xi) into two sub-solutions by only modifying the unit (i)
Output according to its violated prohibited operating zone (j).
(3.2) force each one of the two sub-solutions to adjust unit (I) output to operate in its
Permissible upper and lower limits of the associated prohibited operating
Zone (j).
(3.3) evaluate both sub-solutions, and select the best based on designated solution method.
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1.2.2Mathematical program based methods
1.2.2.1Maclaurin series based Lagrangian method.
The refined sinusoid function in the cost equation is represented by Maclaurin sine series
approximation and then solved by using Lagrangian method [9].
The maclaurin series expansion of the sine function is given as
Sin x= 𝑥 −𝑥3
3!+
𝑥5
5!−
𝑥7
7!… (7)
The fuel cost function is given as
Fi(Pi)=(ai+bi Pi+ci Pi2)+(eisinxi) ………(8)
Where Xi=fi(Pi min- Pi) ……….(9)
The problem complexity increases if higher order terms in equation (7) are used. So the first two
terms of Maclaurin series are considered.
Sin x≃ 𝑥 −𝑥3
3! ……..(10)
Substituting equation 10 in equation 9 we get the cost equation which is given as;
Fi(Pi)=(ai+bi Pi+ci Pi2)+|(eI fi{
pifi2
6−
pimin fi2
6 Pi
2 +[ pi2fi2
2 -1]pi+ pi min-
(pimin )^3fi2
6}|…..(11)
The derivation of equation (11) which is the incremental cost equation is given as:
d fi(pi)
𝑑𝑝𝑖 =bi+ 2cipi +|eipi[0.5 fi
2pi2- fi
2Pi min pi +0.5 fi2pi min
2-1]|….. (12)
We can rearrange the terms of equation 12
dFi(Pi)
𝑑𝑝𝑖=bi + 2cipi +|-eifi(1-(xi
2/2!))| …. (13)
The sine term is approximated, hence there will be an approximation error and the
solution may not converge and to compensate for this approximation, an initialization,
factor.
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yi is multiplied to the right hand side of equation (13)
yi= real (cos-1(1-(xi2)/2!+(xi
4/4!))/xi….(14)
The generated power, pi is initially unknown and it can be chosen by the equation
Pi= ( Pi min +Pi)/v …(15)
Where
v= normalizing factor that normalizes the value of yI between 0 and 1.
Selection of a proper value of V minimizes the error and guarantees an optimal solution
For the DED problem. If the v value is fixed properly for any type of system, then optimal
or near optimal solution can be obtained by the proposed method.
With the initialization factor, Yi in equation 9 and then substituting in equation 13 we get;
dFi(Pi)
𝑑𝑝𝑖=bi + 2cipi +|-eifi(1-(yi fi(pi min – pi))
2/2!)) …(16)
1.1.1 Hybrid algorithms
These are combinations of two or more of the mathematical and dynamic programming algorithms.
The disadvantages of one algorithm are improved by the other algorithm hence they are one of the
most effective optimization methods [1, 9].
Such methods include:
a) EP-SQP.
b) PSO-SQP.
c) HNN-QP.
d) EP-PSO-SQP.
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1.2.9 Summary
The DED problem is a multi-objective and multi-criteria decision making process since the
engineers have to compute the loading levels of each generator so as to use the lowest fuel while
taking into consideration the various generating unit constraints, equality constraints and
inequality constraints.
The advantage of the AHP method is that it takes into account the qualitative and quantitative
aspect of analysis.
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1.3Problem statement.
To solve the DED problem with valve points and ramp rates using the analytic hierarchical process.
1.3.1 Project objectives.
The aim of the project is to optimize the DED problem with valve points and ramp rate constraints
(which is the accurate model of the DED problem) by using the Analytic Hierarchical Process and
compare the results with other methods. From the comparison, it is to be determined whether the
DED problem was optimized or otherwise.
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CHAPTER 2
LITERATURE REVIEW
2.1 LITERATURE REVIEW ON DYNAMIC ECONOMIC DISPATCH
Power systems are built and operated with the following goal: to achieve a reliable and economical
electrical supply. For the consumer to have a reliable and economic electrical power supply, a
complex set of engineering analysis and design solutions need to be done. Optimal system
operation is a wide area of study which involves consideration of the economy of operation, system
security, emissions at certain fossil fuel plants and optimal release of water at hydropower
generating plants. All considerations may make for conflicting requirements and usually a
compromise has to be made for optimal system operation.
In this project, we consider the cost of operation and thus we are dealing with the economic
dispatch problem. The main aim of the economic dispatch problem is to minimize the total cost of
generating real power (production cost) while satisfying the loads and losses in the transmission
links.
