An optimum load management strategy for stand-alone photovoltaic power systems

Solar Energy Vol. 46, No. 2, pp. 121-128, 1991 Prinled in the U.S.A.

0038-092X/91 $3.00 + .00 Copyright © 1991 Pergamon Press plc

AN OPTIMUM LOAD MANAGEMENT STRATEGY FOR STAND-ALONE PHOTOVOLTAIC POWER SYSTEMS P. P. GROUMPOS and G. PAPEGEORGIOU Energy Research Center, Department of Electrical Engineering, Cleveland, OH 44115, U.S.A. Abstract--The subject of load management for stand-alone photovoltaic (SAPV) power systems is addressed. The objective is to minimize the total life-cycle cost of the system while, at the same time, the battery, is protected and the load priorities are observed. The first step in this approach involves a general load classification. The idea is to manipulate the controllable loads in order to reduce battery size. For this reason, optimum curves are obtained for the controllable loads. Then an optimum load management strategy is mathematically formulated. Finally, a tracking algorithm has been devised in order to implement the optimum load management scheme. The previously described method yields cost optimum SAPV systems. An illustrative example using data similar to the first village PV power system of Schuchuli, Arizona shows the practical application of the proposed optimum load management strategy.

1. I N T R O D U C T I O N

2. LOAD CLASSIFICATION

Photovoltaic ( P V ) power generation has been shown to be a technical reality. Indeed, the technical feasibility of PV power generation has been a demonstrated fact for many years from the early space applications to today's many useful terrestrial applications [ 1]. Today, there is a lot of hope and some indications point to the fact that PV power generation has a great potential for providing a substantial portion of most countries' anticipated future energy needs with the most promise existing in the countries with suitable environmental conditions. In particular, today it is estimated that more than two million villages around the world need electrification. Photovoltaics could be one way of meeting this worldwide need. A major obstacle to the widespread acceptance and use of PV power systems has been the high cost associated with them. System cost reductions must be obtained through the use of more efficient and less costly system components and from advanced control strategies, some yet to be developed. Advanced control strategies should be seriously considered for the following reasons. First, from a theoretical point of view they can advance the stateof-the-art of photovoltaics technology. Second, they can reduce the total life-cycle cost, so photovoltaics can be come more economical and competitive with other energy alternatives. Third, they can easily be used in a SAPV which has a microprocessor, improving its overall performance, without any major additional cost other than the cost of the software program. In this paper we limit ourselves only to the theoretical benefits, hoping that prompt, future studies address the other benefits. In this paper, the subject of load management for stand-alone photovoltaic (SAPV) power systems is addressed. The objective is to minimize the total lifecycle cost of the system while, at the same time, the battery is protected and the load priorities are observed.

In order to size a SAPV power system, it is important to carefully estimate the energy needs for the intended application. We must determine how much energy is needed by the total SAPV system. Due to the nature of SAPV systems, a careful load classification is required. This is dictated since the sunlight availability is limited in a particular place and because the battery that is used as a back-up system must be protected so a number of loads will not be deprived of power. Today the classification of loads for SAPV applications has not been standardized. A load classification has been attempted in [2] and [3], and is provided here. Three major categories are proposed: operational classification; system classification; and priority classification.

2.1 Operational classification It seems that for various advantages and disadvantages of the DC and AC power, loads of both types need to be considered.

DC loads: 1. Resistive elements drawing constant power for a given operating voltage. 2. Devices or motors which depend upon the mechanical torque requirements of the driven loads to determine voltage and current inputs. AC loads: the ones widely used today; include clocks, fluorescent lights, microwave ovens, radios, toasters, televisions, telephones and other c o m m o n appliances. 2.2 System classification In order to develop sophisticated control strategies based on modern system theories, the following three classes need to be defined: Uncontrollable loads: those which require power at random times. These loads are either constant or demand instant power. 121

