Malaviya National Institute of Technology Jaipur

Renewable Energy

Project Report on

UNIT COMMITMENT WITH V2G USING PSO

By:

LOKESH KUMAR PANWAR

(2013PCV5172)

SRIKANTH REDDY K

(2013PCV5163)

TO:

Dr. RAJESH KUMAR Associate Professor,

Department of Electrical Engineering,

Adjunct Associate Professor,

Centre of Energy and Environment,

MNIT JAIPUR

CONTENTS

S.NO CHAPTER PAGE NO.

1 INTRODUCTION 3

2 VEHICLE TO GRID MODELLING 14

3 OPTIMIZATION TECHNIQUES 21

4 SIMULATION AND RESULTS 27

CHAPTER-1

INTRODUCTION

“Unit commitment (UC) is an optimization problem used to determine the

operation schedule of the generating units at every hour interval with varying loads under different

constraints and environments.”

Many algorithms have been invented in the past five decades for optimization of the UC problem,

but still researchers are working in this field to find new hybrid algorithms to make the problem more

realistic. The importance of UC is increasing with the constantly varying demands. Therefore, there is an

urgent need in the power sector to keep track of the latest methodologies to further optimize the working

criterions of the generating units. In this chapter focuses on providing a clear idea of the latest techniques

employed in optimizing UC problems for both stochastic and deterministic loads.

1.1-Introduction: The unit commitment (UC) problem deals with the optimum amount of time for which a

generating unit must be operated at a per hour basis in order to meet the load requirements effectively. With

the help of this optimization, it is possible to supply power with least possible losses and minimum fuel

consumption, in order to maximize the profit. Besides achieving minimum total production cost, a

generation schedule needs to satisfy a number of operating constraints. These constraints reduce freedom in

the choice of starting-up and shutting-down of generating units. The constraints to be satisfied are usually

the status restriction of individual generating units, minimum up time, minimum down time, capacity limits,

generation limit for the first and last hour, limited ramp rate, group constraint, power balance constraint,

spinning reserve constraint, etc. The high dimensionality and combinatorial nature of the UC problem

curtails the attempts to develop any rigorous mathematical optimization method capable of solving the

whole problem for any real-size system.

1.2-General background about UC: The off-peak and on-peak demands of electricity may vary for

different purposes. If the units consumed are properly monitored, it may be possible to save some units

when the demand is less, for instance the load is lesser at night compared to the day time. Thus, the main

objective of this paper is to plan the operating time of different generating units such that it satisfies

constraints. The UC problem is applied for both deterministic and stochastic loads [1]. The deterministic

approach provides the definite and unique conclusions. However, the results obtained for stochastic loads

may not be exact. For the deterministic loads data envelopment analysis (DEA), the principal component

analysis (PCA) approach is employed. DEA is a nonparametric method, in which first the input and output

variables are defined. In the PCA, the numbers of variables used are reduced to the minimum. However, in

stochastic models, the constraints are changed into determinate constraints and then the formulation can be

solved by any of the usual algorithms. The various kinds of objective functions for various environments are

as follows.

1.2.1-Conventional fuel based environment [2]: In Eq. (1), there are three costs to minimize. The first one

is P(i, t) which is generation of unit i at time t, and C(P(i, t)) is fuel cost of unit i at time t. The second one is

the start-up cost and the third, shutdown cost.

𝑚𝑖𝑛 ∑ ∑[C-i-(P(i, t))I(i, t) + SU(i, t) + SD(i, t)]

𝑁0

𝑖=1

𝑁𝑡

𝑡=1

… . . (1)

1.2.2- Stochastic environment [3]: Stochastic environment is one in which randomness is included either in

the objective function or to the constraints. In Eq. (2), the second part creates randomness due to the addition

of wind generation. Nowadays, uncertainty occurs in power systems due to the large scale integration of

renewable resources like solar, wind, EV etc. Hence, the demand and supply may also differ for the

successful and reliable operation of the system within an uncertain environment which is also called a

stochastic environment.

𝑚𝑖𝑛 ∑[∑ C-i-(P(i, t))I(i, t) + SU(i, t) + SD(i, t)]

𝑇

𝑖=1

𝑁𝑡

𝑡=1

+ 𝑀 ∑ ∑(𝑤′(𝑘, 𝑡) − 𝑤(𝑘, 𝑡))

𝑇

𝑡=1

𝑁

𝑘=1

. . . (2)

Objective function (2) consists of operation and start-up/ shutdown (STSD) costs of thermal generators, as

well as the expected wind power spillage.

1.2.3 Profit based environment [4]: The profit based environment is one in which the main objective is to

maximise the profit of an individual generation company. The UC schedule has an indirect effect on the

price and a direct effect on the average cost, thus it is an important part of any bidding strategy. Also there is

flexibility in the UC schedule. Objective function (3) can be defined by maximising F(i,t) which is the profit

of the GENCO.

max ∑ ∑ F(i, t)

ti

… … … … … (3)

where F(i,t) = Revenue – cost.

1.2.3.1-Time based constraint for UC: Under this constraint, the main challenge is to minimize the time

taken to obtain the optimal architecture. The time constrained UC (TCUC) problem comprises of non-linear

constraints such as the minimum up and down time for each unit [5]. Once the unit is running, it should not

be turned off immediately, and when it is turned off, there is a minimum time before which it cannot be

turned back on. Thus to meet this constraint, the operation scheduling should be done according to the up-

time and down-time of the generating units. Here the stopping criterion, Total CPU time greater than or

equal to Allowed time is added to stop the algorithm exactly at a moment where the total CPU time taken

exceeds the allowed time [6].

1.2.3.2- Emission based constraint for UC: More than 50 percent of the emission comes from the fossil

fuel based power systems [7]. Hence it can be concluded that the maximization of the profit of GENCOs and

the minimization of the emissions are two conflicting objectives. Therefore, a multi-objective approach can

be followed to achieve sub-optimal solutions if not optimal solutions [1]. If both the cases are considered, it

will lead to a small saving of cost which will bring down the fuel consumption and hence the emission rate.

1.2.3.3- Fuel based constraint for UC: The present scenario requires efficient power generation by

utilizing the resources properly. The cost of fuel is also a major economic concern. During off-peak load

conditions, if the generating units can be managed, then fuel requirements will be able to be reduced.

1.2.4-Ramp based constraint: The ramp based constraint determines the maximum range, by which the

power generated can be increased in a particular duration of time.

P(i, t) − P(i, t − 1) ≤ UR(i)

P(i, t − 1) − P(i, t) ≤ DR(i)

where UR(i) and DR(i) are the maximum ramp-up and ramp-down rates of unit i.

