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This paper presents a method for optimal sizing of an off-grid hybrid microgrid (MG) system in order to achieve a certain load demand. The hybrid MG is made of a solar photovoltaic (PV) system, wind turbine (TW) and energy storage system (ESS). The reliability of the MG system is modeled based on the loss of power supply probability (SPSP). For optimization, an enhanced Genetic Algorithm (GA) is used to minimize the total cost of the system over a 20-year period, while satisfying some reliability and operation constraints. A case study addressing optimal sizing of an off-grid hybrid microgrid in Nigeria is discussed. The result is compared with results obtained from the Brute Force and standard GA methods.

Providing access to clean, reliable and affordable energy by adopting microgrid (MG) power systems is important for countries looking to achieve their sustainable development goals as the extension of the grid is time and capital expensive. These MGs are small electrical power systems that connect several electricity users to some distributed power generators and energy storage systems, which are mainly interconnected by power converters [_{2} emission, rising price and delivery cost of diesel fuel, renewable energy sources are becoming more and more popular in MG development, especially in remote areas. Both solar and wind power sources are intermittent as they depend on weather and climate changes; however, hybridizing the two sources can overcome this drawback [

Previous studies have investigated various methods of optimally sizing in different scenarios of hybrid MG applications. [

This paper aims to find the optimal size of PV array, WT and ESS for an off-grid MG by using an enhanced GA, proposed by [

The reliability of the MG system is modeled based on the loss of power supply probability (LPSP). For optimization, an enhanced Genetic Algorithm (GA) is used to minimize the total cost of the system over a 20-year period, while satisfying some reliability and operation constraints. A case study addressing optimal sizing of an off-grid hybrid MG in Nigeria is discussed.

As the main power supply in this off-grid solar hybrid MG system, the output power of a PV module is estimated from (1) based on the solar irradiation at time t, and the efficiency of the PV module is given by (2).

P P V ( t ) = η P V ⋅ A P V ⋅ G ( t ) (1)

η P V = η S T C ⋅ η M P P T [ 1 − α ( T C − T S T C ) ] (2)

where A P V is the area of a PV module in (m^{2}), G ( t ) is the hourly total solar irradiance in (W/m^{2}), η P V is the efficiency of the PV array, η S T C is reference efficiency of the PV cell at standard temperature condition (STC), η M P P T is the efficiency of the maximum peak power tracker, T C is the temperature of the PV cell in (˚C), T S T C is the reference temperature of the PV cell at STC (25˚C), and α is the temperature coefficient of the PV cell (typically 0.4%/˚C - 0.6%/˚C for silicon cells). The temperature coefficient is given by the PV cell manufacturer and can be obtained from the PV panel datasheet.

The cell temperature can be obtained from Equation (3).

T C = T a + [ N O C T − 20 800 ] ⋅ G ( t ) (3)

where T a is the ambient temperature in (˚C) and NOCT is the nominal operating cell temperature (45˚C - 47˚C).

The output power from a WT at time t depends on the wind speed and can be obtained from (4) [

P W T ( t ) = { 0 V ( t ) < V c i a ⋅ V 3 ( t ) − b ⋅ P W T r V c i ≤ V ( t ) < V r P W T r V r ≤ V ( t ) < V c o 0 V ( t ) ≥ V c o (4)

where a = P W T r ( V r 3 − V c i 3 ) ; b = V c i 3 ( V r 3 − V c i 3 ) ; V ( t ) is the wind speed at time t in

(m/s); P W T r is the rated power of the WT in (W); V r is the rated speed in (m/s); V c i is the cut-in speed in (m/s) and V c o is the cut-out speed of the WT in (m/s).

Depending on its state of charge (SOC), the ESS can supply the load when there is lack of electricity (discharge) and store surplus power when the generated power exceeds the load demand (charge). The discharging and charging energies of the ESS at time t can be obtained from (5) and (6), respectively [

E E S S d ( t ) = E E S S ( t − 1 ) − [ E L o a d ( t ) − E P V ( t ) − E W T ( t ) ] / η d (5)

E E S S c ( t ) = E E S S ( t − 1 ) + [ E P V ( t ) + E W T ( t ) − E L o a d ( t ) ] ⋅ η c (6)

where E E S S ( t − 1 ) is the energy at time t − 1 in (kWh); E P V , E W T , E L o a d are the PV energy, WT energy and load energies, respectively; η d and η c are the discharge and charge efficiencies of the ESS, respectively.