122

P. P. GROUMPOSand G. PAPEGEORGIOU

Controllable loads: those with which a concept of state can be associated and which can provide a way to develop state space models and subsequently to which modern space techniques can be applied. Loads that can also be directly controlled can be considered as controllable loads and they are often referred to as deferrable loads, for example, washing and sewing machines. Semicontrollable loads: those which are intermediate energy "storage loads" [ 5 ]. 2.3 Priority classification For an application where multiiale loads are to be used, the case in a village electrification, some kind of priority for the loads is needed and must be specified from the beginning of the study. It seems that the following four-class priority classification would cover most practical applications of today's village needs. UseJul or convenient loads: routine loads. Their nonoperational status due to power loss would cause only small inconvenience. Power should be supplied by the PV array only. Essential loads; which are needed for the essentials of everyday life or for the operation of a needed system. Power can be supplied by the PV array a n d / o r the energy storage system, as a last resort. Critical loads; which are needed for an overall subsystem's or component's performance and "continuous" power is required. However, the load can tolerate power loss for a finite period of time and power loss for this time period would only lead to a less efficient system. Power now must be supplied by the PV array a n d / o r the energy storage system. Emergency loads'; which require uninterrupted power and are absolutely essential to life support and the safety of the environment. These loads cannot tolerate temporary power loss for any period of time. Their power loss would produce catastrophic effects to life a n d / o r any industrial process. Power must be supplied by the PV and any possible energy storage source or any other back-up emergency power system which might be available to the overall power system.

3. CLASSICAL AND OPTIMAL LOAD MANAGEMENT STRATEGIES

Today, load management has become a buzzword, encompassing so many things and techniques that it is hard to define and use the term concisely. However, in our study here, we will concentrate on the load management concept that deals directly with customer controllable loads. For our discussion, load management is the strategy that involves the concept of controlling customer controllable loads to favorably modify system load curves in correspondence with economically available generation [ 5,6 ]. In other words, load management can be thought of as the deliberate reshaping of the customer's load profile in order to improve the efficiency of the power system.

The different forms of load management addressed by utilities commonly include: (i) direct load management where the deferrable loads are controlled automatically; and (ii) indirect load management where the control of the end use loads is left to the consumer. In SAPV systems, the load management problem is more complex than that of the utilities. A utility can control its system by bringing on line or turning off generating capacity as needed to match the load demand. An SAPV system, on the other hand, is desired to operate at full capacity irrespective of the load. This requires batteries to uncouple the source pattern from the load pattern and/or modification of the load pattern to better use the available power[7-9]. A utility seeks to level the load to reduce the peak demand while a SAPV system should peak the load to match the available energy. In both cases, control over even small portions of the load can have significant effect on total system performance [ 10 ]. The above observations indicate the need to address the problem with a totally different perspective than that of the utilities.

Methods.lor SAPV systems in use Load management in SAPV systems has two objectives: (i) protection of batteries from excessive discharge, and (ii) maintenance of operation of more critical loads at the expense of less critical loads. The latter objective involves the setting of load priorities based on some load priority classification scheme. These goals have been sought in the design of the first village SAPV power system in the world located in the remote Indian village of Schuchuli, Arizona [ 79]. Four load priorities (L~, Lz, L3 and L ) and a maximum depth of discharge (DOD) were set. The last priority loads, washing and sewing machines, are disconnected at DOD = 50%; the lights at 60%; the water pump at 70%; and the refrigerators at 80%. As the batteries are recharged, loads are sequentially reconnected. The depth of discharge is sensed by a parallel arrangement of four pilot cells[7]. It is obvious that priority load operation in the allowable range of battery DOD is accomplished by the previous load management scheme. A flow diagram of this load management control algorithm is shown in Fig. 1. The same strategy was employed in the design and operation of the SAPV system in the village of Hammam Biadha Sud in Tunisia[11]. The village load was divided into the following three sectors: (i) domestic: 22 houses; (ii) commercial: 2 stores, 1 mill, 1 hairdresser; and (iii) public: clinic, mosque, cultural center, primary school, and weaving school. The public sector was given the highest priority and the residential sector, the lowest. At 60% DOD, the residential sector load is shed; at 70%, the commercial sector load is shed; and at 80%, all loads are shut down. This load management strategy is similar to that of Schuchuli. The only difference is that the priorities now involve whole sectors of loads instead of individual loads. The battery is again protected by setting a maximum DOD of 80% at which all loads are shed.