1.2.5 Transmission constraint: With the UC schedule, GENCOs have to satisfy customer load demands

and maintain transmission flows and bus voltages within permissible limits. It is not possible to satisfy all

these constraints using a single optimization technique, therefore a hybrid algorithm approach can be used to

achieve it.

1.2.6 Spinning reserve: The spinning reserve is the unused capacity, which can be activated on decision of

the system operator and which is provided by devices that are synchronized to the network and able to affect

the active power.

1.2.7 System operating system requirements: The system operating reserve requirement (R0) gives the

time required for the quick start capability of a particular unit from the off state. During the on time, it will

consider the spinning reserve capacity.

1.3-UC problem solving techniques: UC is the problem of determining the schedule of generating units

within a power system subject to device and operating constraints. Several optimization techniques have

been applied to find the solution to the thermal UC problem. The solutions available are classified into

conventional techniques, non-conventional techniques and hybrid algorithm.

1.3.1 Conventional techniques: Conventional techniques include exhaustive enumeration, priority listing,

dynamic programming, branch and bound, integer programming, simulated annealing, Lagrangian

relaxation, tabu search, and interior point optimization.

The exhaustive enumeration method is the simplest of the combinatorial optimization techniques.

The principle of this method is to evaluate all combinations of the discrete variables. It assures the global

optimum of the objective function, but the computational time is very huge [8]. Branch and bound (BB or

B&B) is a general algorithm for finding optimal solutions of various optimization problems, especially in

discrete and combinatorial optimization. This method was first proposed by Land and Doig [9]. Dynamic

programming is a methodical procedure, which systematically evaluates a large number of possible

decisions in a multi-step problem. When the existing conventional dynamic programming method is utilized,

although its solution is correct and has the optimal value, it takes a lot of memory and takes a lot of time in

getting an optimal solution [10]. Mixed Integer Linear Programming (MILP) helps reduce the solution

significantly and also the nonlinear constraints can be easily linearised. The MILP UC function developed

by Chang et al. [11] can be used for very large systems which also supports bidding strategies in the power

market.

The stimulated annealing (SA) method facilitates in searching the work space rapidly, but it does not

work properly in the case of wide temperature variation. In the original SA method, a large share of the

computation time was spent in randomly generating and evaluating solutions that turned out to be infeasible

[12]. In order to improve the performance, a hybrid of SA and local search was developed in 2003 by

Purushothama et al. [13] and the results were verified using C++. This simple modification makes it possible

to reduce the number of iterations required at each temperature, and it generates solutions with lower cost

than that obtained by using previous algorithms. Lagrangian relaxation was first applied to UC in 1977 by

Muckstadt et al. [14]. The problem is formulated in terms of a cost function, that is the sum of terms each

involving a single unit, a set of constraints involving a single unit, a set of coupling constraints (the

generation and reserve requirements), one for each hour in the study period, involving all units. When the

Lagrangian relaxation based methods are applied to solve power system UC, the identical solutions to the

sub problems associated with identical units may cause the dual solution to be far away from the optimal

solution [15]. Tabu search is based on the hill climbing method that iteratively evaluates a best solution each

time the neighborhood is updated and it stops if the solution is not improved to minimise the cost function.

The major drawbacks are that it gets stuck in the local minima. To overcome the problems above, the

parallel tabu search (PTS) is developed [16,17]. The idea makes it possible to find out the better solution from

different directions [18].

1.3.2 Non conventional (Non classical) techniques: In Expert systems, for a particular load pattern, a

priority list based heuristic model in the form of interface rules has been proposed by Ouyang et al. [19]. An

expert system consisting of dynamic load pattern matching interface and a commitment schedule database

has been designed by Ouyang et al. [20]. In Fuzzy system, the use of fuzzy logic in the UC problem was

demonstrated by Saneifard et al. [21]. Using this, the characteristics of the system and the response can be

found out without any mathematical calculations. Unexpected load variation leads to insufficient

commitment capacity, therefore Zhai et al. [22] has demonstrated a technique for examining the effect of load

variation on UC. The Hopfield neural network was used by Sasaki et al. [23] to solve UC problems and it

obtained satisfying results, but the accuracy was a major concern. An extension of the mean field annealing

neural network approach was demonstrated by Liang and Kang [24]. Walsh and Mallry [25] designed a new

way of interconnecting neurons to produce an energy equation involving both discrete and continuous terms.

Kurban and Filik [26] proposed a method to reduce the production cost by combining load forecasting with

UC problem using artificial neural network (ANN) model with auto regression (AR) in 2008. The Ant

system was developed by Salam et al. [15] and is based on the idea that a colony of ants is able to succeed in

a task to find the shortest path between the nest and food source. This path is reinforced and many ants

follow the trail to achieve the shortest path. The genetic algorithm (GA) is a general purpose, simple and

robust, stochastic and parallel search method based on the mechanics of natural selection and natural

genetics.

The GA works with a population of chromosomes. A chromosome is a string of bits 0 and 1. Ma et

al. [27] suggested an improved GA method using C++ compilation in 2011. This method was tested using

C++ on a 6 unit system over a scheduling period of 6 hours. Abookazemi and Mustafa [28] developed a

parallel structure to handle the infeasibility problem in a structured and improved GA which provides an

effective search and therefore greater economy in 2009. This method was developed and tested by using C#

program. Tests have been performed on 10 and 20 units systems over a scheduling period of 24 hours.

Atashpaz-Gargari et al. [29] first introduced the imperialistic competition algorithm (ICA) in 2007. In the

ICA, the initial population individuals (countries) are in two types: imperialists and colonies that all together

form some empires. The imperialistic competitions among these empires converge to a state, in which there

exists only one empire.

1.3.3 Hybrid algorithms developed (employing both classical and non classical methods):

1.3.3.1 Hybrid priority list ant: Withironprasert et al. [30] devised a new method hybrid ant system priority

list (HASP), in which there are many units with different paths that can be selected at hour t. At this stage,

mth ant probabilistically selects unit i and commits it as ‘1’ status for satisfying the UC constraints. AS

search space for priority list method, considering size of UC search space, the maximum number of paths to

be selected by mth ant at hour t of the proposed approach is N, which is equal to the number of units, while

the maximum number of paths to be selected by using AS algorithm without the priority list method [31] is

2N– 1 combinations. It is clear that the search space of N is much smaller than 2N– 1 combinations, so that

the UC search space based on the proposed approach is significantly reduced This method has greater

flexibility and achieves greater economic saving whereby saving computational time. Sum-im T and

Ongsakul tested their algorithm with 10 unit system and got their total operating cost as $564324 for 24

hours. Also they validate their results by comparing them with LR ($565825), GA ($565825), evolutionary

programming (EP) ($564551) and ICGA ($566404).

