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A Fokker-Planck analysis of photovoltaic systems
Energy. Vol. 3. pp. $1-62.
Pergamon Press 1978. Printed in Great Britain
A FOKKER-PLANCK ANALYSIS OF PHOTOVOLTAIC SYSTEMS LAWRENCEH. GOLDSTEIN Sandia Laboratories, Division 2142, Albuquerque, NM 87115,U.S.A. (Received 30 March 1977)
Abstract-The battery state-of-charge, S(r), of an arbitrary photovoltaic system is analyzed as a Markov process driven by random white Gaussian perturbations of periodic insolation and load-demand profiles. A Fokker-Planck equation for the probability density function of S(r) is derived, and S(t) minus its mean is recognized as a nonhomogeneous Wiener-Levy process. The Fokker-Planck equation is solved under conditions of no barriers, one absorbing barrier, and two absorbing barriers, and the resulting probability density functions are used to obtain bounds on the complementary cumulative distribution function for the first passage time, x(t) = P{T> t}, to the completely discharged or totally charged state. Limiting expressions for these bounds as t+O and t +m are obtained, and their asymptotic values are compared. Finally, a simple system is analyzed to provide insight into the meaning of the equations developed. 1. INTRODUCTION
With the resurgence of interest in photovoltaic systems for terrestrial applications, techniques are needed for predicting the behavior of these systems under a variety of insolation and load conditions. The most common approach taken to this problem has been simulation, using analytical models with varying degrees of sophistication for the component subsystems.‘-3 While this technique does indeed predict system performance with an accuracy dependent upon the complexity of the subsystem, insolation and load demand models used, large amounts of computer time may be required to determine the performance of a given system under a variety of operating conditions or to determine the optimal configuration for a specified set of operating conditions. For exampIe, to ascertain the minimum amount of battery storage needed to maintain the probability of complete discharge in a specified time interval less than a given e, keeping all other parameters constant, would necessitate a sequence of simulations using different capacity battery arrays under typical insolations and load demands. The overall system optimization problem requires the simultaneous optimization of all component subsystems, a procedure that generally requires much more computation than simple sequential optimization of each subsystem. An alternate approach is to analyze photovoltaic system performance statistically, given a probabilistic description of the environment within which the system operates. The general photovoltaic system has insolation as input, electrical power as output and a battery array to smooth out random fluctuations in the difference between power available from the solar cell array and power demanded by the load. In this view, intermediate power conditioning equipment merely adds inefficiency at the battery input and output terminals. The most important indicator of system performance at any given time is the battery state-of-charge, If insolation and load demand are considered to be basically periodic processes with superimposed random, white-noise, fluctuations, the battery state-of-charge S(t) will be a Markov process with periodic mean and a variance that increases with time. Thus, S(t) can be analyzed as a random walk starting from some initial S0 at time to, and limited by barriers at S = 1, corresponding to full charge, and S = 0, corresponding to complete discharge. This paper uses a statistical approach to analyze photovoltaic systems. In Section 2, the system is described, assumptions about subsystems and operating conditions are elucidated, and the differential equation for the battery state-of-charge is derived. Section 3 develops the Fokker-Planck equation for the probability density function of S(t), p(S, tlSo,to). In Section 4, the Fokker-Planck equation is solved under conditions of no barriers, one absorbing barrier and two absorbing barriers. These barriers can be placed to correspond with the battery states of full charge and complete discharge. Section 5 derives bounds on the complementary first passage time distribution function, 4(t) = P{ T > t}, where T is the first passage time to S = 0 or S = 1, and asymptotic expressions are obtained for these bounds as t +O and t +m. Finally, 51
52
L. H. GOLDSTEIN
Section 6 works through an example to illustrate the use of the equations developed in Sections 2-5. 2. SYSTEM
DESCRIPTION
The photovoltaic system under consideration is illustrated in Fig. 1, where c(r) = insolation function, suns (I sun = 100 mW/cm*); 0 5 C(t) 5 1 for all time t; A(r) = solar cell array power through the first level of power conditioning equipment, KW; S(r) = battery state of charge, 0 s S(t) 5 1; L(t) = load demand presented to the battery through the second level of power conditioning equipment, KW. Assume that the insolation profile has the form C(t) = c(t)11 -
(1)
v(t)19
where the deterministic function c(t) is periodic with period U (e.g. 24 hr), and y(t) is a random process with mean and magnitude between zero and one. The motivation behind eqn (I) is that insolation at a particular point on the earth’s surface during a specified season, assuming clear weather, will be approximately periodic (with period one day). However, atmospheric phenomena such as clouds, dust, and air pollution with decrease the amount of light reaching the solar cell array. These perturbations in insolation can be modeled as a random process multiplying c(r). The solar cell array and its associated power conditioning equipment provide an approximately linear transformation of light intensity into electrical power, viz.
A(r) = KC(r),
(2)
where K is in units of Kwsun. The load presented to the battery through the second level of power conditioning equipment is L(r) = l(r)+ A(r), (3) where I(r) is the periodic deterministic component, and A(r)is an additive random process. If L(t) is considered to be an average load profile, then L(r) will have a 24 hr period, and will exhibit peaks at breakfast and dinner time, and lulls during the late evening and early morning hours for a residential profile. Deviations from the average load demand can be represented by a zero mean additive random process which, since it can be considered as an average of a large number of independent perturbations, can be treated as white Gaussian noise.’ If B is the battery-array capacity in kilowatt-hours, M(r) is the energy stored in the battery at time 1. Assuming that the battery is 100% efficient (i.e., no energy is lost during the charge/discharge cycle), the battery energy stored at time r + At can be computed from the energy stored at time 1, the solar array power, and the load demand as follows:
BS(r +Ar) = BS(r)+[A(r)-L(r)]Ar.
(4)
Using eqn (2) and taking limits, lim
Al-4
B
W + W - SO)= KC(r) - L(r). At
(5)
Therefore, y
=i
[KC(r)- L(r)].
