Optimization of hybrid standalone renewable energy ...

Optimization of hybrid standalone renewable energy systems by linear programming K. Kusakana1, H.J. Vermaak1, G.P. Yuma2 1

Department of Electrical Engineering and Computer Systems Central University of Technology, Free State 20 President Brand Street, Bloemfontein 9300, South Africa 2 School of Electrical and Electronic Engineering, Mangosuthu University of Technology, Durban, South Africa (Received/Accepted: xxx)

Hybrid systems utilizing various renewable energy sources are well suited to satisfy the energy needs of rural and isolated areas not supplied directly by the electrical grid, especially in developing countries. This paper presents a mathematical formulation of Renewable Hybrid Sources connected together in order to build and economic System. This problem is expressed as a Linear Program where the objective function is to minimize the capital investment cost of each Renewable Energy components subject to the energy resources, size of components and energy demand. This method is general and can be applied to different locations as well as energy scenario. For illustration purposes, a numerical example combining Photovoltaic, Wind and Hydrokinetic Energy systems is provided. Keywords: Hybrid Renewable Energy Systems, Optimal Cost, MatLab Linear Programming.

1.

INTRODUCTION

For sustainable development, reliance on the use of renewable energy sources has become essential [1]. Knowing the problems and the high cost of transporting electricity in remote areas, it would be wiser to consider the use of hybrid renewable energy systems in rural and remote regions where the resources are available [2]. The most important advantage of hybrid systems is that when the renewable energy productions are used together, the reliability of the system is improved. Furthermore, the size of the storage system can be reduced slightly as there is less reliance on one unique energy source. The appropriate selection of renewable energy sources as well as the optimal sizing is essential and challenging tasks in the design of isolated hybrid systems [3], because a system oversized means additional costs, while an undersized system would not meet the load demand and be less reliable. This can be expressed as an optimization problem which can be solved by the use of appropriate optimization tool. Our intent is to minimize the total investment costs of the hybrid system while meeting the load energy requirements. Several research works have been done using numerical methods for the hybrid system components sizing and cost optimization according to the load demand and the energy

resources available from the sites [4, 5, and 6]. All the works listed in the references above are limited to only two renewable sources. These methodologies are time consuming and their level of complexity increases exponentially with the number of energy sources or variables considered in the architecture of hybrid systems. Many practical hybrid system designs and implementations often use conventional approaches such as “Rule of thumbs methods” [7] and the “Paper-based methods” [8]. These methods are based on progressive experience and trials including errors. Other approaches are mathematical such as “Graphic method” [9], “Probabilistic techniques” [10] and “Iterative method” [11]. These Approaches are derivative-based and have confirmed their efficacy in handling many types of optimization problems but they are not applicable to certain advanced optimization problems. Modern optimization methods were recently developed and based on mathematical programming of certain characteristics and behavior of biological, molecular, swarm of insects, and neurobiological systems [12]. Genetic algorithms, Simulated annealing, Ant colony optimization, Fuzzy optimization, Neural-network and Particle swarm optimization are part of modern methods of optimization.

In this paper, the initial capital cost optimization of standalone hybrid renewable systems is considered using linear programming implemented into MatLab. This method is general and can be applied to different locations as well as energy scenario. For illustration purposes, an example of a hybrid Wind/Photovoltaic/Micro-hydro energy system has been provided. 2.

OVERVIEW OF LINEAR PROGRAMMING

Linear programming is an optimization method applicable for the solution of problems in which the objective function and the constraints appear as linear functions of the decision variables [13]. The general linear programming problem can be stated in the following standard matrix form: Finds the minimum of a problem specified by:

min f xT

(1)

x

Such that:

Ax b

EW

PW

Where f, x, b, beq, lb,and ub are vectors and A and Aeq are matrices. The characteristics of a linear programming problem, stated in standard form, are: The objective function is of the minimization type. All the constraints are of the equality type. All the decision variables are nonnegative. The standard form above can be modified according to the specifications of the objective function and type of variable constraints of optimization problem to solve. 3. SYSTEM MODELING 3.1. Photovoltaic The energy produced by a Photovoltaic (PV) generator is estimated using data from the global irradiation, the ambient temperature and from the manufacturer of the PV module. The energy generated from the PV generator is given by the equation [14]:

EPV

A.

gen

.Pf .I

1 3 . .CeW . AW .VW 2

3.3. Micro-hydro The micro-hydropower considered in this study is the river current turbine type also called hydrokinetic or in-stream turbine [16]. Most of the operating principles of the hydro kinetic turbines are based upon wind turbines, as they work in a similar way [17]. The power available can be calculated using the following formula.