1.3.3.2 Hybrid ant colony optimization: Yu et al. [32] suggested a hybrid method of ant colony optimization

(ACO) and lambda iteration method for the UC problem in 2010. The ACO algorithm is used to optimize

the on/off status planning of units in the upper level, and the lambda iteration method is used to optimize

economic load dispatch in the lower level. To verify the effectiveness of the method proposed, it is applied

to a test system [33] that consists of 10 units. For the systems of 20, 40 and 60 units, the basic 10 unit system

is duplicated. The proposed method provides better results than the GA, EP and PL. For a 10 unit system the

total operating cost was $562989 as compared to $564950 (PL), $565825 (GA) and $564551 (EP).

1.3.3.3 Hybrid Lagrangian relaxation: The hybrid Lagrangian relaxation method (LR) was devised by

Zhang et al [34]. The advantages of LR in dealing with large-scale power systems and GA in making up the

shortage of Dynamic Programming are used. By using Lagrangian multipliers to relax system-wide demand

and reserve constraints, the UC problem is decomposed and converted into a two-level optimization

problem. The optimal commitment of a single unit is solved by using GA in the low-level problem.

Crossover and mutation are very important operations on which the convergence of the GA depends. It has

good convergent rate and was faster, and provides high quality solution. The developed algorithm was tested

with 10 unit system and the total operating cost was $557726.4, which was comparatively less when

compared to $565825 (LR), $565825 (GA) and $564551 (EP).

1.3.3.4 Hopfield neural network: Kumar and Palanisamy devised a method that employs a linear input-

output model for neurons, which is extremely different from all Hopfield methods previously reported as the

previous methods apply the iterative procedures requiring a large quantity of computation to converge to

accurate solutions. However, based on the formulations developed, the proposed method computes its

solutions analytically, and no iteration is needed in the solving process. Consequently, computational efforts

are greatly reduced which was demonstrated by its use in the 10- and 20-unit systems. Although it works on

a neural network, it does not require any training. It requires 32 times less CPU time than the LR [35].

Therefore, for a 10-unit system, the total operating cost was $564959 as compared to $565508 (ALR),

$565825 (GA) and $565475 (LR).

1.3.3.5 Hybrid EP and particle swarm optimization (PSO): This hybrid intelligence technique proposed

by Lal raja Singh and Christober Asir Rajan [36] in June 2001 for UCP utilizes PSO algorithm and EP. PSO

is used to determine the units and their optimum generation schedule for a particular demand with minimum

cost. Evolutionary programming (EP) assisted by PSO is used to determine the UC that minimizes the cost

for different possible demands. Therefore, for a 7-unit NTPS, the total operating cost (in p.u.) is 0.7516 as

compared to 0.93690 (PSO), 0.97483 (LR) and 0.93461 (EP).

1.3.3.6 Hybrid GA: Chang and Luo [37] used the binary-coded genetic algorithm incorporating a priority list

ordering scheme to solve the “unit scheduled” decision. The genetic algorithm incorporates the solution

produced by the priority list unit scheduled as part of its initial population; this injects domain knowledge

into its search space. Since the merit order unit scheduled forms part of the initial population, the hybrid

genetic algorithm is guaranteed to provide a global solution. The feasibility of the proposed method was

demonstrated for a 10-unit system, and the result is a hybrid GA (HGA) that produced better results than

those obtained by the simple GA (SGA), PSO and BPSO. Therefore for a 10-unit system, the total operating

cost is $556760 as compared to $565804 (BPSO), $581450 (PSO) and $609023 (SGA).

1.3.3.7 Hybrid PSO: Alshareef devised an advanced PSO, in which some parts of PSO are modified (e.g.,

includes bacterial foraging operations) to converge the system in 2011. Besides, a repair method is applied

for the fast convergence of the system. The hybrid PSO was tested on a 135-unit system for a scheduling

period of 24 hours. The execution time is growing rapidly considering the size of the problem [38].

1.3.3.8 Hybrid fuzzy logic: The hybrid fuzzy logic was developed by Mantawy and Abdel-Magid [39] in

2002, in which the UCP is formulated in a FL (fuzzy logic) frame to deal with the uncertainties in the load

demand. The SA algorithm is used to solve the combinatorial optimization of the UCP. This hybrid fuzzy

logic was tested with a 10-unit system and a total operating cost of $536260 was obtained which validated

the result with $536622 (SA).

1.3.3.9 Hybrid ACO Lagrange: The hybrid ACO Lagrange methodology proposed by Nascimento et al. [40]

in 2011 was proven to be competitive in relation to the optimization techniques biologically inspired in the

behavior of the ant colony found in literature, conciliating quality solutions and a reduced colony; the use of

Lagrange multipliers associated with discrete variables of the Thermal UC problem act as a source of

information for the ant colony algorithm. However, the computational simulation time increases

considerably, making this alternative unfeasible, mainly in medium-to-large sized systems and/or with large

programming periods. The hybrid ACO Lagrange methodology was proven to be competitive in relation to

the optimization techniques biologically inspired in the behavior of the ant colony found in literature,

conciliating quality solutions and a reduced colony. It was tested with a 10-unit system and the total

operating cost is $563937 and this cost is less when compared to $564049 (ACSA), $563977 (DACO) and

$563977 (RACO).

1.3.3.10 Shuffled frog leaping algorithm: The shuffled frog leaping algorithm (SFLA) was developed by

Ebrahimi et al. [41] in 2011. The SFLA is a metaheuristic optimization method which is based on observing,

imitating, and modeling the behavior of a group of frogs when searching for the location that has the

maximum amount of available food. The SFLA has been applied to ten up to 100 generating units,

considering oneday and seven-day scheduling periods. The most important merit of the SFLA is its high

convergence speed. The simulation results of the SFLA have been compared with the results of the

algorithms, such as Lagrangian relaxation, genetic algorithm, particle swarm optimization, and bacterial

foraging [14]. Therefore, for a 10-unit system, the total operating cost is $564769 as compared to $566404

(ICGA), $564842 (BF) and $565475 (LR).

