Sizing of a stand-alone hybrid wind-photovoltaic system using a three-event probability density...

Pergamon 0038-092X(95)00116-6

Solar Energy Vol. 56, No. 4, pp. 323-335, 1996 Copyright © 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0038-092X/96 $15.00 + 0.00

S I Z I N G O F A S T A N D - A L O N E H Y B R I D W I N D - P H O T O V O L T A I C S Y S T E M

U S I N G A T H R E E - E V E N T P R O B A B I L I T Y D E N S I T Y A P P R O X I M A T I O N

A. D. BAGUL *, Z. M. SALAMEH ** and B. BOROWY ** * Department of Energy Engineering, University of Massachusetts-Lowell, James B. Francis College of

Engineering, 1 University Avenue, Lowell, MA 01854, U.S.A. and ** Department of Electrical Engineering, University of Massachusetts-Lowell, James B. Francis College of Engineering, 1 University

Avenue, Lowell, MA 01854, U.S.A.

(Communicated by VOLKER WITTWER)

Abstract--The PV-array and battery storage sizing of a stand-alone hybrid wind-photovoltaic system is being addressed here. A probabilistic approach is used to arrive at the results. A new technique using a three event probability density instead of the traditional two event approximation is developed. This technique is then used to determine the optimum relationship between the number of PV panels and the number of storage batteries required for the stand-alone hybrid wind-photovoltaic system, to meet a certain loss of power probability. This method uses long term data of wind speed, irradiance and ambient temperature taken every hour for 30 yr and the load specifications for a "typical New England house" that is going to be powered by the hybrid system. Copright © 1996 Elsevier Science Ltd.

1. INTRODUCTION

The use of renewable energy technology to meet our energy demands has been steadily increasing over the years. However, the important drawback associated with renewable energy systems is their inability to guarantee reliable, unin- terrupted output at a cost that can compete with the conventionally produced power.

It is evident that renewable energy-based electricity generation systems can better compete economically with power from the grid in remote locations, where the grid is either not feasible or nonexistent. This is documented by Godley and Leishout (1987), Bishop (1988) and Payne and Sheehan (1979). This is due to high cost of transmission lines and higher transmission losses that accompany distribution of centrally generated power to remote areas. "Stand alone" systems could be a possible economical alternative to running the grid all the way to these remote places.

Among renewables, wind turbine generators are free from major environmental concerns and, at the same time, cost competitive. Photovoltaics, though environmentally benign, is still an expensive option. The maintenance required for these systems is very little when compared with a diesel generator. A drawback, however, common to wind and solar options is their unpredictable nature and dependence on weather and climatic changes. Both of these systems due to lesser predictability would have to be oversized to make them completely reli-

able, resulting in even higher total cost. Providing back-up generation, say, with a diesel generator as an alternative to oversizing would however retain the drawbacks of using diesel generators.

Combining wind and solar energy into a "hybrid" generation system could solve the problem. The two random sources of energy which are individually less reliable could as a whole have a higher reliability. It has been found that a hybrid wind-photovoltaic system is better than an individual wind or photovoltaic power system as documented by Castle et al.

(1981). Thus an improved performance and dependability can be achieved without much increase in the cost wherever both these resources are available.

The design of such a system would involve the determination of opt imum values for the rated capacity of the wind turbine, the capacity of the photovoltaic arrays and the capacity of batteries for storage that would meet the required reliability conditions for the system. The method for determining the opt imum size of the wind turbine is as discussed by Salameh and Safari (1992). The problem addressed here is, given the rated capacity of the wind turbine and the load demand on the system, to determine the relation between the number of photovoltaic arrays and the number of batteries to meet a given reliability. An older method by Bucciarelli (1984), which was developed for the design of "photovoltaics-only" systems was modified to include the additional input energy

323

324 A.D. Bagul et al.

~-u~= t~,~r r ~ , average wind speed during an hour in m/s,

_ ~ ~ v j L~ global horizontal extraterrestrial insolation and the direct normal extraterrestrial irradiance

I I I (W/m2), V- global, horizontal terrestrial insolation and

/ ~ ~',,~N,X the direct normal terrestrial irradiance also r t t ~ - - - (W/m2),

input { ambient temperature (o). ] ~ v,~,co.~l~__/ ~ The data is in the form of average values for

pv,~ every hour of all the days for 30yr from

Fig. 1. Schematic diagram of the system.

from the wind turbine. However a new technique was developed and used to design the system instead of that proposed in the original method. The new technique provides for more accurate results as is shown by the comparison of the two methods. The detailed schematic diagram of the system as in Borowy and Salameh (1994) is shown in Fig. 1.