(6)
~~~~~~~~~~~
Fig. 1. Schematic diagram of a photovoltaic
system.
A Fokker-Planckanalysis of photovoltaicsystems 3. DERIVATION OF THE FOKKER-PLANCK
53
EQUATION
Assume that the random components of C(t) and L(t), y(t) and A(t), respectively, are independent white Gaussian noise processes with Ey(t) = m, and EA(t) = 0, where E(a) denotes expectation. Then S(t) will be a Markov process.’ Since S(t) is Markov, the ChapmanKolmogorov equation is satisfied and since the driving processes are Gaussian, the probability density function @df) of S(t), p(S, t/S,, to) satisfies the Fokker-Planck equation?
v
+$ MS)p(S,
t(So, to)1-
;$ [Az(S)p(S, tlSo. to)]= 0,
(7)
where
I
,+A
Ak(y) = lim A-O I
p(x,t+A]y,t)qdx,
(k=l,2),
(8)
and the initial condition is P(So*toI&, to) = 6(S - So)
(9)
with S(a) representing the delta function. A,(y) and Ady) can be evaluated through the use of eqns (6) and (8). Thus,
(10) From eqn (6), E[S(t + A) - S(t)] = E
jl’+‘$ [KC(T) - L(T)] dr t+A
=
II
;
[Kc(~)]1 - m,] - f(7)] d7,
(11)
where the order of integration and expectation is interchanged, and eqns (1) and (3) are used to ascertain that EC(t) = c(t)]1 - m,]
(12)
and EL(t) = f(t). By performing the integration in eqn (11) and taking limits, the following result is obtained: lim E S(t + A) - S(r) S(t) = y = $ [Kc(t)]1 - m,] - I(t)]. A I I
(13)
A,(y) = $ [Kc(t)[l - m,] - f(t)] = v(t).
(14)
A-00
Therefore,
Similarly, v(t) is seen to be the effective average supply or demand presented to the battery. Also, AZ(y)
=
lim A-4
E
.
(15)
54
L.H. GOLDSTEIN
From eqns (6) and (1l), E[S(t
+
t+A
A) - S(t)]’ = E
It I
,““+
[KC(r) - Ur)l]KC(5) - L(t)1 dt d7
t+A [zPEc(T)c(r)
+ EL(T)L(()]
-
KEC(T)L(5)
-
KEL(T)C(~)
dt dr.
(16)
The expectations in eqn (16) can be evaluated from the definitions of C(t) and L(t) in eqns (1) and (3):
my],
EL(T)ctt) = 6T)dt)[l-
(19)
ECtT)Ltt) = C(T)&%1- &I,
(20)
where %‘JT, 5) is the autocovariance function of y(t) and R**(T, 6) is the autocorrelation function of h(t). If both y(t) and A(t) are Gaussian white noise processes,
and
By performing the integration in eqn (16) and taking limits, A*(y) = 5
c*(t)& ++ NA= a*(t).
Inserting eqns (14) and (23) into eqn (7), the Fokker-Planck obtained, namely, MS, ~I&, to) at
4.
+ n(t)
@(S, tlS0, to) u*(t) aS -2
a*P(s,
(23) equation for p(S, tlS,-,,to) is
tpo,
as*
to1 = o
’
(24)
q(t) = ; ]K(l - m&(t) - WI,
(25)
CT*(t) = + [K*N,C*(t)+ NJ.
(26)
SOLUTIONOFTHEFOKKER-PLANCKEQUATION
Equation (24) can be solved using techniques described by Papoulis’ and Feller.’ The equation to be solved can be written as
ap a*(t) a*p $+llto,s-s=o’ 2
where p = p(S, tlSo, to), to and So are fixed, and t > to. The initial condition is PG
tlso,to)+W-
So)
for
t-j to;
(27)
A Fokker-Planck p(S,
tlS,,,
to)
analysis
of photovoltaic
systems
55
is the conditional density of a process S(t) satisfying the stochastic differential
equation
Wt) -dt
q(t) =
dW(t)
(28)
7
dt
where W(t) is a continuous process with independent increments such that E{dW)=O
E{(d W)? = o’(t) dt.
(29)
Using a nonlinear time transformation, T = r(f), the variance of d W can be set equal to d7, i.e. E(d W)’ = c’(t) dr = d7; eqn (30) implies
$= a2(t)
and
7=
where T(&,)= 0 is assumed. Therefore, W(T) is a Wiener-Levy process’ and the conditional pdf p( W, ~1W,,, 0) satisfies the equation
w,71wo,0) 0) _ 1 a2pt %-Gw71 wo, -2
al
aw2
.
(32)
Equation (28) can be integrated from f. to f, as follows: S(1)-So=l’n(Z)d[+ 10
W(t)-
W,.
(33)
(a) No barriers
It is assumed in this case that S(t) starts at So, and there are no absorbing or reflecting barriers. Although this solution is not physically meaningful since it allows a negative battery state-of-charge, bounds on first passage time probabilities and a relatively simple solution to eqn (24) are obtained. The solution density for eqn (32) is -(W-W&P/27 ~(WdW0,O)=~~~,~e
.
(34)
By expressing T in terms of t through eqn (31) and using eqn (33), the following result is obtained:
(35)
Thus, S(t) - ES(t) is a nonhomogeneous Wiener-Levy process. (b) One absorbing barrier Consider the process obtained by placing an absorbing barrier at w = d. Using the reflection principle, the solution to eqn (32) under this condition can be obtained as5 Pd(w 71% 0) =
&
~e-W”2’_e-‘2d-WP/2q*
(36)
L. H. GOLDSTEIN
56
The absorbing barrier at w = d translates into a time-varying absorbing barrier for S at &(f) = So+
‘+$)d[+
I IO
d.