1.3.3.11 Fuzzy tuned PSO (FTPSO): PSO is the mathematical modeling and simulation of the food

searching activities of a flock of birds. Each particle moves with a different velocity towards the optimal

point. The velocity of a particle is calculated by three components: inertia, cognitive, and social. The

particles move around the multidimensional search space until they find the optimal solution. A fuzzy

system is utilized to tune the inertia weight and learning factors with the best fitness (BF). This FTPSO has

been applied to a 10- and 20-bus system in MATLAB and is proven to increase the reliability of the system.

It is faster than ACO and BP and is capable of solving both small scale and large scale problems [42].

Therefore for a 10-unit system, the total operating cost is $557952 as compared to $565825 (GA),

$564551(EP) and $ 565475 (LR).

1.3.3.12 Memetic algorithm: A memetic algorithm (MA) is a hybrid computational model of two sources.

The first source is modeled by a GA that mimics biological or Darwinian evolution and the second source is

modeled by a local search algorithm that mimics cultural evolution or the evolution of ideas. The unit of

information in a GA is termed as a gene whereas in a MA it is termed as a meme. Genes are improved by

crossover and mutation operators that are part of a GA and memes are improved by a local search operator.

This method is useful for the price based UC [43]. The method has been implemented on test systems of up to

110 units and the results show that in every case examined the MA converges to higher profit price based

UC (PBUC) schedules than the genetic algorithm, the simulated annealing, and the Lagrangian relaxation

method. Therefore for a 10-unit system, the average profit was $1899.39 as compared to $1898.85 (SA) and

$1899.21 (GA).

1.3.3.13 Binary/Real coded PSO: The PSO can produce higher quality solution within a short interval of

time and stable convergence characteristic than other stochastic optimization methods. The PSO model

consists of a swarm of particles moving in a D-dimensional real-valued space of possible problem solutions

[44]. In this particular algorithm, the tanh function is implemented to enhance the particle searching

performance of binary PSO. The binary/real coded PSO methodology is tested and validated on a 3-, 17-,

26- and 38-generating unit system for 24 hour scheduling and hence it is concluded that it can be

implemented for a large scale power system. The algorithm was tested for a 38 unit system and the total

operating cost (in million Dollars) is 196.10 as compared to 196.73 (FAPSO), 207.8 (SA) and 209 (LR).

1.4 Environments for UC:

1.4.1 Price based UC: In PBUC, satisfying hourly loads is no longer a restriction and the objective is to

maximize the payoff. Thus in the price based approach, the deciding parameter resulting in the on/off state

of a generating unit would be the price including the fuel purchase price, energy sale price and so on. Li and

Shahidehpour [45] suggested the use of mixed integer programming as compared to Lagrangian method for

the PBUC problem. In the present context, it is more advisable to go for individual UC separately based on

the market forecasted price and then combine these individual schedules to optimize the scheduling of the

whole group of generators [46].

1.4.2 Profit based UC: In today’s competitive scenario, GENCOs are no longer bound to serve the given

demand in the open electricity market. The problems under a deregulated environment are more complex

and competitive than traditional problems. GENCOs solve economic dispatch and UC not to minimize the

total production cost as before, but for maximizing their own profit. Many evolutionary programming

models were developed in the literature for profit based UC [47].GENCOs can now consider a schedule that

produces less than the predicted load demand but creates a maximum profit.

1.4.3 Security based UC: Under this constraint the main objective is not only to minimize the generation

production cost but also to meet the other constraints for the overall operating period [48]. The security based

UC planning involves determining whether a generating unit is functioning efficiently, accordingly the

decision has to be taken, that it should be turned on or off at a particular time. Despite market related

pressures, system security should always be the foremost priority. While formulating the solution, power

flow constraints and generator maintenance constraints should be taken into consideration.

1.5 UC in deregulated environment: In the present scenario of deregulated markets, it is required from the

GENCOs to submit their power bids separately. Each bid consists of a cost function and a set of parameters

that define the operative limits of the generating unit [49]. Cost suboptimal solutions that result in lower

prices may exist and therefore the applicability of cost minimization UC models for power pool auctions is

questioned [46]. In Aug. 2001, Madrigal and Quintana [50] investigated the existence, determination and

effects of competitive market equilibrium on UC power pool auctions to avoid the conflict of interest and

revenue deficiency. A new formulation to the UC problems suitable for an electric power producer in a

deregulated market has been provided by Valenzuela and Mazumdar [51] and Lasen et al. [52] in 2001.

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CHAPTER-2

VEHICLE TO GRID POWER

2.1. The concept of V2G:

The basic concept of V2G is that they were charged and discharged to the grid when parked as they

were used for transportation for only 4% of time and the remaining 96% time they can be used for secondary

purpose like power scheduling since they were parked [1]. To minimize the cost of dispatch of overall

system charging is done at low/off peak hours and discharged when peak power hours are encountered. The

V2G can be from Plug-in-hybrid EDV’s, Fuel cell EDV’s with gaseous fuel and hybrid EDV with ICE and

battery.

The EDV should contain three essential elements in order to participate in either V2G or G2V

application they were namely (1). Special converter topologies that allow the power flow between EDV and

grid (2). Communication system with in the EDV and with local grid operator to schedule the charging and

discharging the EDV effectively and reliably (3).controls for on-board control of EDV power sources like

battery SOC, fuel level etc. In most of the cases the convertor topologies required for V2G and vice-versa

are in built in the vehicle and a part of tariff includes cost of those. There are two modes of power transfers

in the system equipped with EDV namely one way which flows from generating stations to the users and

two way in which power flows from and to the EDV or vehicle fleets schematic of the system is shown in

fig.1 depicting the power flows.

Fig. 1. Schematic of power and communication systems for V2G and G2V power flow [2].

2.1.1. Battery EDVs:

These store energy electrochemically and several types of batteries are available which can be

equipped into the EDV. Lead-acid batteries are the cheapest option whereas Ni-Mh batteries provide

relatively long life with moderate cost compared to lead-acid batteries. Long life Li-ion batteries are

becoming popular and reasonably comparative costly with Ni-Mh batteries but they were yet be

standardized in the safety point of view.

2.1.2. Fuel cell EDVs:

Fuel cell EDVs typically store energy in molecular hydrogen (H2), which feeds into a fuel cell along

with atmospheric oxygen, producing electricity with heat and water as by-products. Multiple options for on-

board storage or production of hydrogen are under development, including

Pressurizing the H2 gas, binding it to metals, and on-board production of H2 from natural gas,

methanol, gasoline or another fuel. Currently, distribution infrastructure, on-board storage of hydrogen, and

conversion losses are all substantial problems that leave open the question as to whether fuel cell light

vehicles will be practical and cost-effective[3,4].