When designed to operate at a reliability equal to that at which an electric utility operates, the system has to function at a loss of load probability (LOLP) of 1 day in 10 yr. Thus the system has to be designed for a r r E of 0.00027. However it should be noted that such high reliability is usually not expected of a system and even a slight reduction in nE will reduce the cost of the system dramatically.

The load on the system is an experimental residential load: "a typical New England home". This, as supplied by the NEPSCo., included typical hourly values of the load for four seasons of the year. However the design technique used here needs to consider only the total daily energy demand and supply. Also it can be reasonably assumed that the load requirement of a residence is constant for any particular season, and changes only from season to season over the entire year. The values assumed here are shown in Table 1.

The data used for the design of this system were mainly obtained from the Nor th Carolina Meteorological Center. The data are for the Logan International Airport, Boston, Mass., U.S.A. and include:

Table 1. Load on the hybrid system

Season Months Daily load on the system

Fall September-November 15.825 kWh Winter December-February 22.165 kWh Spring March-May 16.545 kWh Summer June-August 18.845 kWh

1961-90. The system can be designed using either a

deterministic or probabilistic approach. In the deterministic approach, the state of the system is analyzed for each time unit which could either be an hour or a day (depending upon the accuracy intended). For the time unit chosen, say an hour, the energy generated by the system, the load on the system, the energy stored, etc. during that particular hour are calculated exactly and calculations are repeated for each and every hour in a particular month or a season or even for the whole year. Also the chronological sequence of the data is taken into account. The worst case is then identified and the system parameters are selected accordingly to meet the design requirements.

In the probabilistic approach the energy generated by the wind turbine and the photovoltaic array are looked upon as two random variables. Calculations are not performed with the actual values of daily or hourly energy balance. Instead these values are reduced to statistical parameters, like mean and standard deviation which are then used to arrive at the system design. It is assumed that all the daily values of "surplus energy" in a given month or season are distributed randomly over the entire span of that month or season. This method does not take into consideration the chronological sequence of the data which does bear some importance. This makes the probabilistic method, to some degree, an approximate method.

2. METHODOLOGY

2.1. Basic methodology

Using the weather data and manufacturer 's specifications the daily surplus (or deficit) energy generated, D, is obtained. The same calculation is performed for each day in a particular month or a season to obtain the probability density curve, p(D), for the daily energy flow into the storage, D. The statistical information from this density function is then used to determine the

Hybrid wind-photovoltaic system 325

relation between the storage and the array capacities.

The energy generated by the wind turbine is a function of the wind speed only. However the optimum number of PV modules has to be determined. Therefore the values of p(D) are determined for many different values of Npv. Also/~o and aD of the p(D)s are determined for those different values of Npv.

The actual p(D)s are then approximated by a simpler probability density function. The storage which is, in reality, a continuous variable is assumed to be made up of a certain number of discrete levels. Using the approximated probability density function and the values of /~o and tr o, the probabilities of the storage occupying any of the possible levels are determined. These along with the LOLP requirements are then used to determine the value of N b corresponding to each value of Npv. With these values of Npv and Nb a curve giving the relation between these two parameters is plotted. Any point on the curve satisfies the prescribed LOLP condition. To obtain the optimum combination, conditions of economics are then applied.

2.2. Two event approximation technique

The two event approximation technique by Bucciarelli (1984), has been used traditionally for sizing "photovoltaics-only" systems. In that approach the load, L on the system is assumed to be constant and the probability density for D is obtained merely by shifting the probability density for the array power output to the left by an amount equal to L. From this density /~o and ao are determined.

To simplify the calculations this probability density of D is replaced by tha' for two events, namely an increase and a decry. ~se in the stored energy by an amount A with prooabilities p and q respectively. The values of p, q and A are determined from moment equations so that the mean and variance of the modified probability density are the same as those for the original one. The condition p + q = 1 is also used. The storage is assumed to be divided into N levels each of size D. Then the probabilities of transition of the storage between these N levels are determined.

Finally the analytical expression for the probability of the storage occupying any state is obtained. The probability of the system losing power is then the product of the probability of the system being in its lowest level and q, the probability of depleting the storage over the

course of the day. Thus given the loss of load probability required, one could perform the calculations backwards and arrive at the system design.

2.3. Drawbacks of the two event approximation

Though the original method greatly simplifies the design procedure, it cannot be directly applied to hybrid wind-photovoltaic systems, as such a system has two randomly changing parameters as opposed to one in the "photovoltaics-only" system. Thus the probability density for D cannot be obtained merely by shifting the density for the array output. Either convolution has to be used or the values of D have to be calculated for each day and then the probability density for D obtained.

The two event approximation technique assumes that the storage only increases or decreases, while in reality there is a possibility of the storage staying in the same level over the course of a day. It was found that the storage can be considered to be remaining in the same level for more than 50% of the observations. Thus the assumption, though intended to make the calculations simple, usually oversizes the system.