(37)
Thus, the probability density function for S(t), given the absorbing barrier in eqn (37), is (38) where m(r) is the time varying mean of the process given by (39) 0 < Im(f)l < 1 is assumed for all t L lo, and o’(t) is the time varying variance
By adjusting d such that sup [d + m(r)] = 0,
(41)
approximations to the condition of a battery with infinite capacity which is disconnected upon complete discharge are obtained. Tighter bounds than those computed using eqn (35) are thereby achievable on the probability distribution of the first passage time to complete discharge.
(c) Two absorbing barriers
If absorbing barriers are placed on the W(T) process at w = a and w = d, the transition densities poad(W, ~10,O) must satisfy the differential eqn (32), together with the boundary conditions P&O, ~10,O)= 0 and pJd, rlO,O) = 0. The solution obtained by a method of successive approximations by repeated reflections is
P..d(w 4tO) = __&
[Z.
+ “i.
W
(e-IW+2nb-dw21_
I W+Zn(d-o)l*/Z+
e-12P+2nb-dwlY23
_ e -[2d+2n(d-o
_
j- WI’/2
1.
-w/21
3e
(42)
These absorbing barriers on W(7) translate into absorbing barriers for the S(t) process of the form S.(r) = a + m(t)
and (43)
S‘,(f) = d + m(t).
Using eqns (31) and (33), the solution to eqn (24). given the two absorbing barriers in eqn (43), is
Pad(SMO3 to)=
VUnMQ
’
ci
{e-[S-m(t)+2n(rr-d)l*~u*~~)
[S-m(?)+2n(d-o)lz/2u~~) +“go+-
_ e -(2~+2n~a-d)-S+m~1~1*/2v~f~ I
n-0
_ et2d+2n(d-~)-S+m(f))2/20~r)}
_
e-tS-mU)~1202(l)
WI
A Fokker-Planck analysis of photovoltaic systems
57
The two absorbing barriers can be adjusted to approximate a battery with initial charge So disconnected upon attaining full charge or complete discharge. These values for a and 4 are:
d
such that
sup [d + m(t)1 = 0 j d = - sup m(t),
c1
such that
inf [a + m(t)] ,
,
I
= i j
a =
1- inf m(t). I
5. BOUNDS ON THE FIRST PASSAGE TIME-DISTRIBUTION
One quantity of interest in the design of photovoltaic systems is the time required for the battery array to charge completely or discharge fully. Neither condition is desirable in a well-designed system. When the battery discharges, the load must obtain power from another source, usually at a penalty in cost. Further, a battery that is frequently fully charged suggests excess generating capacity available in the solar cell array for the specified battery storage and load demand. Since solar-array costs are relatively high, excess storage or generation capacity is not as economical as balanced arrangement of subsystems. Thus, it is desirable to maximize the time for the battery array to discharge or charge fully. The problem can be analyzed as a first passage time of the battery state of charge to S = 0 or S = 1. If T is the first passage time, then $(t) = P{T > t} is the complementary first passage time distribution. An upper bound to $(t) can be obtained by integrating the conditional density in eqn (35) from 0 to 1, viz. cL(t)5 41(t) = o’ PG +$I, to)dS. I Here 4,(t) is an upper bound because it includes all S(t) paths crossing 0 or 1 before time t and then recrossing into the interval (0,l); 41(t) which is merely the probability that an unrestricted sample function of the random process S(t) will lie between 0 and 1 at time r, is presented because of its relative simplicity and relevance to deriving other bounds,
MC) =I,’ q(zl)u(r) e
-IS-m(f)l*/ZvW
dS_
(45)
Setting y = [S- m(t)J/u(t),
where N(x) =
dcilr,I _beey2”dy
is the normal distribution function. Since N(- x) = I - N(x), (47) The periodicity of c(t) and I(t) implies the periodicity of q(t) by eqn (25). It is assumed the system is well-designed in the sense that the expected change in battery state-of-charge over one period is zero. Hence, m(r) is periodic. It is further assumed, as part of the “well-designed” criterion, that 0 < m(t) < 1 for all 1. Bounds on a’(r) can be obtained from eqn (23), i.e. 0 I C’(t) I 1 for all r implies
L.
58
H. GOLDSTEIN
Therefore, y
v/(~ - to)
5
o(t) =
,/(j-1c&f) d5> 5 ‘@;
+ N”)v’(t - to).
(49)
Since Im(t)l< 1,
An upper bound on N(x) that is a good approximation for small x is
Using eqn (SO)in eqn (47), an upper bound on 4(t) is obtained which approaches 4,(t) as t + ~0, i.e.
(51) Inserting the lower bound on o(t) from eqn (49), a final coarse upper bound on 4(t) is obtained as
(52) An asymptotic expression for N(x) is 1 x V(2r)
-**L?
l-A--e
N(x)-
x+m.
as
However, as t + to, +,(t)+ 9(t), m(t)/u(t) +Q), and [1-m(t)]/~(t)+~. eqn (47), the following result is obtained as t+to:
W--l-~
[
1 m(t)e
--mw2u*(I)
1
+
(53) Substituting eqn (53) into
e-[l-m*u)1/2u*v)
1 - m(t)
(54)
I.