2.1.3. Hybrid EDVs:

Contemporary hybrid vehicles use an internal combustion (IC) engine whose shaft drives a

generator. A small battery buffers the generator and absorbs regenerative braking. The battery and generator

power one or more electric motors that drive the wheels, possibly in conjunction with direct shaft power

from the IC engine [5]. More conceptually, a hybrid has one power system with large energy storage—for

range—and a second with high power output and discharge-recharge capability—for acceleration and

regenerative braking.

2.2. Power Markets:

Based on the operating mechanism, time of operation the power markets are classified into four

major types base load, peak shaving, spinning reserves, regulation [6,7]. Of these markets spinning reserves

and regulation were been critical since they have to have a quick response time thus effecting the system

stability. The control of these will be done by “grid operator” however the terminology differs from country

to country.

2.2.1. Base load power:

Base load power plants provides power throughout the day irrespective of load. Normally plants with

low kWh price like nuclear and thermal units were operated as base load power plants.

Due to high cost of kWh compared to that of existing plants V2G is not an attractive choice for acting as

base load power plant.

2.2.2 Peak power:

Peak power comes into picture when the load is low or high w.r.t the base load like hot summers and

cold winters. This type of markets usually employs buying power at off pea time and selling it during the

peak hours. This gives an opportunity for thermal units to operate at maximum/optimum efficiency at off

peak times and avoids turning on the expensive small units like gas plants at peak hours. V2G gives an

attractive option for these as large number of EDV’s are available during peak power hours i.e. summer

afternoons and winter nights.

2.2.3. Spinning reserves:

This is referred to the surplus capacity that could be turned on for short duration like 10 to 15

minutes in case extra demand is requested by the grid operator. Normally 10 to 15 % of total capacity is set

as reserve. High the spinning reserve higher will the reliability. The EDVs at parking can provide an option

for spinning reserves as they can incur some revenue without actually the energy being spent.

2.2.4. Regulatory services:

These services are important in supplying the balancing power between load and base load powers

there by regulating the frequency of the system. These are in general called more frequently (400 times a

day) for shorter durations(less than a minute per call) compared to that of spinning reserves. Regulation can

be “up” (where the load is more than generation) or “down” (where the load is less than generation).

In the EDVs fuel cell systems are not suitable for regulation down services. The actual power

dispatched is just a fraction of power capacity and time it is contracted for as given in following equation,

which is “dispatch to contract ratio”

𝑅𝑑−𝑐 =𝐸𝑑𝑖𝑠𝑝

𝑃𝑐𝑜𝑛𝑡𝑟𝑡𝑐𝑜𝑛𝑡𝑟

Where Rd-c is the dispatch to contract ratio (no dimension) , Edisp is the dispatched power (MWh), Pcontr

is the contracted power (MW) and tcontr is the contract time. Its estimation is around 0.1.

2.3.1. Power capacity of V2G:

There are two factors which will affect the capacity of V2G power namely (1). Line limited capacity

and (2). Capacity limited by vehicles stored energy.

𝑃𝑙𝑖𝑛𝑒 = 𝑉𝐴

Where Pline is expressed in kW and V is supply voltage and A is maximum current carrying capacity of

line. Typical values of 250V and 50A are considered which gives a limit of 15kW as the Pline.

2.3.2. Power limited by vehicle’s stored energy:

A part form the line capacity the EDV capacity is also limited by the available on board stored

energy that could be dispatched. The capacity limit by storage can be calculated by dividing the energy

available divided by the dispatch time where the available energy is given by energy stored less used energy

and energy required to drive multiplied by conversion efficiency.

Where Pvehicle capacity of vehicle, Es is on board storage, dd is driven distance since the battery is last

charged and drb is he distance to be driven following the V2G operation and ηveh is the conversion

efficiency of fuel (miles per kWh) which is 2.86 miles/kWh and ηinv is the inverter efficiency of 90%. The

values of dd and drb depends on the driving patterns in this paper dd is considered as 32 miles and drb of 16

miles is taken as average.

2.4. Revenue versus cost of V2G:

The economic value of V2G is the revenue minus the cost.

2.4.1. Revenue equations:

The values and formulas for calculating the revenue and cost depends on the type of market that the

energy is sold into. Revenue is simply the energy multiplied with time of dispatch.

𝑟 = 𝐸𝑑𝑖𝑠𝑝𝑃𝑒1 = 𝑃𝑑𝑖𝑠𝑝𝑡𝑑𝑖𝑠𝑝𝑃𝑒1

On an annual basis, peak power revenue is computed by summing up the revenue for only those hours that

the market rate (pel) is higher than the cost of energy from V2G (cen, discussed later).

For spinning reserves there exists two types of revenues “capacity payment” and “energy payment”.

Capacity payment refers to the payment made for the contract on hourly basis even though EDV is not

discharged. Energy payment refers to the payment for actual used energy. Capacity payment is made on

MW-h i.e one MW capacity contract for one hour (note MWh is for used energy whereas MW-h is for

capacity).

𝑟 = (𝑃𝑐𝑎𝑝𝑃 𝑡𝑝𝑙𝑢𝑔) + (𝐸𝑑𝑖𝑠𝑝𝑃𝑒1)

Where Pcap is the capacity price $/Kw-h and Pe1 is the electricity price $/kWh, 𝑡𝑝𝑙𝑢𝑔 total time for which

the capacity is available and plugged in. For spinning reserves the Edisp can be calculated as sum of all

dispatches,

𝐸𝑑𝑖𝑠𝑝=∑ 𝑃𝑑𝑖𝑠𝑝𝑡𝑑𝑖𝑠𝑝𝑁𝑑𝑖𝑠𝑝

𝑖=1

Where Ndisp is the number of dispatches, Pdisp is power per dispatch and tdisp is the time per dispatch. A

typical spinning reserves contract sets a maximum of 20 dispatches per year and a typical dispatch is 10 min

long, so the total Edisp will be rather small.

For regulation services also there would be two types of revenues as in case of spinning reserve.

Typically it may have as many as 400 to 500 dispatches per day.

𝑟 = 𝑃𝑐𝑎𝑝𝑃 𝑡𝑝𝑙𝑢𝑔 + 𝑃𝑒1𝑅𝑑−𝑐 𝑃 𝑡𝑝𝑙𝑢𝑔

2.4.2. Cost Equations:

The cost for EDV during V2G operation consists of three component (1). Energy purchase cost

(charging, hydrogen or gasoline) (2) Wear of converters and battery (3) capital cost. On annual basis cost

can be expressed as

𝐶 = 𝐶𝑒𝑛𝐸𝑑𝑖𝑠𝑝 + 𝐶𝑎𝑐

Where C total cost per year, 𝐶𝑒𝑛 is the cost of energy per unit (kWh), 𝐸𝑑𝑖𝑠𝑝 is the energy dispatched per year

and 𝐶𝑎𝑐 annualized capital cost.