1 . .CeH . AH .VH3 2

3.2. Wind generator The energy generated from the wind generator is given by the equation:

(5)

Where A is the turbine area (m2), ρ is the density of water (1000 kg/m3), V is the velocity of water (m/s), Ce is the power coefficient 0.593 (theoretical maximum power available). But a small-scale river turbine has its own losses which will reduce the power coefficient to around 0.25. The energy produced by micro hydropower plant is expressed by: E H PH . t (6) 3.4. General model formulation The monthly energy produced by the system per unit area is EPV,m (kWh/m2) for photovoltaic, EW,m (kWh / m2) for wind energy and EH,m (kWh/m2) for the micro hydropower plant (where m = 1, ..., 12 represents the month of the year). The worst month is a function of the monthly load demand, the renewable energy resources and of system component‟s performance. The size (m2) of the generator needed to ensure full coverage (100%) load (ELoad) during a month is given by:

(2)

Where A is the total area of the photovoltaic generator (m2), ηgen is the generator efficiency, Pf is the packing factor, and I is the hourly irradiance (kWh/m2).

(4)

Where A is the area traversed by the wind (m2), ρ is the air density (1.225Kg/m3) and V the wind speed (m/s). Ce is the efficiency factor, which depends on wind speed and system architecture.

PH

x ub

(3)

The wind generator can recover only some of that wind power representing the power generated given by the following formula [15].

Aeq .x beq lb

PW . t

Ai

max

E Load ,m Ei , m

(7)

Ai represents the size in m2 of the PV, wind or microhydro component respectively. The total energy produced by the photovoltaic, wind and micro-hydropower generators supplied to the load is expressed as:

Ei Ai

E Load

(8)

f i .ELoad

(9)

fi

With:

Ei . Ai

With

0

1

(10)

fi 1

CREN

(13)

With:

E Load ,m

Where fi are the fraction of load supplied by the PV, Wind and micro-hydro sources respectively. At every moment, the sum of the fractions of energy contribution from each components supplied to the load must be equal to 1.

fi

ELoad ,m Ci Ei ,m

Ei , m

Ci ki fi

ki

CREN

(14)

In equation (14) the variables or design vectors are the renewable fractions fpv, fW and fH, they can be replaced by x1, x2 and x3 in the case of a hybrid Wind/Photovoltaic/Micro-hydro energy system for example.

3.5. Most unfavorable sizing method In this method [18], the sizes Ai of the renewable generators are derived from the worst monthly energy contribution of each component Ei. Similarly, the load ELoad is represented by the average monthly load. Therefore, the sizes of the generators are given by:

Ai

f i . max

E Load ,m Ei , m

4.2. Number of units After the optimal total investment cost CREN has been found, the integer numbers Ni of units from each renewable energy source integrated in the hybrid‟s optimal configuration can also be found. This can be expressed as a linear equation as follow:

(11)

Ci .N i With

In this case, Eload is constant.

Ni

C REN

(15)

0 and

N i Integers The actual real size is calculated according to the surface of The value of N i must be rounded off to the nearest 10.