A tighter upper bound on $(t), ~,+~(f), can be achieved by integrating eqn (38) from 0 to 1. In this case, all S(t) paths that touch the lower barrier are absorbed, making $(t) 5 ~j~(t)5 $i(t), since the lower absorbing barrier is below S = 0 for all time; also -tS-“E0)12/2V2(1) ds
+2(r) = I,’ d(2i)n(t)
e-[S-m(t)-2dlt/2u*(r)
_
e
dS
(59
As in eqn (46), the above expression reduces to $2(t) = “[z]
+
I$-]
- N[’- “u;- ‘“I-
N[ “‘:::, ‘“1
(56)
TO obtain the best bound on G(t), d = -sup m(t) should be chosen [see eqn (41)]. However, m(t) < 1 j d = - 1 will always provide a valid bound. Using d = - 1 and a lower bound on N(x) given by
N(&+L---
2 V(27r)
1 x3 V(2r) 6 ’
(57)
A Fokker-Planck
analysis
of photovoltaic
59
systems
a bound for G*(t) is obtained, viz. 13-m(t)]’ 4.5 ~~‘v’o. u3(t)
$*(C)( &
1
(58)
Since v(t) 2 (~NA/B)(~ - lo)“* from eqn (49). the final bound on Jlz(r) assumes the form (59) Notice that the coarse upper bound given in eqn (52) varies inversely with (t - IO)“* while the better bound presented in eqn (59) decreases as (t - 1,,)-3’2.An even tighter bound on 4(t) can be obtained from eqn (44). However, the solution is difficult to use because of convergence problems for the infinite series. 6. EXAMPLE
The following simple illustrative example is presented to provide some insight into the equations developed in Sections 2-5. Assume a deterministic insolation function, c(t), of the form indicated in Fig. 2. Time t = 0 corresponds to sunrise. The sun is up for 12 hr and then sets for 12 hr. Let the mean of the random insolation component, m,, equal l/2. Therefore, the average insolation, c(t)[l - m,], will be a sequence of pulses as in Fig. 2 with the pulse height equal to l/2. Let the load demand, l(1) be a constant 1 kW and assume that K = 4 kW/sun. From eqns (25) and (39), ?(+2c(l)-
and m(t) = So+
I]
(60)
,
~(7) dr,
(61)
where m(r) = ES(t), the time varying mean of the battery state-of-charge S(t). Equations (60) and (61) are graphed in Fig. 3(a) and (b), respectively. Bounds on So and B can be readily derived for the example from the curve in Fig. 3(b), i.e.
o
(62)
thus, O
(6%
and 12 -
r, U
I
12
24
Fig. 2. Description
36 of a deterministic
(64
1.
48
72
insolation
function.
HRS
60
L.
O-
12
24
H.
GOLLSTEIN
36
48
60
*
72
HRS
-l/B ..
MltkE S(t) ’
1 --
0
12
24
36
48
60
72
w HRS
tb) Fu. 3.
(a) q(r) vs t: (b) m(r) = ES(r) vs I.
Equation (40) indicates that the variance of qt) equals o’(t), where I
u2(t) =
I ~~(7)
dr
0
(65)
and a2(t) = + [Ngc2c2(t) + NJ.
VW
Curves for a2(t) and 02(t) are graphed in Fig. 4(a) and (b), respectively, where k, = (NJ’+ NJB2 and k2 = NAlB2. It is seen that S(t) is a Gaussian random variable with periodic mean given in Fig. 3(b) and increasing variance as illustrated in Fig. 4(b). The upper bound $,(t) on +4(t), the complementary first passage time-distribution of S(t) to 0 or 1, is graphed in Fig. 5, where k, = 11144,k2 = l/576, So = l/4, and B = 24 are chosen. For large times, (1,(t) decreases proportionally with the inverse square root of time [eqn (52)]. The tighter upper bound on $(t), Ilz(t) can be obtained for this example by setting d = - sup m(t) = - 3/4. Using eqn (56). I
= $,(C) -
(N[S’yt)] - $3’2;(y]).
As indicated in Fig. 5 and by eqn (59), $2(t) decreases more rapidly with time than 4,(t).
(67)
61
A Fokker-Planck analysis of photovoltaic systems
12
24
36
48
60
72
48
60
72
. HRS
(a)
36(kl+k+ 36kl + 24k2
24kl +-12k,
I
t
:
12
24
36
t
HRS
(b) Fig. 4. (a) a’(t) vs I; (b) u’(t) = var S(r) vs f.
-o! : : 0 12 24
!
:
:
!
36
48
60
72
o
84
:
96
:
!
’
!
-
qt,
---
G2ct,
!
*
!
!
!
:*
108 120 132 144 156 168 180 192 204
Fig. 5. Bounds on the complementary distribution function of the first passage time of S(r) to 0 or 1.
7. CONCLUSIONS
A general photovoltaic system has been analyzed from the statistical point-of-view, using the Fokker-Planck equation to derive time-varying probability density functions for the battery state-of-charge, S(t), under conditions of no barriers, one absorbing barrier, and two absorbing barriers. S(t) is treated as a random walk with barriers at S = 0 (complete discharge) and S = 1 (full charge). Bounds on the complementary distribution function of the tist passage time of
62
L. H. GOLDSlWN
S(t) to S = 0 or S = 1 are obtained, and a simple example is analyzed to provide insight into the equations developed. If accurate statistics for the insolation and load conditions for a photovoltaic system can be obtained, this approach will provide the system designer with a tool for estimating the behavior of various system configurations without extensive simulation.
REFERENCES 1. M. W. Edenburn. G. R. Case and L. H. Goldstein, Computer simulation of photovoltaic systems. IEEE 12th Phorouolfaic Specialists Conference, Baton Rouge, Nov. 1976. 2. A. Kirpich. N. F. Shepard, Jr. and S. E. Irwin. Performance and cost analysis of photovoltaic power systems for on-site residential applications. I I th Infersociety Energy Conversion Engineering Conference, State Line, Nevada, Sept. 1976. 3. L. H. Goldstein and G. R. Case. PVSS-A photovoltaic system simulation program. Solar Energy, to be published. 4. W. Feller, An Introduction lo Probability Theory and Ifs Applicofions, Vol. 1, p. 244. Wiley, New York (1%8). 5. A. Papoulis, Probability, Random Variables. & Stochastic Processes. McGraw-Hill, New York (1965). 6. A. 1. Viterbi. Principles of Coherent Communication. McGraw-Hill, New York (1966). 7. W. Feller, An Introduction fo Probability Theory and 11s Applications, Vol. II. pp. 340-345. Wiley, New York (1971).