Equations (5) and (6) can be rewritten as (7) and (8) [

E E S S d ( t ) = E E S S ( t − 1 ) − P E S S d ( t ) ⋅ Δ t / η d (7)

E E S S c ( t ) = E E S S ( t − 1 ) + P E S S c ( t ) ⋅ Δ t ⋅ η c (8)

where P E S S d ( t ) = P L o a d ( t ) − P P V ( t ) − P W T ( t ) is the power discharged by the ESS; P E S S c ( t ) = P P V ( t ) + P W T ( t ) − P L o a d ( t ) is power charged into the ESS; and Δ t = 1 since the time interval is 1 hour.

The load profile determines the requirements of power supply from the hybrid MG power system. The load profile is modeled according to the dynamic load power demands P L o a d at times t. In off-grid power system design, the load profile is the driver.

・ Most residents wake up in the morning to prepare for work and school (i.e. taking hot showers, preparing breakfast, etc.).

・ People then leave for work and school typically from hours 7 to 18 on weekdays.

・ Most household members are cooking/warming food, eating dinner, watching TV from hours 18 to 22 then go to bed.

The curve shown here in

The power balance of the system is illustrated in

L P S ( t ) = P L o a d ( t ) − [ P P V ( t ) + P W T ( t ) + P E S S d ( t ) ] ⋅ η i n v . (9)

where η i n v is the efficiency of the inverter.

The Loss of Power Supply Probability (LPSP), which is the reliability index of an MG system, for a given time period T can be defined as the ratio of all LPS(t) values for that period to the sum of the load demands [

L P S P = ∑ t = 0 T L P S ( t ) ∑ t = 0 T P L o a d ( t ) = ∑ t = 0 T Power failure time T (10)

The objective of this optimization problem is to minimize the capital and operating costs of the off-grid hybrid MG over a total life period of 20 years, while satisfying some reliability, operational and stability constraints. This optimization problem is expressed in (11) and (12) [

min f ( N P V , N W T , E E S S ) = C P V ⋅ N P V + C W T ⋅ N W T + C E S S ⋅ E E S S (11)

{ C P V = ( C c a p P V + 20 ⋅ C o p P V ) ⋅ P P V C W T = ( C c a p W T + 20 ⋅ C o p W T ) ⋅ P W T C E S S = C c a p E S S + y E S S ⋅ C c a p E S S + ( 20 − y E S S − 1 ) ⋅ C o p E S S (12)

where N P V is the number of PV panels; N W T is the number of wind turbines; E E S S is the storage capacity of the ESS in (Wh); C P V and C W T are the total costs in (US$) of a PV and a WT, respectively; C E S S is the per-unit cost of the ESS in (US$/Wh) of the ESS; C c a p P V , C c a p W T and C c a p E S S are the capitol costs of the

PV in (US$/W), WT in (US$/W) and ESS in (US$/Wh), respectively; C o p P V , C o p W T and C o p E S S are the operating costs of the PVin (US$/W), WT in (US$/W) and ESS in (US$/Wh), respectively; and y E S S is the expected number of ESS replacement over the 20 years. The capital cost of each component includes the purchase and installation cost of that component. The operation cost of each component includes the maintenance and operation cost of that component.