3.6. Real size of components

the component unit (manufacturer) available on the market. (APV,U = 0.3m2, AW,U = 0.65m2 and AH, U = 1.65m2) [19] [20]. 4. MODELING OF THE HYBRID SYSTEM COST IN TERMS OF LINEAR PROGRAMMING 4.1. Cost modeling In this paper only the initial investment costs of the renewable components (CREN) will be considered as the objective function to minimize. The total cost of the system is given by:

Ai Ci

CREN

(12)

Where Ci represents the unit prices of the PV, Wind and micro-hydro modules. The initial costs of the renewable components (CREN) in equation (12) can be further developed using the equation (11) as:

5. Case analysis This example considers three renewable energy sources such as Photovoltaic, Wind and Micro hydrokinetic as modeled in section 3 above. 5.1. Load estimation For this study, we have considered that the average household will use electricity more for lighting and then to supply basic equipments such as iron, radio and TV for entertainment [21]. The specific cases of domestic appliances, power demand and running times for an average typical household in rural South Africa have been considered. The average household daily energy consumption of 1.97kWh is the sum of the daily consumption of the household appliances. 5.2. Resources assessment Table 1 shows the amount of solar, wind and water energy available, the specific energy production of the hybrid system components and the monthly load energy requirements. As explained above, the hybrid system is

designed according to the September which is the worst month [18]. Table 1: Monthly energy produced by the renewable energy components Solar Energy Output (kWh/m2) 26.072 22.037 21.803 18.063 15.944 13.486 15.149 17.953 20.574 22.818 24.300 26.544 20.395

Months January February March April May June July August September October November December Average

Wind Energy Output (kWh/m2) 8.552 9.756 10.801 12.937 14.397 18.037 22.62 28.148 24.216 24.064 16.072 11.701 16.775

Hydro Energy Output (kWh/m2) 731.791 1636.893 1097.923 376.295 97.8240 52.773 44.644 32.173 18.018 40.183 113.374 664.433 408.860

Load Energy demand (kWh) 61.07 55.16 61.07 59.1 61.07 59.1 61.07 61.07 59.1 61.07 59.1 61.07 59.34

5.3. Costs of components The initial costs of the different renewable generators from the manufacturers are given bellow. PV module: $129.24 for 0.3m2 area [22], Wind turbine: $487.5 for 0.65m2 swept area [23], Micro-hydro: $495 for 1.65m2 swept area [24]. The constants (specific costs) are calculated as follows: 5.4. Expression of the problem in terms of linear programming From the data above, we can calculate the specific costs represented by the constants CiKi in equation (15).

C pv k pv CW kW CH k H

129.24 59.1 $1237 .5 / m 2 0.3 20.574 487.5 59.1 $1830 / m 2 0.65 24.216 495 59.1 $984 / m 2 1.65 18.018

Finally it is found that the initial cost linked to the size to minimize can be interpreted as a linear optimization problem with the objective function and constraints expressed as follow: The function to minimize: (16) CREN 1237.5x1 1830 x2 984 x3 Subject to:

x1 x2 x3 1 0 x1 1 0 x2 1 0 x3 1

(17)

5.5. Results and discussion To find the solution to the above problem, we have wrote an interface using the command “linprog” for soling linear programming problems in MatLab as indicated below [25]: f=[1238;1830;984]; Aeq=[1 1 1]; beq=1; lb=[0;0;0]; ub=[1;1;1]; x0=[0;0;0]; options=optimset; [x,fval,exitflag,output]=linprog(f,[],[],Aeq,beq,lb,ub,x0, options) This produces the solution as follows: x= 0.0000 0.0000 1.0000 fval = 984.0000

The optimal combination obtained from the data of the selected site gives a micro-hydro system alone (fPV = 0, fW = 0, fH = 1), for a total investment cost of $ 984. Using the linear equation (15), the integer optimal number Ni of hydrokinetic modules is found as 2 modules. As these modules are purchased from the manufacturer, the initial cost will be $ 990. This linear programming method is simple, faster and reliable. It can be used to minimize the size and initial cost for hybrid system with greater numbers of renewable sources. 6. CONCLUSION In this paper, general method for optimal sizing of hybrid renewable energy systems to supply loads in rural areas of developing countries has been presented. The approach is based on minimizing the total investment cost subject to availability of the renewable resources as well as the load energy demand. The simulations have been implemented into MatLab using the linear programming optimization toolbox. This method is general and can be applied to different locations as well as energy scenario. Optimization by linear programming method is faster, reliable and suitable to minimize the size and cost of hybrid systems with high number of renewable sources. For future work, the fluctuations of the energy resources as well as the load, the hybrid system operation control using reliability indexes will be taken into account in the sizing and total cost optimisation process.

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