Pergamon Press 1978. Printed in Great Britain
A FOKKER-PLANCK ANALYSIS OF PHOTOVOLTAIC SYSTEMS LAWRENCEH. GOLDSTEIN Sandia Laboratories, Division 2142, Albuquerque, NM 87115,U.S.A. (Received 30 March 1977)
Abstract-The battery state-of-charge, S(r), of an arbitrary photovoltaic system is analyzed as a Markov process driven by random white Gaussian perturbations of periodic insolation and load-demand profiles. A Fokker-Planck equation for the probability density function of S(r) is derived, and S(t) minus its mean is recognized as a nonhomogeneous Wiener-Levy process. The Fokker-Planck equation is solved under conditions of no barriers, one absorbing barrier, and two absorbing barriers, and the resulting probability density functions are used to obtain bounds on the complementary cumulative distribution function for the first passage time, x(t) = P{T> t}, to the completely discharged or totally charged state. Limiting expressions for these bounds as t+O and t +m are obtained, and their asymptotic values are compared. Finally, a simple system is analyzed to provide insight into the meaning of the equations developed. 1. INTRODUCTION
With the resurgence of interest in photovoltaic systems for terrestrial applications, techniques are needed for predicting the behavior of these systems under a variety of insolation and load conditions. The most common approach taken to this problem has been simulation, using analytical models with varying degrees of sophistication for the component subsystems.‘-3 While this technique does indeed predict system performance with an accuracy dependent upon the complexity of the subsystem, insolation and load demand models used, large amounts of computer time may be required to determine the performance of a given system under a variety of operating conditions or to determine the optimal configuration for a specified set of operating conditions. For exampIe, to ascertain the minimum amount of battery storage needed to maintain the probability of complete discharge in a specified time interval less than a given e, keeping all other parameters constant, would necessitate a sequence of simulations using different capacity battery arrays under typical insolations and load demands. The overall system optimization problem requires the simultaneous optimization of all component subsystems, a procedure that generally requires much more computation than simple sequential optimization of each subsystem. An alternate approach is to analyze photovoltaic system performance statistically, given a probabilistic description of the environment within which the system operates. The general photovoltaic system has insolation as input, electrical power as output and a battery array to smooth out random fluctuations in the difference between power available from the solar cell array and power demanded by the load. In this view, intermediate power conditioning equipment merely adds inefficiency at the battery input and output terminals. The most important indicator of system performance at any given time is the battery state-of-charge, If insolation and load demand are considered to be basically periodic processes with superimposed random, white-noise, fluctuations, the battery state-of-charge S(t) will be a Markov process with periodic mean and a variance that increases with time. Thus, S(t) can be analyzed as a random walk starting from some initial S0 at time to, and limited by barriers at S = 1, corresponding to full charge, and S = 0, corresponding to complete discharge. This paper uses a statistical approach to analyze photovoltaic systems. In Section 2, the system is described, assumptions about subsystems and operating conditions are elucidated, and the differential equation for the battery state-of-charge is derived. Section 3 develops the Fokker-Planck equation for the probability density function of S(t), p(S, tlSo,to). In Section 4, the Fokker-Planck equation is solved under conditions of no barriers, one absorbing barrier and two absorbing barriers. These barriers can be placed to correspond with the battery states of full charge and complete discharge. Section 5 derives bounds on the complementary first passage time distribution function, 4(t) = P{ T > t}, where T is the first passage time to S = 0 or S = 1, and asymptotic expressions are obtained for these bounds as t +O and t +m. Finally, 51
52
L. H. GOLDSTEIN
Section 6 works through an example to illustrate the use of the equations developed in Sections 2-5. 2. SYSTEM
DESCRIPTION
The photovoltaic system under consideration is illustrated in Fig. 1, where c(r) = insolation function, suns (I sun = 100 mW/cm*); 0 5 C(t) 5 1 for all time t; A(r) = solar cell array power through the first level of power conditioning equipment, KW; S(r) = battery state of charge, 0 s S(t) 5 1; L(t) = load demand presented to the battery through the second level of power conditioning equipment, KW. Assume that the insolation profile has the form C(t) = c(t)11 -
(1)
v(t)19
where the deterministic function c(t) is periodic with period U (e.g. 24 hr), and y(t) is a random process with mean and magnitude between zero and one. The motivation behind eqn (I) is that insolation at a particular point on the earth’s surface during a specified season, assuming clear weather, will be approximately periodic (with period one day). However, atmospheric phenomena such as clouds, dust, and air pollution with decrease the amount of light reaching the solar cell array. These perturbations in insolation can be modeled as a random process multiplying c(r). The solar cell array and its associated power conditioning equipment provide an approximately linear transformation of light intensity into electrical power, viz.
A(r) = KC(r),
(2)
where K is in units of Kwsun. The load presented to the battery through the second level of power conditioning equipment is L(r) = l(r)+ A(r), (3) where I(r) is the periodic deterministic component, and A(r)is an additive random process. If L(t) is considered to be an average load profile, then L(r) will have a 24 hr period, and will exhibit peaks at breakfast and dinner time, and lulls during the late evening and early morning hours for a residential profile. Deviations from the average load demand can be represented by a zero mean additive random process which, since it can be considered as an average of a large number of independent perturbations, can be treated as white Gaussian noise.’ If B is the battery-array capacity in kilowatt-hours, M(r) is the energy stored in the battery at time 1. Assuming that the battery is 100% efficient (i.e., no energy is lost during the charge/discharge cycle), the battery energy stored at time r + At can be computed from the energy stored at time 1, the solar array power, and the load demand as follows:
BS(r +Ar) = BS(r)+[A(r)-L(r)]Ar.
(4)
Using eqn (2) and taking limits, lim
Al-4
B
W + W - SO)= KC(r) - L(r). At
(5)
Therefore, y
=i
[KC(r)- L(r)].