Further the value of 3 chosen in this technique is fixed using moment equations. However one should be able to fix the value of A independently in order to obtain the closest possible approximation of the actual distribution. If a small value of 3 is chosen, it results in the storage getting divided into a large number of levels each of a smaller size. This would increase the accuracy of the calculations. However for the assumed distribution to correctly match the original the value of 3 has to be relatively higher. Thus by choosing the value of A independently a proper tradeoff can be struck between these conflicting requirements.

Also the original method uses only the mean and standard deviation of the array output to design the system. Though simple, this approach cannot take into consideration the effect of temperature on the performance of the photovoltaic array which was found to be particularly pronounced in this case.

2.4. Three event approximation technique

This technique also uses the same basic con- cepts of probability theory as used for the two event approximation. The starting point for this method is the probability density of D for the time period under consideration which could


either be a month or a typical season. Or in terms of statistical parameters, the mean and the standard deviation of D must be known before this method can be applied. The way in which these parameter values are arrived at would depend on how much accuracy is expected of the calculations and also the time and effort that can be afforded by the designer. However for the development of this technique the way in which the parameters were calculated is immaterial.

Figure 2 shows a sample p(D) curve for the month of December. This curve has been drawn for one particular combination of number of PV panels and batteries. There are different curves for different combinations but the basic shape of the curve remains the same. The curve is shifted either to the left or to the right depending upon the combination used. The discussion that follows is for a probability density curve for one particular combination.

2.4.1. Obtaining the approximate probability density function. We begin with the probability density, p(D), of D (in kWh). This also includes the values of t o and an- The probability density for D is now replaced by an approximate three event probability density. The three events are as described below:

Over the course of a day, Event 1. An increase in the stored energy by an amount A (kWh), with a probability, p.

Event 2. A decrease in the stored energy by an amount A, with a probability, q.

Event 3. Neither an increase nor a decrease in the stored energy, with a probability, t.

Thus in contrast with the original method, which approximated the probability density by a two event Markov process and treated it to

0.09

0.O8

0.07

"~ o.os

0.0'3 o = ~o.~ / o

0 . ~

0 . ~

0.01

0 -20 -10

\

0 10 D (KWH/doy)

Case: NI; v = 68 NI = 111 D( (:ember

\ \

30 40

Fig. 2. Probability densities of D (for December).

0.09

0.08

0.07 ?

~" 0.0¢

0.05

N o.o4

ft. 0.03

0.01

t•/ : NI~

h m l

nott¢

o r ~ I ' I .20 -10 0 10 20 30

o (KWH/clay)

Fig. 3. Ac tua l a n d a p p r o x i m a t e p r o b a b i l i t y (for December ) .

/ = 6 8 = 111

cember

ed sca~}

\ 4O

densi t ies

be a case of random walk, the technique suggested here takes into account the possibility of the stored energy remaining at its current value. The three events stated above are shown in Fig. 3.

The three event distribution has to be equiva- lent to the actual distribution of D. To obtain a reasonable approximation the following three conditions are sufficient:

(1) The three events should completely represent the original distribution,

p + q + t = l . (1)

(2) The mean of the new distribution must be same as that of the original one,

(p--q)@A =Pn. (2)

(3) The second moment of both the distributions should be the same,

(p+q)e,a2=~ +o~o. (3)

To select the value of A a fourth condition can be used, which is to equate the third moment of the two distributions:

(p--q)QA 3 =kt 3' (4)

where #3' the third moment of D about zero, and p, q, t and D are as explained earlier and are unknowns in the above equations.

However it is not necessary to determine the value of A using eqn (4) or any other similar condition. On the contrary the three event approximation makes it possible for the designer to chose the value of D independently taking into consideration the original probability density for D.

As mentioned earlier, the value of A has to


be optimized to satisfy two counteracting conditions. The approach suggested here to determine the optimum value of A is as follows:

Assume that 99% of all the data points of the probability density of D are spread over a span of (n%%). Since the assumed distribution has exactly three events, this total span of the original distribution is divided into three equal zones. The center one is associated with Event 3, the remaining portion of the distribution which is to the left of the central zone is associated with Event 2 and the portion on the right with Event 1, respectively. The mid points of these zones will then correspond to 0, - A and + A, respectively. This is clear from Fig. 3. Each zone will have a span of (n" ao/3). This gives a value of .4 as

n ' o - .4 - (5)

3

Thus selecting the value of .4 in this manner ensures that the proportion of observations we associate with each of the three events is accurate. For the sizing calculations a value of .4 = 1.5~v was found to be the accurate value. This is explained in further detail later.

2.4.2. Determination of transition probability matrix. The process of daily energy flow into the storage can be assumed to be stationary as the fluctuations are usually distributed normally about the constant mean value and hence the expressions used for #n and an hold.