Fuel cost is meaningful in the case of nuclear and thermal stations. When a power system is
operated economically, the invested capital is returned, profits are made to pay the workers and
maintain the system and also the cost per kilowatt hour is reduced hence the consumer pays less.
For a fuel –fired station, the input-output curve of the generating units is given below;
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Figure1. 1Input output characteristics of a generating unit
The Pgimin is the minimum loading limit below which, operating the unit proves to be
uneconomical (or may be technically infeasible) and Pgimax is the maximum output limit.
Generally the equation of the input output characteristics has been formulated as a mathematical
optimization problem. The mathematical equations are then solved for minimum operating costs
with a set of equality and inequality constraints.
In general terms, power systems problems are mathematically modelled as;
Min f(X)…(30)
Subject to
g(x)=0
h(x)≤0
Where;
x is a set of decision variable vector
f(x) is the objective function
g(x) is the equality constraint
h(x) is the inequality constraint
power output of fossil plants is increased sequentially by opening a set of valves to its steam turbine
at the inlet. The throttling losses are large when the valve is just opened and small when the valve
is fully opened.
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A simple model of a fossil power plant is as below;
Figure1. 2 A simple model of a fossil power plant
Classical economic dispatch attempts to minimize the cost of supplying energy subject to
constraints that focus in the static behavior of the generating units. It is assumed that the amount
of power to be supplied by a given set of units is constant for a given interval of time.
To extend the life span of the generating units, thermal gradients inside the generating units are
kept within safe limits.
This results into a mechanical constraint that is translated to a limit on the rate of the electrical
output. This is the ramp rate constraint that distinguishes the classical dynamic dispatch and
economic dynamic dispatch.
The fundamental objective of DED problem is to schedule the committed generating unit outputs
in order to meet the predicted load demand with minimum operating cost, while satisfying all the
system inequality and equality constraint.
As earlier stated, the fuel cost function with valve point loadings in the generating units is the
accurate model of the dynamic economic dispatch problem.
The total cost function is given as;
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F= mi𝑛 ∑𝑇𝑡=1 ∑ 𝑓𝑖(𝑝𝑖)
𝑁𝑖=1 …(31)
And the simplified cost function for each generator can be represented as;
* …(32)
Where
F= total generation cost
Fi= cost function of generator i
Ai , bi , cI are the cost coefficients of generator i
PI = power of generator i
N= no of generators
Ei and hi coefficients of generator I reflecting valve point coefficients.
Figure1. 3 Cost function with five valves
We solve the above problem considering the following constraints;
(1). Equality constraints
∑ 𝑃𝑔𝑖𝑘𝑖=1 –𝑃𝑑 − 𝑃𝑙=0…(33)
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(2). Inequality constraints
Pi min≤ Pi
t≤ Pi max…(34)
(3) Ramp rate limits
-DRi.T≤ Pi T+1≤-URi.T …(35)
2.2 LITERATURE REVIEW ON ANALYTICAL HIERARCHICAL PROCESS
AHP is a technique used to solve multi-criteria decision problems requiring paired comparison
judgments concerning the dominance of one element over another for each n elements with respect
to an element on the next higher level using a 1-9 scale.
The advantage of AHP is that it has both the qualitative and quantitative parts (makes decisions
when alternatives depend on criteria with simple interactions).
The scale for pairwise comparisons is given as;
Table 2. 1 Pairwise comparisons
1 equally preferred
3 One is moderately preferred over another.
5 One is strongly preferred over another.
7 One is very strongly preferred over another.
9 One is extremely preferred over another.
2,4,6,8 are intermediate values.
Reciprocals of the above scale are used for inverse comparisons.
Steps for the AHP algorithm
1. Set up the hierarchy model.
2. Form judgment matrices. The value of elements in the judgment matrix reflects the user’s
knowledge about the relative importance between every pair of factors.
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3. Calculate the maximum Eigen vector values and corresponding Eigenvector of the
judgment matrix.
4. Hierarchy ranking and consistent check of results.
The consistency index of hierarchy ranking [CI] is defined as;
CI=( λ max- n) /(n-1) …(36)
Where
λ max = maximal Eigen value of the judgment matrix.
n=dimension of the judgment matrix
Stochastic consistency ratio
CR=CI/RI …(37)
For n=3,
The required consistency ratio should be less than 0.05
For n=4, the required consistency ratio should be less than 0.08
For n≥5 should be less than 0.10 to get a sufficient consistency matrix otherwise, the matrix should
be revised.
RI=a set of given average stochastic consistency indices and CR is the stochastic consistency ratio.