(6)
~~~~~~~~~~~
Fig. 1. Schematic diagram of a photovoltaic
system.
A Fokker-Planckanalysis of photovoltaicsystems 3. DERIVATION OF THE FOKKER-PLANCK
53
EQUATION
Assume that the random components of C(t) and L(t), y(t) and A(t), respectively, are independent white Gaussian noise processes with Ey(t) = m, and EA(t) = 0, where E(a) denotes expectation. Then S(t) will be a Markov process.’ Since S(t) is Markov, the ChapmanKolmogorov equation is satisfied and since the driving processes are Gaussian, the probability density function @df) of S(t), p(S, t/S,, to) satisfies the Fokker-Planck equation?
v
+$ MS)p(S,
t(So, to)1-
;$ [Az(S)p(S, tlSo. to)]= 0,
(7)
where
I
,+A
Ak(y) = lim A-O I
p(x,t+A]y,t)qdx,
(k=l,2),
(8)
and the initial condition is P(So*toI&, to) = 6(S - So)
(9)
with S(a) representing the delta function. A,(y) and Ady) can be evaluated through the use of eqns (6) and (8). Thus,
(10) From eqn (6), E[S(t + A) - S(t)] = E
jl’+‘$ [KC(T) - L(T)] dr t+A
=
II
;
[Kc(~)]1 - m,] - f(7)] d7,
(11)
where the order of integration and expectation is interchanged, and eqns (1) and (3) are used to ascertain that EC(t) = c(t)]1 - m,]
(12)
and EL(t) = f(t). By performing the integration in eqn (11) and taking limits, the following result is obtained: lim E S(t + A) - S(r) S(t) = y = $ [Kc(t)]1 - m,] - I(t)]. A I I
(13)
A,(y) = $ [Kc(t)[l - m,] - f(t)] = v(t).
(14)
A-00
Therefore,
Similarly, v(t) is seen to be the effective average supply or demand presented to the battery. Also, AZ(y)
=
lim A-4
E
.
(15)
54
L.H. GOLDSTEIN
From eqns (6) and (1l), E[S(t
+
t+A
A) - S(t)]’ = E
It I
,““+
[KC(r) - Ur)l]KC(5) - L(t)1 dt d7
t+A [zPEc(T)c(r)
+ EL(T)L(()]
-
KEC(T)L(5)
-
KEL(T)C(~)
dt dr.
(16)
The expectations in eqn (16) can be evaluated from the definitions of C(t) and L(t) in eqns (1) and (3):
my],
EL(T)ctt) = 6T)dt)[l-
(19)
ECtT)Ltt) = C(T)&%1- &I,
(20)
where %‘JT, 5) is the autocovariance function of y(t) and R**(T, 6) is the autocorrelation function of h(t). If both y(t) and A(t) are Gaussian white noise processes,
and
By performing the integration in eqn (16) and taking limits, A*(y) = 5
c*(t)& ++ NA= a*(t).
Inserting eqns (14) and (23) into eqn (7), the Fokker-Planck obtained, namely, MS, ~I&, to) at
4.
+ n(t)
@(S, tlS0, to) u*(t) aS -2
a*P(s,
(23) equation for p(S, tlS,-,,to) is
tpo,
as*
to1 = o
’
(24)
q(t) = ; ]K(l - m&(t) - WI,
(25)
CT*(t) = + [K*N,C*(t)+ NJ.
(26)
SOLUTIONOFTHEFOKKER-PLANCKEQUATION
Equation (24) can be solved using techniques described by Papoulis’ and Feller.’ The equation to be solved can be written as
ap a*(t) a*p $+llto,s-s=o’ 2
where p = p(S, tlSo, to), to and So are fixed, and t > to. The initial condition is PG
tlso,to)+W-
So)
for
t-j to;
(27)
A Fokker-Planck p(S,
tlS,,,
to)
analysis
of photovoltaic
systems
55
is the conditional density of a process S(t) satisfying the stochastic differential
equation
Wt) -dt
q(t) =
dW(t)
(28)
7
dt
where W(t) is a continuous process with independent increments such that E{dW)=O
E{(d W)? = o’(t) dt.
(29)
Using a nonlinear time transformation, T = r(f), the variance of d W can be set equal to d7, i.e. E(d W)’ = c’(t) dr = d7; eqn (30) implies
$= a2(t)
and
7=
where T(&,)= 0 is assumed. Therefore, W(T) is a Wiener-Levy process’ and the conditional pdf p( W, ~1W,,, 0) satisfies the equation
w,71wo,0) 0) _ 1 a2pt %-Gw71 wo, -2
al
aw2
.
(32)
Equation (28) can be integrated from f. to f, as follows: S(1)-So=l’n(Z)d[+ 10
W(t)-
W,.
(33)
(a) No barriers
It is assumed in this case that S(t) starts at So, and there are no absorbing or reflecting barriers. Although this solution is not physically meaningful since it allows a negative battery state-of-charge, bounds on first passage time probabilities and a relatively simple solution to eqn (24) are obtained. The solution density for eqn (32) is -(W-W&P/27 ~(WdW0,O)=~~~,~e
.
(34)
By expressing T in terms of t through eqn (31) and using eqn (33), the following result is obtained:
(35)
Thus, S(t) - ES(t) is a nonhomogeneous Wiener-Levy process. (b) One absorbing barrier Consider the process obtained by placing an absorbing barrier at w = d. Using the reflection principle, the solution to eqn (32) under this condition can be obtained as5 Pd(w 71% 0) =
&
~e-W”2’_e-‘2d-WP/2q*
(36)
L. H. GOLDSTEIN
56
The absorbing barrier at w = d translates into a time-varying absorbing barrier for S at &(f) = So+
‘+$)d[+
I IO
d.