𝐶𝑒𝑛 is the energy cost which includes two terms one for energy purchased and other for degradation cost of

converters and battery/ ICE / Fuel cell thus is given as follows

𝐶𝑒𝑛 =𝐶𝑝𝑒

𝜂𝑐𝑜𝑛𝑣+ 𝐶𝑑

Where 𝐶𝑝𝑒 is the cost of energy purchase (electricity for battery EDVs, hydrogen for fuel cell EDVs and

gasoline + electricity for hybrid EDVs), 𝜂𝑐𝑜𝑛𝑣 is the conversion efficiency of fuel/storage to electricity or

vice-versa and 𝐶𝑑 is the degradation cost of battery/fuel cell/ICE due to V2G use. Thus 𝐶𝑒𝑛 thus gives the

cost of energy per kWh whatever the vehicle category. The degradation cost of battery can be found by

dividing the cost of battery by total energy throughput of battery along the life.

𝐶𝑑 =𝐶𝑏𝑎𝑡

𝐿𝑒𝑡

Where 𝐶𝑏𝑎𝑡 is the cost of battery per kW and 𝐿𝑒𝑡 is the life time throughput energy (both charged and

discharged energies are to be considered i.e total processed energy). Thus 𝐿𝑒𝑡 is given as follows

𝐿𝐸𝑇 = Lc Es DoD

Where 𝐿𝐶 is the lifetime given as number of cycles at a given depth of discharge (DOD) per cycle and Es is

the energy storage of battery in kWh. DOD of 80% is considered here to define cycle life and lifetime

throughput.

The capital cost of battery and other essentials that are need for V2G application but not driving are

considered on annual basis which is given as follows, CRF is the capital recovery factor

𝐶𝑎𝑐 = 𝐶𝑐 𝐶𝑅𝐹 = 𝐶𝑐

𝑑

1 − (1 + 𝑑)−𝑛

where 𝐶𝑎𝑐 is the annual cost and Cc is the capital cost, d is the discount rate and n is the lifetime.

Issues need to be addressed w.r.t the costing:

1.Degradation costing:

The degradation cost per kWh is defined for a particular DOD but in actual case the things are

different, since the number of cycles and DOD are not linear rather exponential cost of degradation depends

on the DODend. For example from a 100kWh battery 40kWh is discharged. There are two cases,

1. If the DODin is 90% and DODend is 50% i.e (90-50/100)*100kWh energy is drawn i.e 40kWh

2. Ifthe DODin is 50% and DODend is 10% i.e (50-10/100)*100kWh energy is drawn i.e. 40kWh

In the two cases same amount of energy is drawn but the second case will have more degradation so the Cd

would also be high for second case.

2. Energy lost due to battery degradation

Costing for degradation is done which is for replacement of battery but for every cycle some amount

of capacity is lost. From the next cycle onwards the amount of energy that the battery can store will store

also reduces. This should also the compensated in cost since the degradation cost is for battery replacement

and not for this energy lost due to the degradation. This cost can be formulated as follows

𝐶𝑑𝑒 = 𝐶𝑙𝑑𝑖𝑠𝑝(𝑁 − 𝑁𝑢) 𝐶𝑝𝑒

Where Cde is the cost of energy lost due to degradation, 𝐶𝑙𝑑𝑖𝑠𝑝 is the capacity (kWh) lost for particular

discharge, N is the nominal number of cycles and Nu is the number of used cycles and Cpe is the nominal

cost of energy purchase cost.

References:

[1] W. Kempton, J. Tomic, S. Letendre, A. Brooks, T. Lipman, “Vehicleto-grid power: Battery, hybrid and fuel cell vehicles as

resources for distributed electric power in California, Davis, CA,” Institute of Transportation Studies, Report # IUCD-ITS-RR 01-

03, 2005.

[2] W. Kempton and J. Tomic, “Vehicle to grid fundamentals: Calculating capacity and net revenue,” Journal of Power Sources,

vol. 144, no. 1, pp. 268-279, 1 Jun 2005.

[3] National Academy of Sciences (NAS), The Hydrogen Economy: Opportunities, Costs, Barriers, and R&D Needs, By the

Committee on Alternatives and Strategies for Future Hydrogen Production and Use,

National Research Council, Report, 2002, p. 394.

[4] Daniel Sperling, James S. Cannon, The Hydrogen Energy Transition: Moving Toward the Post-petroleum Age in

Transportation, Elsevier Academic Press, Amsterdam & Boston, 2004.

[5] M. Duvall, Infrastructure roadmap for plug-in hybrid electric vehicles—update, in: Presentation at National Electric

Transportation Infrastructure Working Council (IWC), EPRI–IWC Hybrid Electric Vehicle Working Group, San Francisco, CA, 6

October, 2004.

[6] Federal Energy Regulatory Commission (US), Notice of Proposed Rulemaking: Remedying Undue Discrimination through

Open Access Transmission Service and Standard Electricity Market Design, Docket No. RM01-12-000, Washington, DC, 31 July,

2002.

[7] North American Electric Reliability Council, NERC Operating Manual, Princeton, NJ, 21 November, 2002.

Chapter-3

Optimization Techniques

3.1-PARTICLE SWARM OPTIMIZATION:

Particle swarm optimization [1,2], first introduced by Kennedy and Eberhart, is one of the heuristic

optimization algorithms which is derived from the social-psychological theory. PSO has been found to be

robust for solving problems featuring non-linearity and non-differentiability, multiple optima and high

dimensionality through adaptation. It has several advantages over other optimization algorithms such as easy

to implement and potential to achieve a high-quality solution with stable convergence characteristic.

PSO does not use any evolutionary operator to manipulate the individual as in other evolutionary

computational algorithms. Instead, each individual in PSO flies in the search space with a velocity which is

dynamically adjusted according to its own flying experience and its companions' flying experience. Each

individual is treated as a volume-less particle in a d-dimensional search space. Each particle keeps track of

its coordinates in the problem space which are associated with the best solution (fitness) it has achieved so

far. This value is called pbest. Another best value that is tracked by the global version of the particle swarm

optimizer is the overall best value and its location obtained so far by any particle in the population. This

location is called gbest. At each time step, the particle swarm optimization concept consists of velocity

changes of each particle toward its pbest and gbest locations. Acceleration is weighted by a random term

with separate random numbers being generated for acceleration toward pbest and gbest locations. For

example, the i th particle is represented as x=(xi1,xi2,……….xid) in the d-dimensional space. The best

previous position of the ith particle is recorded and represented as pbest=(pbesti1, pbesti2………………

pbestid). The index of the best particle among all the particles in the population is represented by the gbestd .