Stand-alone photovoltaic power systems

•

123

N

t

Enable washing & sewing machines

Disable washing & sewing machines

SOC battery state of charge

I I lights isable

I Enable lights

]

j water Disable pump

Enable water pump

J ....

N

Disable efrigerators

Enable refrigerators ] _I

Fig. 1. The load management strategy for the Schuchuli Village PV Power System.

New approach One more general objective can be added to the two that have been already considered in the design and operation of village SAPV power systems[ 7,8,11 ]. That objective is the minimization of the total lifecycle cost of the system, through load management techniques, based only on system load classification, an approach that has not been previously explored. Here we will consider only the two main classes: (i) controllable and (ii) uncontrollable loads. The first are deferrable and have flexible hours of operation. Some of them even involve energy storage. In this division, there are loads such as water pumps, washing machines, water heaters, space heaters, etc. The second category includes loads that are either constant (or can be assumed constant) throughout the day (refrigerators) or demand instant energy (lights). To minimize the total life cycle cost of the system, the optimal sizing results of ref. 13 are used. There it is shown that the total life cycle cost depends on the array size and the required nominal battery capacity Q (kwh). Q is actually determined from

Q = QI + Q2

(1)

where Q. is the long-term capacity and Q2 is the shortterm capacity. Furthermore, Q, = (CF)(C)(DL)

(2)

Q2 = (CF)(NSR)(DL).

(3)

and

where CF is a factor which depends on the allowable depth-of-discharge (DOD)max and the battery etficiency, C is the storage requirement in days of load, DL in kwh/day is the estimated daily demand for the month and no-sun-ratio (NSR) is the ratio of the night load to the total daily load [ 13 ]. The idea is to manipulate the controllable loads curve in order to save battery size by reducing the NSR. This load curve modification should also be aimed at saving battery cycles, so that battery life is extended, thus reducing battery

P. P. GROUMPOSand G. PAPEGEORGIOU

124

replacements, and further decreasing the total life-cycle cost[13]. These objectives are attained by minimizing the integral of absolute differences between the controllable loads curve x ( t ) ( k W ) and the available power for the controllable loads curve w(t) ( k W ) over the 24-hour period. This is accomplished by minimizing the following objective function: in this way and using the optimal sizing method ofref. 13 the total life-cycle cost is further reduced using the proposed load management strategy. 24

min J =

J~0

[x(t) - w(t)] 2 dt

(4)

The necessary conditions for the optimal solution are:

Oga

~x =°

or

2[x(t)-w(t)]

Og~ Opl 0p2

0

-

or

0

Oeo d (Ogo]

Oz ~\Oz]=O

+p,(t)+p2(t)=O

(11)

x ( t ) - 72(t) = 0

(12)

x(t) - J(t) = 0

(13)

or

or p 2 ( t ) = 0 or

with the constraints

(b)

0g~

(5)

(a) x ( t ) >_ 0 x ( / ) d t = DL -

yo

c ( t ) d t = CL

(6)

0v

(7)

is the available power for the controllable load curve (kW) where S ( t ) = available array output curve (kW). The constraint eqn (5) is obvious since the load cannot assume negative values. The isoperimetric constraint eqn (6) expresses the balance of loads during the 24-hour period. The problem can be described as an optimization problem with an inequality constraint. There are two methods to solve such problems: the slack-variable method and the penalty-function method [12 ], which are briefly presented here as they apply to our SAPV optimization problem. Slack-variable method. In applying this method, we introduce a new variable:

(8)

~,2(t) = x ( t ) .