For matrices with 1-9 dimensions respectively, the value of RI will be as follows;
Table 2. 2 Values of RI for various matrices of dimension 1-9
N 1 2 3 4 5 6 7 8 9
RI 0.00 0.00 0.58 0.90 1.12 1.24 1.32 1.41 1.45
Basic principle of AHP is to calculate the Eigen vector of the alternative for each criterion.
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For the qualitative factors such as the relative importance of units and criteria, the corresponding
Eigen vectors can be obtained by the judgment matrix.
Generally, the judgment matrix A has the following characteristics;
aij >0 …(38)
aji =1/ aij i≠j…(39)
aii=1 i,j =1,2,….,n …(40)
aij the element in the judgment matrix
n the dimension of the judgment matrix.
Judgment matrix A is positive and irreducible matrix
Consistency of a judgment matrix
We say a matrix A is consistent if there exists
aij= aik/ajk , for all I,j,k
If a positive matrix A is consistent, it has the following properties;
(a) aji =1/ aij aii= 1 I,j =1,2,…,n …(41.a)
(b) The transpose of A is also consistent.
(c) Each row in A can be obtained by multiplying a row of positive numbers
(d) The maximal Eigen value of A is λmax = n. The other Eigen values of A are all zero.
(e) If the Eigen vector of A corresponding to the largest Eigenvalue λmax is
x=[x1, x2 ,..., xn]T,..(41.b)
Aij= xi/ xj ; I,j =1,2 ,…, n …(42)
The following methods are used to compute the maximal Eigen value and the corresponding Eigen
vector.
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Root method.
(i) multiply all elements of each row in the judgment matrix
Mi=𝛱 I xij i=1…n; j=1…n, …(43)
Where
N= the dimension of the judgment matrix A.
Xij = an element in the judgment matrix A.
(ii) calculate the nth root of Mi .
Wi*= √𝑀i
𝑛 , i=1,…, n. …(44)
We obtain the vector W* =[ W1*, W2*,….,, Wn*]T…(45)
(iii) normalize the vector w*
Wi = wi*/∑ Wi ∗
𝑛𝑗=1 i= 1,…, n. …(46)
In this way, we obtain the Eigen vector of the judgment matrix A, that is,
W=[w1 , w2, …, wn]T …(47)
(iv) calculate the maximal Eigenvalue λmax of the judgment matrix.
λmax=∑ (𝐴𝑊)𝑖/(𝑛𝑤𝑖)𝑛𝑖=1 …(48)
j=1, …, n
where (AWi) represents the ith element in vector AW.
Sum method
(i) Normalize every column in the judgment matrix
Xij*=Xij/∑ 𝑋𝑖𝑗
𝑛𝑘=1
I,j = 1, … , n …(49)
Now the judgment matrix A is changed into a new matrix A* , in which the column has
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been normalized.
(ii) Add all the elements of each row in the matrix A*
Wi*=∑ 𝑋𝑖𝑗
𝑛𝑗=1 , i=1, … ,n …(50)
(iii) Normalizing the vector W* we have;
Wi= Wi*/ ∑ 𝑊𝑗 ∗𝑛
𝑗=1 i=1…,n …(51)
Hence we obtain the Eigen vector of the judgment matrix A
W=[w1 , w2, …, wn]T …(52)
(iv) Calculate the maximal Eigen value λmax of the judgment matrix.
λmax=∑ (𝐴𝑊)𝑖/(𝑛𝑤𝑖)𝑛𝑖=1 …(53)
j=1, …, n
Where (AWi) represents the ith element in vector AW
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CHAPTER3
SOLUTION OF THE DYNAMIC ECONOMIC DISPATCH PROBLEM USING THE
ANALYTICAL HIERARCHICAL PROCESS.
3.1 FORMULATION OF DED PROBLEM AHP SOLUTION.
The hierarchy tree for the dynamic economic dispatch is as follows;
Figure3.1 Hierarchy tree of the dynamic economic dispatch problem.
Dynamic economic dispatch
problem
Criteria 1. (How
efficient are the
generators depending
on their location on
the test-network)
Criteria 2.
How much load can
each generator
handle? (Quantitative
analysis)
Optimal
cost
Gener
ator1
Gener
ator2
Gener
ator3
Gener
ator4
Gener
ator5
Gener
ator6
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The first pairwise comparison matrix is formed by considering the positioning of the generators
(criteria 1). The load was assumed to be at bus 2 of the 30 bus IEEE test network and pairwise
judgment matrices generated by considering the position of the generator relative to bus 2. The
greater the distance from bus 2 generator is, the less efficient the generator is. From the above
assumption, the following comparisons matrices were generated;