The rate of position change (velocity) for particle i is represented as v=(vi1,vi2,………………..vid). The

modified velocity and position of each individual particle can be calculated using the current velocity and

the distance from pbestid to gbestd which is shown in the following formula.

vi,dk+1 = wvi,d

k + c1 ∗ rand1 ∗ (pbesti,dk − xi,d

k ) + c2 ∗ rand2 ∗ (gbestdk − xi,d

k ) … … … . . (1)

xi,dk+1 = xi,d

k + vi,dk+1 . . … … . (2)

where, vi,dk is the velocity of individual i at iteration k ,vd

min ≤ vi,dk ≤ vd

max , w is the inertia weight

factor, c1 and c2 are the acceleration constants, rand1 and rand2 are the uniform random number between 0

and 1, xi,dk is the current position of individual i at iteration k, pbesti is the particle best of individual i and

gbest is the generation best of the group

In the above procedures the parameter vmax determines the resolution or fitness with which regions

between the present position and target position are searched. If vmax is too high, particles may fly past the

good solutions. If vmax is too small, particles may not explore sufficiently beyond local solutions. In

previous experience with PSO, vmax was often set at 10-20% of the dynamic range of the variable on each

dimension. The constants c1 and c2 represent the weighting of the stochastic acceleration terms that pull each

particle toward pbest and gbest positions. Low values of c1 and c2 allow particles to roam far from target

regions before being tugged back. On the other hand, high values result in abrupt movement toward or past

the target regions. Hence, the acceleration constants c1 and c2 were often set to be around 2.0 according to

past experiences. For a suitable selection of inertia weight w, the equation (3) provides a balance between

global and local exploration, and exploitation. As originally developed, w often decreases linearly from

about 0.9 to 0.4 during a run. In general, the inertia weight w is set according to the following equation.

w = wmax −wmax − wmin

Itermax× Iter … … … … … … … . . (3)

where , Itermax is the maximum iteration number (generations), Iter is the current iteration number

and, wmax and wmin are the maximum and minimum values of inertia weight respectively.

3.2-QUANTUM-INSPIRED BPSO ALGORITHM [3]:

The proposed QBPSO combines the conventional BPSO with the concept of quantum computing such as a

quantum bit and superposition of states. It adopts a Q-bit individual for the probabilistic representation,

which replaces the velocity update procedure in the conventional PSO. It combines the advantages of both

quantum theory and classical PSO and achieves a better balance between exploration and exploitation of the

solution space and obtains better solutions, even with a small population, compared with the conventional

PSOs, thus well solving the combinatorial optimization problems.

In the proposed QBPSO, the position vector of the ith particle is updated by probability of |𝛽𝑖𝑗|2

stored in the ith Q-bit individual and a Q-bit may be in any superposition of the two states: 0 and 1. The state

of a Q-bit is represented as follows:

|𝜑𝑖𝑗⟩ = 𝛼𝑖𝑗|0⟩ + 𝛽𝑖𝑗|1⟩, |𝛼𝑖𝑗|2

+ |𝛽𝑖𝑗|2

= 1 … … … … … (4)

Where 𝛼𝑖𝑗 and 𝛽𝑖𝑗 are complex numbers that specify the probability amplitudes of the corresponding states.

Here |𝛼𝑖𝑗|2

𝑎𝑛𝑑 |𝛽𝑖𝑗|2 are the probability that the Q-bit is in “1” state and “0” state respectively.

(A) The QBPSO procedure is as follows:

a) Q-bit initialization: 𝛼𝑖𝑗0 𝑎𝑛𝑑 𝛽𝑖𝑗

0 of all Q-bit individuals are set to be 1/ 2 , which means a Q-bit individual

can be all possible states with the same probability. The jth element of the ith particle takes a value by the

following:

xij = { 1 if rnij < |βij|2

0 other … … … … … … … … (5)

Where i=1,2,…,NP, j=1,2,…,n. Here, rnij is the uniformly distributed random number between [0, 1] and

NP is the population size.

The initial Pbest of each particle is set as its initial position, and the initial Gbest is determined as the

position of the particle with the minimum cost.

b) Q-bit update through the rotating gate:

z = zmax − (zmax − zmin) ×k

itermax … … … … … … … … . . (6)

∆𝑧𝑖𝑗𝑘+1 = 𝑧 × {𝛾1𝑗

𝑘 × (𝑥𝑖𝑗𝑃,𝑘 − 𝑥𝑖𝑗

𝑘 ) + 𝛾2𝑗𝑘 × (𝑥𝑗

𝐺,𝑘 − 𝑥𝑖𝑗𝑘 )} … … … … … . . (7)

Where z takes the value of 0.01𝜋 ~ 0.05𝜋 , itermax is the maximum iteration number and k is the current

iteration number. 𝛾1𝑖 and 𝛾2𝑖 are obtained as follows:

γ1i = {1 other0 if f(xi) ≥ f(Pbesti)

… … … … … . (8)

γ2i = {1 other0 if f(xi) ≥ f(Gbest)

… … … … … . (9)

A new Q-bit can be computed by the following:

(𝛼𝑖𝑗

𝑘+1

𝛽𝑖𝑗𝑘+1) = [

cos(∆𝑧𝑖𝑗𝑘+1) − sin(∆𝑧𝑖𝑗

𝑘+1)

sin(∆𝑧𝑖𝑗𝑘+1) cos(∆𝑧𝑖𝑗

𝑘+1)] (

𝛼𝑖𝑗𝑘

𝛽𝑖𝑗𝑘 ) … … … . . (10)

c) The position vectors of the particles update by (5).

d) Pbest and Gbest update:

f(x) = {xi

k+1, if f(xik) ≥ f(Pbesti

k)

Pbestik, other

… … … … . . (11)

In addition, Gbestk+1 is the optimal value among𝑃𝑏𝑒𝑠𝑡𝑖𝑘.

e) Stopping criteria: the algorithm is end when the iteration reaches a pre-specified maximum iteration.