The following constraints should be satisfied:

・ Reliability:

L P S P ≤ L P S P s e t (13)

・ PV power limits:

P P V min ≤ P P V ( t ) ≤ P P V max ; N P V = 0 , 1 , 2 , ⋯ (14)

・ WT power limits:

P W T min ≤ P W T ( t ) ≤ P W T max ; N W T = 0 , 1 , 2 , ⋯ (15)

・ ESS stored energy and power limits:

E E S S min ≤ E E S S ( t ) ≤ E E S S max (16)

E E S S min = ( 1 − D O D ) ⋅ E E S S max (17)

0 ≤ P E S S d ( t ) ≤ P E S S max d = min { E E s s max ⋅ η d , ( E E S S ( t − 1 ) − E E S S min ) ⋅ η d } (18)

0 ≤ P E S S c ( t ) ≤ P E S S max c = min { E E s s max / η c , ( E E S S max − E E S S ( t − 1 ) ) / η c } (19)

In addition to the stored energy constraint, the charge and discharge powers of the ESS must be kept within a certain limit at any point of time. In Equations (18) and (19), the energy terms are divided by the time internal Δ t = 1 to get the powers.

・ Energy balance (starting and ending limits):

E E S S ( 0 ) = E E S S ( T ) = E E S S s t o r e d (20)

・ Power balance

P P V ( t ) + P W T ( t ) + P E S S ( t ) = P L o a d ( t ) (21)

In the past few decades, various nature-inspired computational methods have been developed to solve complex engineering problems [

were introduced (multiple weighted roulettes, multiple cross over points, multiple mates, utilizing the Daubechies wavelets (D4) and using normal distribution for selecting the initial population). The convergence velocity of the algorithm is also improved thereby reducing the time taken for the algorithm to reach the sought solution.

A basic part of the selection process is to stochastically select from one generation to another in order to create the basis of the next generation. The underlining requirement is that the set of fittest individuals would have a greater chance of survival than the set of weaker ones. This inheritance nature in those fitter individuals will tend to have a better probability of survival and will go on forward to form updated mating pool for the next generation. Weaker individuals are not left without a chance though. In nature those individuals may have genetic coding that may also prove useful to future generations.

Selection is the first genetic operation in the reproductive phase of GA. It helps the GA by directing the genetic search towards promising regions in the search space. Selection pressure is a crucial factor that determines the efficiency of the algorithm and is reduced in our proposed algorithm. The first enhancement proposed is to use multiple weighted roulettes, each designed to complement the others. This will further distribute the selection pressure for one generation to another. The job that GAs have in this case is to mate sets of individuals and then replicate this selection process. The usual implementation is by crossover. The only general requirement is that the offspring carry forward the important genetic material from the parents, whilst introducing enough variation that they survive. The crossover method emulates this process by exchanging chromosome patterns between individuals to create offspring for the next generation.

The second and third enhancements are to use multiple cross over points as well as using multiple mates as a function of results of mating individual parents creating some offspring. Those offspring will have of the genetic material of both parents. There are three options regarding the fitness of the offspring, they can be weaker, the same or fitter than their parents. If they are weaker they will tend to die out―if they are stronger their chances of survival are better. It is of general note that the stronger the parents are in terms of fitness then the fitter the offspring will be. The variation caused by this process allows the offspring to search out different available niches, i.e. find better fitness values and subsequently better solutions.

The fourth enhancement proposed is to utilize the Daubechies wavelet (D4), which is named after its discoverer the mathematician Ingrid Daubechies, as a preprocessing step. The D4 transform has four scaling function coefficients and can be extended to multiple levels as many times as the signal length can be divided by 2. D4 was compared to other wavelets. The HAAR wavelet for instance is simple, memory efficient and computationally cheap. It uses two scaling and wavelet function coefficients, thus calculates pair wise averages and differences. Daubechies wavelet family is the most popular wavelet family used for texture feature analysis, due to orthogonal and compact support abilities. Daubechies averages over more pixels, it is smoother than the HAAR wavelet. It is Similar to the well-known Fourier transform, but it takes care of rapid transitions better than Fourier.

The fifth enhancement is used for initializing the initial population and it is done by replacing the uniform distribution with the normal distribution. The normal (Gaussian) distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a standard of reference for many probability problems. That is why it was selected to be the fifth enhancement proposed to deal with the step of selecting the initial population. Some of the characteristics of the normal distribution are that it is symmetric, bell shaped and continuous for all values of X between -∞ and ∞ so that each conceivable interval of real numbers has a probability other than zero. Normal distribution is actually a family of distributions since the two parameters μ and σ determine the shape of the distribution.