The value of .4 obtained is then plugged into the first three equations to get the following relations since/~D and an are already known.

~ n ' . 4 + a ~ + u~ p = 2..42 (6)

q = 2..4 2 (7)

t = l - \ ~-h )" (8)

Let the total required capacity of the storage be C (kWh). Now the entire storage capacity

C=N*.4 (9)

can be assumed to be divided into N equal and discrete levels, each of size .4, which it could occupy.

The process of the storage going from one of its levels to another over the course of a day has to be analyzed in order to determine the LOLP of the system. This process can be considered to be a Markov process as it satisfies the

definition as given by Bhat (1972) for a Markov process.

For the process under consideration here, the transition of the storage from one of its levels to a new level is an event. The number of possible events depends upon the assumptions made and the conditions imposed on the transition of the storage between different levels. If at the beginning of any day i, the storage is in level K (K = 1, 2, 3 ..... N) then the level of the storage at the beginning of the ( i + l ) t h day depends only on the condition of the storage on the ith day. Thus the process of storage is a Markov process.

In the original method it was assumed that the number of possible events is two. One was the transition of the storage to the next lower level with probability q and the other was the transition to the next higher level with a probability p. Thus this process was considered to be a random walk.

In the modified method however there are three possible events. The transition probability matrix for this N state, three event Markov process is as shown Fig. 4. In this matrix any element Pij represents the probability of the storage undergoing a transition from the jth level to the ith level.

Now consider a column vector n such that any of its elements, hi, represents the probability of the storage occupying the ith level. The sum of all the elements of the vector n is equal to unity. The vector n is also shown in Fig. 4.

2.5. Analytical expression for nj This transition probability matrix gives N

equations which relate the probability of occupying any one of the N levels hi, to p, q, t and the probability of occupying the next higher or lower level ni+l and n~_l. Under steady state, the following matrix equation holds true as given by Bhat (1972):

P .g=zc.

It represents N linearly dependent equations, as given by Bhat (1972), which are listed below:

nl =(hi" q) + (nl" t) + (~2" q) ~2 = ( h i " P) + (n2" t) + (n3" q) n3 = (n2" p) + (~3" t) + (rt 4 • q) n4 = (n3" p) + (n4" t) + (ns" q)

rrj=(rcj_l" p)+(nj" t)+(n~+ 1 'q) (10)

~ N - 2 = (rCN - 3" P) + (=N - 2" t) + (ZCN - 1" q) ~ N - 1 = (7~N - 2" P) -~- (nN - 1" t) + (~N" q) ZCN = (rCN- 1" P) + (ZCN" t) + (rrN" p).

SE 56-4-D


q+l

P

0

0

0

0

0 0

q 0 0 0 0 ............. 0 0 0 0

t q 0 0 0 ........... 0 0 0 0

p t q 0 0 ............. 0 0 0 0

0 p t q 0 ............ 0 0 0 0

0 0 0 0 0...p t q....0 0 0 0

0 0 0 0 0 .............. p t q 0

0 0 0 0 0 ............. 0 p t q 0 0 0 0 0 ............ 0 0 p t+p

m

7(

I

a 2 11

11j

% 11,

11

P

-2

[-1

Fig. 4. The transition probability matrix and the vector n.

This set o f eqn (10) can be solved using matrix algebra and the Gaussian elimination method. The fact that p + q + t = 1 is also used during the matrix manipulations• Rearranging the terms in each equat ion and writing in matrix form we get eqn (11) as shown in Fig. 5.

N o w solving this matrix equat ion step by step substituting (t + q - 1 ) by - p, (t + p - 1 ) by - q and performing row operations, divide each row by - p or + p in order to make the first non-zero element in each row equal to unity•

As the left hand side of the equat ion is the product of two matrices, the row operat ion has to be performed only on the pre-multiplier. Thus we perform it on P and the p vector remains unaltered• This gives us the matrix eqn (12) as shown in Fig. 6.

N o w the operat ion R 2 ~ > - ( R 2 - R I ) is performed and the result simplified• This same calculation is repeated for all the rows beginning from the second row upto the ( N - 1 ) t h row. This gives the matrix eqn(13) as shown in Fig. 7.