(15)

p l ( t ) + C2 2

(16)

Finally, the set of eqns ( 1 3 ) - ( 1 6 ) along with the boundary conditions gives a solution to the problem. Penalty-function method. This method is based on the premise that if the performance measure is severely penalized when the inequality constraint is not satisfied, but the performance measure is not penalized when the same constraint is satisfied, then the optimal solution will be forced to satisfy the constraint. I f a functionf~ can be found with the property that: Ji(t) = O

for

x ( t ) >- O

f~(t)>>0

for

x(t)
then this function can be added to the integrand of eqn (4) to form the new augmented functional J~. 24

J, =

~0

{ [x(t) - w(t)] 2 + p(t)[x(t)-

(9)

2 p , ( t ) 7 = 0.

x ( t ) = w(t)

The isoperimetric constraint can be written: x ( t ) = .~(t).

or

~(t)] + f ~ ( t ) } d t .

(17)

Let [2x(t)-

Thus, an augmented functional Ja is formed:

(14)

Equation ( 11 ) becomes:

where x ( t ) = controllable load curve ( k W ) ; c(t) = uncontrollable load curve (estimated or known (kW); DL = daily demand ( k W h / d a y ) , and w(t) = S ( t ) - c(t)

- 0

p2(t) = C2

a] 2

f(t)

(18)

24

J, =

f0

{ [x(t) - w(t)] 2 + p , ( t ) [ x ( t )

where a is a large positive constant. Then

- 72(0]

+ p 2 ( t ) [ x ( t ) - ~(t)] }dt

ga(t) = I x ( t ) -- w(t)] 2 +

[ 2 x ( t ) - a] 2 OL

=

g.(t)dt.

(10)

+ p ( t ) * [ x ( t ) - z((t)].

(19)

Stand-alone photovoltaic power systems The necessary conditions for optimal solution are

Oga = 0 3X

or

2 [ x ( t ) - - w(t)]

+

ogo Oz

412x(t) - a

a2

+ p(t) = 0

(20)

/~(t)=0

(21)

d (Ogo]

dt\dz]=O

or

~o24 x(t)dt = CL.

(22)

termination is not uniform between systems and depends on the primary needs of the people for w h o m the system is intended. Priority setting is helpful in cases where the system cannot generate enough power to satisfy all the demand. Such a case will exist in the future U.S. space station power system which will be powered using solar energy. Then the least priority loads are shed so that the most important loads can still be operational [ 3,7-9,11 ].

4.1 Daily demand determination The daily demand (DL) of the SAPV system is now comprised of the controllable (CL) and uncontrollable ( U L ) daily demands.

Equation (20) becomes

2a2w(t) + 4a x(t) =

125

-

a2p

2a 2 + 8

(23)

DL = CL + UL.

(29)

The controllable daily demand CL is equal to

Substituting eqn (23) into eqn (22), we obtain

r

2a 21+ 8

foo2, [2a2w(t)

+ 4a - a2p]dt = CL.

CL = ~ W ( i ) T ( i ) N ( i )

(24)

(30)

i=1

But w(t) = s(t) - c(t) is known since the sun curve s(t) can be predicted and is provided below and the uncontrollable load curve c(t) can be estimated. Thus, we can solve eqn (24) with respect to p. Finally, we substitute this p into eqn (23) to obtain x ( t ) . Assume

c(t)=const

for

t~
(25)

where ts = sunrise time. Also, c(t) can assume any value for t < ts, t > 24 - ts. The array output curve can be approximated by D L {e_U_l~)2/40 s(t) = 7 -

_

e_~ts_12)2/40}

(26)

where W(i) T(i) N(i) r

= = = =

unit power rating of priority i ( W ) unit usage ( h o u r s / d a y ) number of units in priority i number of priorities with i = 1 assuming the highest one.

It is assumed in this study that in a given priority i, all controllable loads have the same unit usage time T and the same unit rated power W. The uncontrollable daily demand consists of the constant daily demand PL and the instant daily demand IL. U L = PL + IL

for ts < t < 24 - ts, and s(t) = 0, for t < ts, t > 24 - ts where

I

=

f24-ts {e - u 127/,0 - - e-(,:-12)214O}dt. (27) ols

Approximation in eqn (26) is verified through available typical insolation data. For a boundary condition x( ts) = O, x( t) becomes

x(t) = ~

{e -u-12)2/40 -- e -u'-12)2/4°}

(28)

forts 24 - t~. This is the optimal curve which the controllable loads should follow.