This enhanced GA was used in radar system application to detect the angle of arrival by analyzing readings from an array of multiple radars [_{PV} N_{WT} E_{ESS}]. To accomplish this, a specific fitness function was developed along with a new data preprocessing program in Java to organize and prepare the data for importation into the enhanced GA search environment to evaluate every individual for 8760 hours per simulation cycle. A GA optimization can have many ways to use as a stop rule. The best one of course is if the optimum solution is known. However, since in this case the optimum solution was unknown, the system used the population size and a maximum number of generations as the stop rule.

The location of Guzape (suburb of Abuja), Nigeria (Latitude 9˚0'31''N, Longitude 7˚30'50''E) is used as site for the case study. The global horizontal solar irradiance

The enhanced GA was used to minimize the total MG cost, while considering different LPSP values for system reliability. The results were compared with

those obtained from the Brute Force and standard GA. A comparison between the three strategies is presented in

Parameter | Value | |
---|---|---|

PV Data | Max power Efficiency at STC Length Width Capital cost Operation cost Life time | 255 W 15.7% 1.64 m 0.99 m $0.80/W $0.04/W 25 yr |

WT Data | Rated power Cut-in wind speed Cut-out wind speed Rated wind speed Capital cost Operation cost Life time | 1500 W 2.5 m/s 17 m/s 12 m/s $0.67/W $0.1005/W 20 yr |

ESS Data | Charge efficiency Discharge efficiency Capital cost Operation cost Life time | 80% 100% $0.15/Wh $0.015/Wh 5 yr |

Brute Force | Standard GA | Enhanced GA | |
---|---|---|---|

No. of Trials_{ } | 35,267,760 | 117,624 | 76,406 |

DOD = 0.8 | DOD = 0.7 | DOD = 0.6 | DOD = 0.5 | |
---|---|---|---|---|

N_{PV } N_{WT } P_{PV } P_{WT } E_{ESS} Total Cost | 164 5 41.8 kW 7.5 kW 200 kWh $255012.00 | 166 5 42.3 kW 7.5kW 200 kWh $255828.07 | 168 5 42.8 kW 7.5 kW 200 kWh $256644.00 | 176 5 44.9 kW 7.5 kW 200 kWh $259908.00 |

found that the optimal size of the site is 176 solar panels, 5 wind turbines with 200 kWh of energy storage for the minimum total 20-year cost of $259,908.00 based on the desired operation and reliability constraints with 50% DOD and LPSP less than or equal to 0.15.

An approach of sizing a PV/WT/ESS hybrid off-grid MG using an enhanced GA is proposed in this paper. A case study involving the optimal sizing of an off-grid MG system site in Nigeria using real field data was discussed. For the case study, a special Java program was developed along with the enhanced GA to obtain the optimal size of the MG.

That is the combination of PV, WT and ESS that met the load demand subject to the operation constraints and the desired LPSP with the minimum total MG cost over a period of 20 years. The result is compared to those obtained from the Brute Force and standard GA. All methods give the same result under the same conditions, showing that the enhanced GA is well suited for optimal MG system sizing, and the proposed method is feasible for sizing PV/WT/ESS hybrid off-grid MG systems.

This work was supported by the Information, Computation, Communication& Control (ICCC) group at Oakland University, USA and Blue Camel Energy Ltd., a solar energy company based in Abuja, Nigeria. We thank Blue Camel Energy for providing site data and details that greatly contributed to the development of this manuscript. We thank Dr. Serge Kruk from the Department of Mathematics and Statistics at Oakland University for providing insight and expertise that greatly improved the manuscript.

Traoré, A., Elgothamy, H. and Zohdy, M.A. (2018) Optimal Sizing of Solar/Wind Hybrid Off-Grid Microgrids Using an Enhanced Genetic Algorithm. Journal of Power and Energy Engineering, 6, 64-77. https://doi.org/10.4236/jpee.2018.65004