N o w these equations can be solved individually beginning from the Nth row. The last two rows are redundant , and cannot be independently solved• Another condit ion which will be later used to solve these N - 1 distinct equations with N variables is that the sum of all elements of the vector p is unity• We assume

2 = P . (14) q

The matrix obtained above is the same that

u

t+q-I q 0 0 0 0 .............. 0 0 0 0

p t-I q 0 0 0 ........... 0 0 0 0

0 p t-I q 0 0 ............. 0 0 0 0

0 0 p t-I q 0 ............ 0 0 0 0

0 0 0 0 0 O . p t-I q.. 0 0 0 0

0 0 0 0 0 0 .............. p t-I q 0

0 0 0 0 0 0 ............. 0 p t - l q

0 0 0 0 0 0 ............ 0 0 p t+p-! n

Fig. 5. Equation (11).

rt I

~3

~j

7t b 11

1I

o l 01

I 0 I 0j

_- i 0

01 _.1


1

1

0

0

0

0

0

0

-clip

(t- 1 )/p

1

0

0

0

0

0

0

q/P

( t - l ) /p

I

0

0

0

0

0 0 0 ......................... 0 0

0 (1 (1 ........................ 0 0

qlp 0 0 .......................... 0 0

( t - l ) /p q/p 0 .......................... 0 0

0 0 ..... 1 ( t - I ) / p q / p . . o o

0 0 0 . . . . . . . . . . . . . . . . . . . . I ( t - I ) / p

0 0 0 . . . . . . . . . . . . . o 1

0 0 o ............ {} 0


m

0 0

0 0

0 0

0 0

0 0

q/p o

(t- I ) /p q/p

1 -q/p

/t I

n 3

r~j

x}

3

I I I I =

I I I

I

'I I

i

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

.q/p

1

o

0 !

0

0

0

0

0 0

q/p o

1 -q/p

o I : !

0 0

0 0

0 0

0 0

0 0 ................... 0 0

0 0 ................... 0 0

o o ................... 0 0

<lip o ................... 0 0

0 0 .... 0 I -clip ..... 0 0

0 0 .................. 0 1

0 0 .................. 0 O

0 0 ................. 0 0


-q/p

0

0

o

0

0 0

0

l -q/p

I -q/p

K I

n 2

/1

74 X

/t:

/t.

" "1 0

0 I |

0

o l

-- o l

2 0

I 0

0

would have been ob ta ined had the two event a p p r o x i m a t i o n been applied• The only difference, which also is an i m p o r t a n t difference, thus lies in the values of p and q and in turn the value of 2. Solving the above equa t ions we get

1 ~ N - 1 - - ~ " 7 I N = 0

which can be wri t ten as

1 7~N-1~ ~ "7~N

and

1 ( ~ ) 2 ~ N - 2 ~- ~ "7~N-1 ~-" "~N

and so on

~ 1 = ( ~ )N--1 "~N . (15)

These N - 1 dis t inct equa t ions have N variables• To solve them we now use the condi t ion tha t

nl + n 2 + % + . . . + n j + . . . + n N = 1. (16) Subs t i tu t ing in terms of PN

[ ( -~ ) N - I + ( ~ ) N - 2 + . . . + ( ~ ) + 1 ]

1 + ~N

• TtN= 1

(17)

(18)


summation from i to N - 1. Solving this summation gives

(19)

Now for any r~j we have

7tj = "r~N. (20)

Substituting and simplifying we get

r t j=2 j-1 . ( 1 - 2 ) / ( 1 - 2 N) j = 1,N. (21)

This is the expression for the probability, re j, of the storage occupying any general level j. This expression is same as that obtained using the two event approximation, however the values of 2 and q are different in this case.

Now suppose that rq is the probability that the storage would occupy the lowest level. The loss of load probability of the system is 7re. Thus loss of load would occur if the system is in the lowest level at the beginning of a day and if the day is a bad day. Hence we can write

rc E =~zl* q. (22)

Hence given the value of n E, one can work backwards and determine the value of N corresponding to the value of Npv. Substituting the value of 7 h from eqn (21) into eqn (22) we get

N=ln [ l + q ' (2 -1 ) (23)

The total capacity of the storage needed can then be determined from eqn (9). This explains the three event approximation technique to determine the required storage capacity, given the density for D. The procedure for arriving at the density of D is explained in the next section.

south and inclined at an angle equal to the latitude which is 42°N. To determine the insolation on the array from the direct and global components of the horizontal insolation, the anisotropic sky model as given by Duffy and Beckman (1991) was used.

The symbols used to represent different variables are explained in the Nomenclature. For the geometry of the solar insolation the relations as given by Duffy and Beckman (1991) were used. The actual calculation is not shown here. The cell operating temperature is determined using an empirical relation as given by Hart (1982). The power generated is then given by

p : v*i

(average value of power assumed constant for the hour being considered).

The series resistance of the panel can be determined using the formula given by Rauschenbach (1980). The value of p is also equal to the energy generated by one photovoltaic panel during that particular hour. The values of p for a given day are then added together to give the daily energy produced by one panel, ps(i).

3.2. Step 2: calculation of the daily energy of the wind turbine

The values for the wind speed are modified for the tower height using the following relation as presented by Justus (1978):

V : V l * ( 2 / z 1 ) 1]'7 .