(B) Heuristic Adjusted Regulations:

Using the QBPSO to determine the unit operation states must consider system spinning reserve and

minimum up/down time constraints, otherwise the solutions may be infeasible due to the randomness of

QBPSO. Therefore, the following adjusted rules are imposed on the particles:

Step 1: Check up/down spinning reserve constraints: For the intervals when the system reserve

constraints are violated, sort all the units which are at the “0” state according to the priority list, that

is, the ascending order of the average full-load cost. Then, take out the units in turn and let it start

until the load demands and spinning reserve constraints are satisfied.

Step 2: Check minimum up/down time constraints: In order to ensure reserve constraints not violated

again, just start units. The operations are as follows: First, for every particle, find the change

moments of the unit state during the scheduling period, that is, to search for the “01” and “10”

combinations in the row vector. Second, calculate the continuous uptime before the state of the unit

changes (for the “10” combination) and continuous shutdown time (for the “01” combination). Third,

compare the continuous up/down time with the minimum up/down time of the unit. If the former is

larger, then go to seek the next change moment; otherwise, make up its up/down time to at least the

minimum up/down time of the unit.

Step 3: Check excessive spinning reserve: Step 2 obviously will produce excessive spinning reserve

which is inefficient, thus it is necessary to close the excessive units. The operations are as follows:

At first, sort all the units which are at the “1” state according to the above-mentioned priority table in

the reverse order. Then, take out the first unit and close it, judging whether all the units at the “1”

state meet the reserve constraints and the shutdown request. If both are met, shut it down and take

out the next one and do the same thing until the reserve constraints are violated if one more unit is

shut down.

Step 4: Check the minimum up/down time constraints again according to step 2.

(C) Overall Solution Procedure:

Based on the determination of unit operation states in the outer layer, the article uses the primal-dual interior

point method for the inner-layer load economic dispatch problem because of its fast calculation speed and

good numerical stability. As a result, the specific procedures of the proposed algorithm for unit commitment

problem are as follows:

Step 1: Initialize the Q-bits corresponding to the optimized thermal units.

Step 2: Create the position vectors of the particles by (5), and then produce the initial population

matrixes ( N ×T × ssize ) representing unit on/off states, where ssize is the population size.

Step 3: Adjust the population matrix according to the heuristic adjusted regulations in the second part

of the section and produce new population matrixes satisfying the constraints.

Step 4: Use quadratic function for load economic dispatch of every unit in system at every interval.

Step 5: Calculate the fitness of every particle using the objective function. According to the

operators’ demand, generation costs and (or) emissions are considered in the fitness function.

Step 6: Evaluate all particles in the population. The initial Pbest of each particle is set as its initial

position, and the initial Gbest is determined as the position of the particle with minimum cost.

Step 7: Update the Q-bits according to (10).

Step 8: Produce the next population by (5).

Step 9: Stopping Criteria: The proposed QBPSO algorithm is terminated if the iteration reaches a

pre-specified maximum iteration.

References:

[1] M. Clerc and J. Kennedy, "The particle swarm: Explosion stability and convergence in a multi-dimensional complex space,"

IEEE Trans. On Evol. Comput., pp.58-73, 2002.

[2] J. Kennedy and R. C, Eberhart, "A Discrete Binary Version of the Particle Swarm Algorithm," Proceedings of the Conference

on Systems, Man, and Cybernetics, Piscataway, pp.4104-4108, 1997.

[3] Xiaoshan Wu, Buhan Zhang, Kui Wang, Junfang Li, and Yao Duan, “A Quantum-inspired Binary PSO Algorithm for Unit

Commitment with Wind Farms Considering Emission Reduction.” IEEE PES ISGT ASIA 2012

CHAPTER-4

SIMULATION AND RESULTS

All the simulations were carried out using MATLAB (7.10.0).A 10 unit system with 10% of the load

is considered as spinning reserve. It is assumed that the cold startup cost is double that of hot startup cost

and no shutdown cost. Load and the plant data is taken form [1]. Load has two peaks at hours 12PM and

20PM which is depicted in following figure.

If the unit commitment is done on hourly basis with all the constraints satisfying for the particular

error some of the constraints were being violated in the later hours of commitment which is 20th hour in with

respect to the given load and plant constraints. This is avoided by optimizing the unit commitment globally

i.e. the violation of constraints in the later hours is also considered there by avoiding the constraint violation

and attaining the best possible solution. As a thumb rule large and cheap units are turned on in the initial

hours so that there exists flexibility and constraint satisfaction is not a problem in later stages/ hours.

0

200

400

600

800

1000

1200

1400

1600

1800

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

po

wer

(MW

)

Hour

Load profileload with out reserve

load with 10% spinning reserve

Unit Commitment Reserve for 24 hours:

The following figure shows the on/off schedule of 10 units for 24 hours. It can be seen that during

the peak hour (12th hour) all the units are on and units 1,2are on for all 24 hours which confirms that chap

and large units suits as base load plants.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Po

wer

Hour

Unit Commitment

U1 U2 U3 U4 U5 U6 U7 U8 U9 U10

0

50

100

150

200

250

300

350

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Po

wer

(MW

)

Hour

Spinning Reserve

Load Sharing:

The equality constraint for load is satisfied for every hour with 100% accuracy unlike the base paper

in which equality constraint is violated in some hours (14 hours out of 24 hours were fall short by 0.1MW

each adding up to 1.4 MW in 24 hours) which may bring some difference in the total cost. First and second

plants status is on from the previous hour so no costs are incurred for them for first hour. In every hour

maximum load is shared by the base load plants i.e. 1 and 2 which gives the minimum cost whereas smaller

and costly (8th, 9th, 10th units). Load sharing between 10 units for 24 hours for the best case (PSO) is given as

follows:

Comparison of cost for Dynamic Programming, Genetic Algorithm, Pattern Search and Particle

Swarm Optimization:

Method

Best cost ($) Worst ($) Average ($)

Dynamic

programming

571778 N/A N/A

Genetic Algorithm

568954 571504 569450

Pattern Search

569500 N/A N/A

Particle Swarm

Optimization

564640 570149 566685

Parameter values and convergence of PSO for UC:

Parameters values are swarm size=30, MaxIter=3000, trust parameters c1=1.4 and c2=2.6. Initial

values for pbest and gbest should be high so that the convergence direction is towards the optimal value

otherwise particles may trap at the initial states if they were assigned with very low values.

Iterations

C

O

S

T

($)

References:

[1]. Ahmed Yousuf Saber and Ganesh Kumar Venayagamoorthy ,“Optimization of Vehicle-to-Grid Scheduling in Constrained

Parking Lots”, IEEE transactions on smart grid, 2009.