4. A N O P T I M A L

LOAD

MANAGEMENT

ALGORITHM

After the load system classification has been performed and the controllable loads have been defined, the next step is to determine their priorities. This de-

(31)

The instant daily demand IL (e.g., lights, etc.) can be estimated from historical data of similar applications. Finally, the daily demand DL computed from eqns (29), (30), and (31) is used to determine the optimal sizes of array and battery for a m i n i m u m life cycle cost, through either today's standard sizing methods [4] or using a new cost optimization method has been developed in [6] and [13]. 4.2 Algorithm formulation The insolation required to exactly meet the demand is DRI -

DER

nA

( k W h / m 2)

(32)

where A = actual array size (m2), r/= system efficiency, D E R = daily energy requirement ( k W h ) . This DRI should be compared with the Actual Estimated Insolation (AEI) ( k W h / m 2) to determine the actions to be taken by the load management scheme.

P. P. GROUMPOSand G. PAPEGEORGIOU

126

If their difference, DRI - AEI, is positive, then the insolation will not be sufficient to cover the demand. The energy deficiency, ED, is equal to ED = rt[(DRl) - ( A E I ) ] A .

(33)

k

CL = W(k)T(k)N(k)+ ~ W(i)T(i)N(i).

Actually, each priority is forced to follow the curve eqn (28) where

We then have to determine the amount of energy that the battery can supply, EFB = Q [ ( S O C ) - SOC min, allowable]

(34)

where Q is the total nominal battery size. I f E D is less than EFB, the available energy, AE, is equal to the demand AE = ~(DRI)A.

(35)

In case ED is greater than EFB, the battery cannot supply the energy needed. Then, the available energy is equal to AE = r/(AEI)A + EFB.

(36)

The available energy for the controllable loads, AECL, is now AECL = AE - U E D

(37)

CL = W(i)T(i)N(i).

(42)

Along with the above described optimal load management, we can have in effect, a classic load management scheme [ 7,8 ]. This would set priorities for all the loads, controllable and not. The need for this supplementary load management is obvious in case the estimated instant load does not match the actual and thus the battery SOC falls below the prespecified levels. 4.3 Algorithm implementation The optimal load management strategy described above has been transformed to a F O R T R A N computer simulation program[6]. Appropriate flow diagrams have been developed outlining the new method and are used for simulation studies[6]. A simple flow diagram of the optimal load management algorithm is shown in Fig. 2. The F O R T R A N program is divided into one main program, one function (X) and two subroutines (TRACK, O U T P U T ) . The main program

where U E D is the uncontrollable energy demand during the 24-hour period (kWh). The next step is to determine the number of completely operational priorities, NCOP, that is the priorities of which all the units will operate. If AECL is greater than the total controllable demand CL, then all the priorities can be supplied with power ( N C O P = r). In case N C O P is less than r a n d AECL is positive, there will always be one priority in which only some of the units can operate. The energy available for this priority is equal to

START )

E

OBTAIN NEEDED DATA

/

COMPUTE TOTAL -I DAILY DEMAND

1

E A ( k ) = W(k)T(k)N(k) - [(LL) - (AECL)] (38)

FIND NUMBER OF PRIORITIES TO BE SERVED

where k

LL = ~ W(i)T(i)N(i)

(39)

i=l

and k is the index number of the partially shedded priority. Then the number of operational loads in this priority is

N(k)-

EA(k)

W(k)T(k)"

(41)

i=l

(40)

Finally, all units of priorities 1 to k - 1 and N ( k ) units of priority k will operate. The total demand of these units should follow the curve eqn ( 28 ), as mentioned in the previous section. In this case, CL is equal to

CALL TRACK R TIMES TO MAKE CONTROLLABLE LOADS FOLLOW CURVES IN X

/

CALL OUTPUT TO PRINT RESULTS

Q

STOP

/

)

Fig. 2. Optimal load management algorithm.