The power curve for the wind turbine is shown in Fig. 8. The exact values of the output power for integer values of wind speed were known from the specifications. The curve was assumed to be linear in between the data point.

The power curve is then used to determine the hourly output of the wind turbine. All the

3. CALCULATIONS

The method that was developed is now 16oo applied to the hybrid system. The formulae used ~- 140o

~., 1200 to convert the data to information needed for - the design are explained step by step in the ~ 10oo a00 next section, o 600

~= 400

3.1. Step 1: calculation of the daily energy output ~ 200 of one PVpanel o

The available data was used to determine the total solar insolation on the plane of the array which was proposed to be mounted facing due

P o w e r c u r v e

2 4 6 8 10 12 14 16 IS 20

Wind speed (m/s)

Fig. 8. Power curve for the wind turbine.


values of the hourly energy generated by the wind turbine are then added to give the total daily energy generated by the wind turbine, P,(i) during that particular day. These calculations are repeated for all the days (i.e. for all values of i) for the 30 yr period and the values are then arranged into four groups for the four seasons and into 12 groups for the 12 months.

3.3. Step 3: calculation of the number of batteries corresponding to a number of P V panels

Now to perform the calculations necessary to arrive at the Np~ versus Nb curve, a certain value of Npv is assumed to begin with. For this value of Npv the total daily energy generated by the PV array, Ps(i) is determined:

Ps(i) = Npv*p~(i).

Now the value of D, the daily surplus energy is determined as

D(i) = P~(i) + Pw(i) - L(i)

where

ps(i)= the daily energy generated by one panel for the ith day

Ps(i) = daily energy generated by an array of Npv number of panels for the ith day

P,(i) =dai ly energy generated by the wind turbine for the ith day

/_(i) = the total daily load on the system for the ith day.

These calculations are repeated for all the days for which the data is available to obtain the probability density for D. From this density the values of/~o and aD are determined. These are used to determine the values of p, q and t. The value of d being fixed to be 1.5ao, the ideal capacity of the storage, that would be necessary to provide the required system reliability, is determined. The efficiency of the batteries and the capacity of a single battery as known from the specifications are then used to determine the actual number of batteries, Nb, required. This gives us one point of the Npv versus N b curve. To obtain the entire curve the above calculations are repeated over a range of values of Np~. These points are then plotted to arrive at the curve. Individual Np~ versus Nb curves are obtained for each season of the year. To obtain the curve for summer for instance only the data points from the summer season are used. Individual curves are also obtained for the 12 months of the year.

3.4. Step 4: construction of the constant cost lines

To determine the point on the curve that would give the least cost combination, a constant cost line is drawn tangent to curve. The point where the curve touches the line (or in other words the point where the curve has the same slope as the constant cost line) is the most economical combination of Npv and N b for a given system reliability. The slope of the constant cost line is determined using the following relation:

Slope -- - ( C o s t of one battery/ Cost of one PV panel).

4. RESULTS, ANALYSIS AND DISCUSSION

4.1. Results obtained using the three event technique

All calculations discussed in the previous section were performed on the system, using both methods. The results that were obtained using the modified method and the new technique are summarized in this section.

To begin with, the data for the whole year was grouped into four seasons and the curve for Npv versus Nb was obtained for each season for the given value of L O L P of the system. These curves are shown in Fig. 9. As is seen from the graph and as would be expected, winter is the worst season of the year as far as energy generation and energy consumption are con- cerned and the design of the system will have to be based on the values obtained for winter. Any point on the curve for winter would satisfy the condition of system reliability. However the final goal is to obtain the opt imum combination

10(1

i + "6 ~ . ~ Summe

10 40 60

Iolp = .00027

r~n Ix)in consideri lg seQso i ( wlrner )

%.-__ Con rtant ~ ~

80 100 120 140 160 No. of batte~es(Nb)

Fig. 9. Npv versus N b curves for four seasons.


of N b and Npv which would result in the least possible system cost. That point is determined from the constant cost line as explained in the previous section. Accordingly the system should be provided with 58 PVpanels and 100 batteries to meet the reliability conditions throughout the year.

This design is however for the whole season which is a fairly large time period. One cannot be sure, therefore, that the design that did apply well for the season as a whole will also hold true for a smaller time span within this season.

To verify this the winter season was broken down into smaller time spans each of one month duration. The calculations were repeated for these individual months of December, January and February and also for November. The results are shown in Fig. 10. Out of the three December is the worst month for the system performance. When compared with the curve for winter as is shown in Fig. I I, it is seen that if the design were to be based on the winter season as a whole it would have failed in December. Hence the system has to be designed for the month of December. The optimum combination is found out in the same way as explained earlier. Thus the system should be provided with 68 PVpanels and 111 batteries for energy storage to meet the reliability conditions throughout the year (Fig. 12).