(37)
Thus, the probability density function for S(t), given the absorbing barrier in eqn (37), is (38) where m(r) is the time varying mean of the process given by (39) 0 < Im(f)l < 1 is assumed for all t L lo, and o’(t) is the time varying variance
By adjusting d such that sup [d + m(r)] = 0,
(41)
approximations to the condition of a battery with infinite capacity which is disconnected upon complete discharge are obtained. Tighter bounds than those computed using eqn (35) are thereby achievable on the probability distribution of the first passage time to complete discharge.
(c) Two absorbing barriers
If absorbing barriers are placed on the W(T) process at w = a and w = d, the transition densities poad(W, ~10,O) must satisfy the differential eqn (32), together with the boundary conditions P&O, ~10,O)= 0 and pJd, rlO,O) = 0. The solution obtained by a method of successive approximations by repeated reflections is
P..d(w 4tO) = __&
[Z.
+ “i.
W
(e-IW+2nb-dw21_
I W+Zn(d-o)l*/Z+
e-12P+2nb-dwlY23
_ e -[2d+2n(d-o
_
j- WI’/2
1.
-w/21
3e
(42)
These absorbing barriers on W(7) translate into absorbing barriers for the S(t) process of the form S.(r) = a + m(t)
and (43)
S‘,(f) = d + m(t).
Using eqns (31) and (33), the solution to eqn (24). given the two absorbing barriers in eqn (43), is
Pad(SMO3 to)=
VUnMQ
’
ci
{e-[S-m(t)+2n(rr-d)l*~u*~~)
[S-m(?)+2n(d-o)lz/2u~~) +“go+-
_ e -(2~+2n~a-d)-S+m~1~1*/2v~f~ I
n-0
_ et2d+2n(d-~)-S+m(f))2/20~r)}
_
e-tS-mU)~1202(l)
WI
A Fokker-Planck analysis of photovoltaic systems
57
The two absorbing barriers can be adjusted to approximate a battery with initial charge So disconnected upon attaining full charge or complete discharge. These values for a and 4 are:
d
such that
sup [d + m(t)1 = 0 j d = - sup m(t),
c1
such that
inf [a + m(t)] ,
,
I
= i j
a =
1- inf m(t). I
5. BOUNDS ON THE FIRST PASSAGE TIME-DISTRIBUTION
One quantity of interest in the design of photovoltaic systems is the time required for the battery array to charge completely or discharge fully. Neither condition is desirable in a well-designed system. When the battery discharges, the load must obtain power from another source, usually at a penalty in cost. Further, a battery that is frequently fully charged suggests excess generating capacity available in the solar cell array for the specified battery storage and load demand. Since solar-array costs are relatively high, excess storage or generation capacity is not as economical as balanced arrangement of subsystems. Thus, it is desirable to maximize the time for the battery array to discharge or charge fully. The problem can be analyzed as a first passage time of the battery state of charge to S = 0 or S = 1. If T is the first passage time, then $(t) = P{T > t} is the complementary first passage time distribution. An upper bound to $(t) can be obtained by integrating the conditional density in eqn (35) from 0 to 1, viz. cL(t)5 41(t) = o’ PG +$I, to)dS. I Here 4,(t) is an upper bound because it includes all S(t) paths crossing 0 or 1 before time t and then recrossing into the interval (0,l); 41(t) which is merely the probability that an unrestricted sample function of the random process S(t) will lie between 0 and 1 at time r, is presented because of its relative simplicity and relevance to deriving other bounds,
MC) =I,’ q(zl)u(r) e
-IS-m(f)l*/ZvW
dS_
(45)
Setting y = [S- m(t)J/u(t),
where N(x) =
dcilr,I _beey2”dy
is the normal distribution function. Since N(- x) = I - N(x), (47) The periodicity of c(t) and I(t) implies the periodicity of q(t) by eqn (25). It is assumed the system is well-designed in the sense that the expected change in battery state-of-charge over one period is zero. Hence, m(r) is periodic. It is further assumed, as part of the “well-designed” criterion, that 0 < m(t) < 1 for all 1. Bounds on a’(r) can be obtained from eqn (23), i.e. 0 I C’(t) I 1 for all r implies
L.
58
H. GOLDSTEIN
Therefore, y
v/(~ - to)
5
o(t) =
,/(j-1c&f) d5> 5 ‘@;
+ N”)v’(t - to).
(49)
Since Im(t)l< 1,
An upper bound on N(x) that is a good approximation for small x is
Using eqn (SO)in eqn (47), an upper bound on 4(t) is obtained which approaches 4,(t) as t + ~0, i.e.
(51) Inserting the lower bound on o(t) from eqn (49), a final coarse upper bound on 4(t) is obtained as
(52) An asymptotic expression for N(x) is 1 x V(2r)
-**L?
l-A--e
N(x)-
x+m.
as
However, as t + to, +,(t)+ 9(t), m(t)/u(t) +Q), and [1-m(t)]/~(t)+~. eqn (47), the following result is obtained as t+to:
W--l-~
[
1 m(t)e
--mw2u*(I)
1
+
(53) Substituting eqn (53) into
e-[l-m*u)1/2u*v)
1 - m(t)
(54)
I.