I

Stand-alone photovoltaic power systems calls subroutine T R A C K to follow the optimal curves stored in function X. Subroutine O U T P U T is for formatting the output and plotting the actual and optimal load curves. The program is versatile in the respect that it can be used when any portion of the load is desired to follow any curve. For best results, the unit usage constants T(i) should not exceed three hours, which is a realistic assumption. Otherwise, the purpose of closely following a continuous curve cannot be achieved. The input data needed for this program are • the number of controllable load priorities r • the array size A (m 2 ) • the battery size Q ( k W h ) • the battery state-of-charge SOC • the system efficiency rt • the estimated instant daily demand IL ( k W h / d a y ) • the constant daily demand PL ( k W h / d a y ) • the controllable unit power ratings W ( i ) ( W ) • the controllable unit usages T(i) (hrs / day) • the number of units in controllable priority iN(i) • the actual estimated insolation AEI ( k W h / m 2). The output includes • the number of completely operational priorities • the number of operational units in partially shedded priority • the time each unit is turned on • the actual and optimal load curves. 4.4 Example of optimal load management

implementation Loads similar to those of Schuchuti can be used as an example of the optimal load management algorithm. The loads include water pumps, lights, washing machines, sewing machines, refrigerators, and control instrumentation. Water pumps, washing machines, and sewing machines are defined as controllable loads. The parameters used for these loads are (a) Water pumps W = 100 Watts T = 1 hour/day N = 50 (b) Washing machines W = 310 Watts T = 1 hour/day N = 50 (c) Sewing machines W = 132 Watts T = 1 hour/day N = 50. Thus, the controllable load daily demand CL is equal to CL = ( 1 0 0 ) ( 5 0 ) ( 1 ) + ( 3 1 0 ) ( 5 0 ) ( 1 ) + ( 1 3 2 ) ( 5 0 ) ( 1 ) or CL = 27.1 k W h / d a y . We also classify water pumps as priority 1 loads, washing machines as priority 2, and sewing machines as priority 3.

127

The village lights are the instant demand 1L of the system. The estimated IL is 14.53 k W h / d a y . The constant load PL consists of the 15 refrigerators and the controls. Let their requirements be 5.03 k w h / day and 0.67 k W h / d a y , respectively. Thus PL is equal to 5.7 k W h / d a y . The required array size for this system is 125.5, with a system efficiency of 0.08. Assuming that the actual insolation is 5.2 k W h / m 2, which is less than the average insolation of the worst month and that battery SOC = 0.7, we performed some simulation studies. The results showed that, although the actual insolation is below average, there is no need for load shedding[6], as would be the case if we used previous control strategies [ 7-9 ]. The total life cycle cost was also smaller than using other techniques because by manipulating the controllable curve, indeed the N S R was reduced [ 6,13 ].

5. CONCLUSIONS

A theoretical investigation of the load management problem for SAPV power systems was peribrmed. A new optimal load management strategy that has been proposed and tested seems suitable for optimal designs of SAPV energy power systems when multiple types of loads are to be seived. The new approach is suitable not only for protecting the batteries and maintaining operation of emergency loads at the expense of less critical loads, but at the same time an optimal cost is obtained. Indeed, by forcing the controllable loads to follow optimal curves, we achieve system cost reduction. There are some interesting future research topics that should be pursued as a natural consequence of the proposed optimum load management strategy. There is a need to compare this approach and the obtained results with hybrid systems [ 14 ] and particularly with the autonomous PV-diesel power plants[15] which are gaining popularity lately. The degree of reliability should also be investigated. It would be interesting to see how reducing the NSR and extending the battery life would affect the overall reliability of the SAPV system. The actual implementation of the proposed algorithm is also a challenging problem. We think that the proposed algorithm would be beneficial if artificial intelligence and expert system techniques are to be used to improve the overall performance of a SAPV power system.

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and G. PAPEGEORGIOU

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