It might be possible that a similar discrepancy might be observed if the month of December itself were to be broken down into yet smaller time spans. However to perform those calculations would be cumbersome and as the number of data points available from the historical data will be significantly small the accuracy of the

1°° I

80 / ~ worst m x~h (Dec~ (111 .e3) z / - ~

- ~ , ~ t &--~

411

7O

Iolp 1 0.00027

/

90 110 130 150 170 NO of Batteries (Nb)

Fig. 11. Nvv v e r s u s N b curves for winter and its three months (three event approximation technique).

probabilistic approach might be lost. Also the figure shows that the months of January and February are much better than December and thus the chances of the system failing in those months are very slim. The month immedi- ately preceding December is an exceedingly good time for system operation and the system cannot fail in the month of November when it is designed for December.

4.2. Comparison of the two techniques

The comparative graphs from the two approaches for winter and December are shown in Fig. 13 and Fig. 14. It is seen in both cases that the original method gives values that are higher than the new technique. It is clear that the original method oversizes the system to some extent. The difference between the two methods may vary depending upon the distribution for D. The reasons behind this difference

100

90

8O

Z z 7o

"~ so

4O

~0

2O 5O

Iolp = O.(X ~27

Nt~Brnb~

70

~ ~ " " - ~ ...__...____

90 110 1~10 150 NO. of b41ttedu (Nb)

,oo8o /X 80 ; ~'~'1~0"0 ~ml~er)

15 ~ ~Jcember

=t e0 ..... 0 , ,~ , ,o~t~ " ~ , ; ~ *inter ( 1 0 , ~ _ _ . ~ _ . , / ~ Wit ~r

50 ~ .

40 70 90 110 130 150 170

No. of Io/'teriee (Nb)

Fig. 10. N~, v e r s u s N b curves for November through Fig. 12. N~, v e r s u s N b curves for winter and December (two February. event approximation).

H y b r i d w i n d - p h o t o v o l t a i c sys tem 333

90 Io/p = 0 00027

5O

(~rEItD 1

4O 70 90 110 130 150 170

NO. of batteries (Nb)

Fig. 13. Npv versus N b curves for win te r u s ing t w o techniques.

z

i & "6 o

100

8090 co~st constan line I ~ 1 Iolp = 0. X)027

( ,~ ,es) 70 -,~

eo c Bo (ptcO ~ " ~ _ . hree event~

5o 70 90 110 130 150 170

No. of batteries (Nb)

Fig. 14. Npv versus N b curves for D e c e m b e r us ing t w o tech- n iques .

can be explained if we take a closer look at the assumptions underlying each technique.

For simplicity the two methods are analyzed at the design point obtained from the new technique. The analysis was done both for December and for winter.

Figure 15 and Fig. 16 show the actual and assumed frequency distributions of D for winter for the original and modified methods, respectively. Here

N p v = 57.5 /~o = 4.335 kWh ao --- 11.3 kWh.

From the original method

A=12.1 kWh N b = l l 0 .

From the modified method

A = 1.5*tr o = 16.95 kWh N b --- 100.

8

6 Z

d e l t a = 12.1 kWh

2000 - Std. deviation = I 1.3 kWh

1800 - M e a n - 4 . 3 3 5 k W h DI No. o f o b s e r v a t i o n s = 2700

H 1600 p

1400

1200 - [ '- '1 Actual

l -del ta L +delta ~1 m A s s u m e d 1 0 0 0 !H! 8 0 0 - r'l q 600 -

400 -

2 0 0 - 2 ~ ~ n _

3

Multiples of standard deviation (s.d.)

Fig. 15. Ac tua l a n d a s s u m e d p r o b a b i l i t y densi t ies for win te r ( two event a p p r o x i m a t i o n ) .

¢J e. o

o

6 Z

1400 -

1200

1 0 0 0

800 -

600 -

400

200

0

delta = 1.5*s.d. = 16.95 kWh

Std. deviation = I 1.3 kWh t Mean = 4.335 k W h r " l Actual

No. of observations = 2700 [7"J] A s s u m e d

P

I _ ~lelta +delta

- 2 -1 0 1 2 3

M u l t i p l e s o f s t a n d a r d d e v i a t i o n ( s .d . )

Fig. 16. A c t u a l a n d a s s u m e d p r o b a b i l i t y densi t ies for win te r ( th ree event a p p r o x i m a t i o n ) .

Similar figures that were plotted for December are not included here.

From the original method

A = 12.518 kWh Nb = 122.

From the modified method

A = 1.5*tr o = 17.599 kWh N b = 111.

It is found that the number of batteries obtained from the modified method is about 10% less than that obtained from the original method in both cases.