A tighter upper bound on $(t), ~,+~(f), can be achieved by integrating eqn (38) from 0 to 1. In this case, all S(t) paths that touch the lower barrier are absorbed, making $(t) 5 ~j~(t)5 $i(t), since the lower absorbing barrier is below S = 0 for all time; also -tS-“E0)12/2V2(1) ds
+2(r) = I,’ d(2i)n(t)
e-[S-m(t)-2dlt/2u*(r)
_
e
dS
(59
As in eqn (46), the above expression reduces to $2(t) = “[z]
+
I$-]
- N[’- “u;- ‘“I-
N[ “‘:::, ‘“1
(56)
TO obtain the best bound on G(t), d = -sup m(t) should be chosen [see eqn (41)]. However, m(t) < 1 j d = - 1 will always provide a valid bound. Using d = - 1 and a lower bound on N(x) given by
N(&+L---
2 V(27r)
1 x3 V(2r) 6 ’
(57)
A Fokker-Planck
analysis
of photovoltaic
59
systems
a bound for G*(t) is obtained, viz. 13-m(t)]’ 4.5 ~~‘v’o. u3(t)
$*(C)( &
1
(58)
Since v(t) 2 (~NA/B)(~ - lo)“* from eqn (49). the final bound on Jlz(r) assumes the form (59) Notice that the coarse upper bound given in eqn (52) varies inversely with (t - IO)“* while the better bound presented in eqn (59) decreases as (t - 1,,)-3’2.An even tighter bound on 4(t) can be obtained from eqn (44). However, the solution is difficult to use because of convergence problems for the infinite series. 6. EXAMPLE
The following simple illustrative example is presented to provide some insight into the equations developed in Sections 2-5. Assume a deterministic insolation function, c(t), of the form indicated in Fig. 2. Time t = 0 corresponds to sunrise. The sun is up for 12 hr and then sets for 12 hr. Let the mean of the random insolation component, m,, equal l/2. Therefore, the average insolation, c(t)[l - m,], will be a sequence of pulses as in Fig. 2 with the pulse height equal to l/2. Let the load demand, l(1) be a constant 1 kW and assume that K = 4 kW/sun. From eqns (25) and (39), ?(+2c(l)-
and m(t) = So+
I]
(60)
,
~(7) dr,
(61)
where m(r) = ES(t), the time varying mean of the battery state-of-charge S(t). Equations (60) and (61) are graphed in Fig. 3(a) and (b), respectively. Bounds on So and B can be readily derived for the example from the curve in Fig. 3(b), i.e.
o
(62)
thus, O
(6%
and 12 -
r, U
I
12
24
Fig. 2. Description
36 of a deterministic
(64
1.
48
72
insolation
function.
HRS
60
L.
O-
12
24
H.
GOLLSTEIN
36
48
60
*
72
HRS
-l/B ..
MltkE S(t) ’
1 --
0
12
24
36
48
60
72
w HRS
tb) Fu. 3.
(a) q(r) vs t: (b) m(r) = ES(r) vs I.
Equation (40) indicates that the variance of qt) equals o’(t), where I
u2(t) =
I ~~(7)
dr
0
(65)
and a2(t) = + [Ngc2c2(t) + NJ.
VW
Curves for a2(t) and 02(t) are graphed in Fig. 4(a) and (b), respectively, where k, = (NJ’+ NJB2 and k2 = NAlB2. It is seen that S(t) is a Gaussian random variable with periodic mean given in Fig. 3(b) and increasing variance as illustrated in Fig. 4(b). The upper bound $,(t) on +4(t), the complementary first passage time-distribution of S(t) to 0 or 1, is graphed in Fig. 5, where k, = 11144,k2 = l/576, So = l/4, and B = 24 are chosen. For large times, (1,(t) decreases proportionally with the inverse square root of time [eqn (52)]. The tighter upper bound on $(t), Ilz(t) can be obtained for this example by setting d = - sup m(t) = - 3/4. Using eqn (56). I
= $,(C) -
(N[S’yt)] - $3’2;(y]).
As indicated in Fig. 5 and by eqn (59), $2(t) decreases more rapidly with time than 4,(t).
(67)
61
A Fokker-Planck analysis of photovoltaic systems
12
24
36
48
60
72
48
60
72
. HRS
(a)
36(kl+k+ 36kl + 24k2
24kl +-12k,
I
t
:
12
24
36
t
HRS
(b) Fig. 4. (a) a’(t) vs I; (b) u’(t) = var S(r) vs f.
-o! : : 0 12 24
!
:
:
!
36
48
60
72
o
84
:
96
:
!
’
!
-
qt,
---
G2ct,
!
*
!
!
!
:*
108 120 132 144 156 168 180 192 204
Fig. 5. Bounds on the complementary distribution function of the first passage time of S(r) to 0 or 1.
7. CONCLUSIONS
A general photovoltaic system has been analyzed from the statistical point-of-view, using the Fokker-Planck equation to derive time-varying probability density functions for the battery state-of-charge, S(t), under conditions of no barriers, one absorbing barrier, and two absorbing barriers. S(t) is treated as a random walk with barriers at S = 0 (complete discharge) and S = 1 (full charge). Bounds on the complementary distribution function of the tist passage time of
62
L. H. GOLDSlWN
S(t) to S = 0 or S = 1 are obtained, and a simple example is analyzed to provide insight into the equations developed. If accurate statistics for the insolation and load conditions for a photovoltaic system can be obtained, this approach will provide the system designer with a tool for estimating the behavior of various system configurations without extensive simulation.
REFERENCES 1. M. W. Edenburn. G. R. Case and L. H. Goldstein, Computer simulation of photovoltaic systems. IEEE 12th Phorouolfaic Specialists Conference, Baton Rouge, Nov. 1976. 2. A. Kirpich. N. F. Shepard, Jr. and S. E. Irwin. Performance and cost analysis of photovoltaic power systems for on-site residential applications. I I th Infersociety Energy Conversion Engineering Conference, State Line, Nevada, Sept. 1976. 3. L. H. Goldstein and G. R. Case. PVSS-A photovoltaic system simulation program. Solar Energy, to be published. 4. W. Feller, An Introduction lo Probability Theory and Ifs Applicofions, Vol. 1, p. 244. Wiley, New York (1%8). 5. A. Papoulis, Probability, Random Variables. & Stochastic Processes. McGraw-Hill, New York (1965). 6. A. 1. Viterbi. Principles of Coherent Communication. McGraw-Hill, New York (1966). 7. W. Feller, An Introduction fo Probability Theory and 11s Applications, Vol. II. pp. 340-345. Wiley, New York (1971).