The actual frequency distribution of D clearly shows a distinct pattern. Almost 99% of the total observations fall in a zone about 4.5~o wide divided equally about the mean. At and


near the design point the mean of D is usually very close to zero. Hence it can be said for all practical purposes that 99% of the observations of D lie in a zone 4.5a o wide divided equally about zero. To observe the distribution more closely, this zone of 4.5a o was further divided into three subzones each equally divided about zero. It was found that about 50% of the observations lie in the central zone which coincided with the 0.750" 0 limits about zero. About 85% of the total observations lie within 1.5ao limits about zero.

Therefore the following conclusions can be drawn about the distribution of D and the selection of A:

It is necessary and sufficient to consider all the observations that lie within 2.25tr o limits about zero.

A significant number of observations (50%) lie within 0.75ao limits about zero.

The remaining 50% lie outside the 0.75tr o limit but fall within the 2.25ao limits about zero.

It is not sufficient to approximate the distribution by only two events of probabilities p and q.

It is necessary to consider that there is a probability, t >0, that at the end of a day the storage is in the same level where it was at the beginning of the day.

Choosing a value of A=l .5ao is justified because it divides the distribution into three zones.

The central zone which is 1.5an wide is associated with probability t. All the observations to the left of this zone are associated with the probability q and those to the right with the probability p.

5. CONCLUSIONS

A new technique for the sizing of PV array and battery storage for a stand-alone hybrid wind-photovoltaic system has been developed here.

This new technique has the following important advantages over the original two event technique:

The three event approximation increases the accuracy of the system sizing without any significant increase in the effort and time consumed for computation. Three events more closely represent the actual distribution of the daily surplus energy than two events.

Being able to select the value of A based upon the reasoning put forth here enables one to obtain a better approximation of the real pro-

cess of day-to-day change in the energy storage levels. As future work the effect of considering more than three events for approximation should be investigated.

6. NOMENCLATURE

Note: The batteries used were 12V, 110 Amphr and 80% efficiency. The photovoltaic array output was 53 W each. The number of batteries and PV arrays as mentioned in the paper represent units of the above mentioned capacity.

D the daily surplus (or deficit) energy generated by the system which is also equal to the daily energy flowing into (or out of) the storage

mo the mean value of D SO the standard deviation of D L the load on the system N number of discrete levels into which the

storage capacity is assumed to be divided Nb number of batteries required Npv number of PV panels required D the size of each of the N level of the

storage C the total capacity of the storage (kWh) p probability of the storage going up one

level over the course of one day q probability of the storage going down

one level over the course of one day t probability of the storage staying in the

same level over the course of one day PE lOSS of load probability of the system i actual output current of each module

(panel) v actual output voltage of each module p output power of each module z the installed hub height of the wind

turbine zl height at which wind speed was

measured V wind speed at the installed hub height V1 measured wind speed n multiples of the standard deviations that

equal the entire span of observations

Acknowledgements--The authors would like to acknowledge the grant from the New England Power Service Company (NEPSCo) which made this research possible.


REFERENCES

Bhat U. N. (1972) Elements of Applied Stochastic Processes. Wiley, New York.

Bishop D. (1988) Wind augments solar power at Mountain- top Radio Site. Mobil Radio Technol. 6, 8-13.

Borowy B. S. and Salameh Z. M. (1994) Optimum photovoltaic array size for a hybrid wind/PV system. IEEE/PES Winter Power Meeting.

Bucciarelli L. L. Jr (1984) Estimating loss-of-power probabilities of stand alone photovoltaic solar energy systems. Solar Energy 32, 205-209.

Castle J. A., Kallis J. M., Marshall N. A. and Moite S. M. (1981) Analysis of merit of hybrid wind/photovoltaic concept for stand alone systems. Proc. 15th IEEE PV Specialists Conf., pp. 734-744.

Duffy J. A. and Beckman W. A. (1991) Solar Engineering oJ Thermal Processes, 2nd edn. Wiley, New York.

Godley W. L. and Van Leishout P. (1987) Wind power in Antarctica. Telephony, pp. 28-34.

Hart G. W. (1982) Residential photovoltaic system simula- tion: electrical aspects. 16th IEEE Photovoltaic Specialists Conf., pp. 281-288, San Diego, California.

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Payne P. E. and Sheehan J. L. (1979) Hybrid alternate energy systems. Am. Chem. Soc. Catalog, No. 8412-0513- 2/79-0779-051, pp. 21-254.

Rauschenbach H. S. (1980) Solar Cell Array Design Hand- book, pp. 55-60. Van Nostrand Reinhold, New York.

Salameh Z. M. and Safari I. (1992) Optimum windmill-site matching. IEEE Trans. Energy Convers. 7, No. 4.

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