LESSONS OF EASTER ISLAND
March 17th, 2016
The following example is based on the approach developed by Sandia National Laboratory, Albuquerque, USA (Chapman, 1987) and automatically incorporates over 23 years of insolation data.
An interesting outcome of the study involved in the formation of this model and approach is that accuracy of system design is not lost by basing the design only on data for the month with the lowest insolation levels over the year. This of course greatly simplifies the design approach. In addition, through the use of calculations similar to those in the approach described above, but over a wide range of possible design values and in conjunction with appropriatelytreated average global insolation data, curves have been generated that facilitate:
1. determination of battery capacity for a specified lossofload probability (LOLP)
2. optimisation of array tilt angle
3. obtaining insolation data at the appropriate tilt angle
4. determination of the array size that, in conjunction with (1), provides the required LOLP.
Four sets of the curves exist, each giving a different battery capacity in (1) for the specified LOLP. When followed through (2)(4), each set of curves provides for a different system design (different array size/battery storage combination) with the same LOLP. These four sets can then be analysed on the basis of cost to determine the leastcost approach to satisfying system specifications.
Step 1—Define sitespecific and applicationspecific parameters
• latitude
• horizontal insolation for worst month (usually June in Australia, December in the northern hemisphere)
• daily energy demand
• LOLP required.
For example:
• latitude—30°N
• average daily horizontal insolation in December—3 kWh/m2
• daily demand—5 kWhac
• LOLP—0.001 (critical load such as a vaccine refrigerator).
If the average daily summer demand exceeds the average daily winter demand by more than 10%, attention needs to be paid to the possibility of battery discharge during summer (which may necessitate reoptimisation of tilt angle).
Step 2—Determine battery storage for each of the four designs
This is read directly from the appropriate nomogram, as a function of LOLP. Fig. G.1 shows this nomogram for design 2, showing that for an LOLP of 0.001, storage (S) is 5.80 days.
Similarly, looking at the appropriate nomograms, designs 1, 3 and 4 give storage values of 3.49, 8.13 and 10.19 days, respectively. From the S values, the actual battery capacity can be calculated from
CAP =——— ——————— (G.2)
DOD x noUt
where S is the number of days of storage, L is the average load per day, DOD is the allowable depthofdischarge for the batteries and щout is the storagetodemand path efficiency.
probability for design 2 (©1987 IEEE, Chapman). 
For qout, since we are using an AC load, we need an inverter. If we allow for a peak load of 1000 Wac, our inverter should be rated about 20% higher, i. e. 1200 Wac. The inverter efficiency will vary as a function of load and needs to be determined in conjunction with the anticipated load profile throughout each day. A typical average daily inverter efficiency of 0.76 is assumed.
The other contributor to щои, is the battery controller which, because of parasitic power drains, will be in the vicinity of 95% efficient. Losses associated with charge leaving the battery need not be considered here, since the rating of battery capacity is in terms of charge obtainable from the battery. Therefore, we consider battery inefficiencies only in terms of charge being stored in the batteries.
nout = 0.95 x 0.76 = 0.72 L = 5 kWh/day S = 5.80 (for design 2)
DOD = 0.8 (using a deepcycle type battery)
Therefore
CAP = 5.80 x 5 / (0.8 x 0.72)
= 50 kWh.
Step 3—Determine array size for each of the four designs
The ‘design insolation in the plane of the array’ (POA) can be determined from the appropriate nomogram. By reading off the POA for different tilt angles (i. e. one nomogram exists for each tilt angle), the tilt angle is easily optimised by selecting the one that gives the maximum POA value. These values are, of course, a function of the latitude. Fig. G.2 shows the nomogram for design 2, for the case where the tilt angle equals the latitude.
Figure G.2. Design insolation in the plane of the array (POA) as a function of the daily global insolation (©1987 IEEE, Chapman). 
For our example, where the horizontal insolation is 3 kWh/m2/day and the latitude is 30°, we get a POA value of 4.3 kWh/m2/day.
The corresponding array area (A) is calculated from
A = L————————————– (G.3)
POA X Vin X Vent
where L is the average daily load (kWh/day), POA is the design insolation value (kWh/m2/day), ^in is path efficiency from insolation to storage and qout is storageto – load efficiency.
The advantage of determining array size in terms of area is that it remains independent of the voltagecurrent configuration. The disadvantage is that the purchase of solar panels involves the specification of currents (and voltages) and/or power rating for standard test conditions.
If a maximum power point tracker (MPPT) is not used, the array voltage is determined by the batteries and each solar module can be assumed to operate at its rated maximum power point current (since excess voltage is built into each module to allow for temperature effects). In this instance, the array size is most conveniently specified in terms of peak current (Ip) according to
where I0 is the peak light intensity (1 kW/m2) under which the array will produce current Ip at the nominal battery voltage Vbat, POA is the design insolation, щЬа, is the battery coulombic efficiency (typically 85% during lowest insolation months), DF is the dust factor, (typically 0.90; i. e. 10% loss owing to dust), L is the average daily load, SD is a factor for selfdischarge of the batteries and щbat is the batterytoload efficiency as dealt with previously.
For deep cycle batteries, in the worst month, selfdischarge can be neglected, as little charge generally remains, unless the LOLP figure is extremely small. For a 24 Vdc configuration
Ip = 5000 x 1 / (4.3 x 0.85 x 0.90 x 24 x 0.72)
= 88.0 A rated current (at nominally 24 V).
As a power rating by the manufacturer using standard test conditions (25°C), this will typically correspond to
88 A x 34 V (25°C)
= 3.0 kW as a manufacturer’s rating.
However, if an MPPT is used, we cannot make assumptions about the operating voltage of the array, since the MPPT will adjust it to the maximum power point. Accordingly, we need to look at the efficiency of the array at the maximum power point (^mp), which is a function of the array’s cell temperature (Tc) and the array efficiency (qr), which should be given by the manufacturer. That is
nmp = nr[1 – C {Tc – Tr)] (G.5)
where Cr is the maximum power coefficient of variation with temperature (typically 0.005 °C1) and Tr is the reference temperature.
Depending on the site, at latitude 30°N, a typical ambient temperature during a winter’s day will be 10°C and the array will operate at approximately 20°C above ambient. Assuming the manufacturer quotes and array efficiency of 10% at a reference temperature of 25°C, then
nmp = 10.0 x (1 – 0.005 x (30.0 – 25.0))
= 9.75% efficiency.
The resulting array area required will be given by Eqn. (G.3), where щіп (the insolationtostorage efficiency) is given by
Vin Vmp X Vbat X Vmppt X D X SD (G.6)
where qmppt is the MPPT efficiency (typically 95%). Therefore
rjin = 0.0975 x 0.85 x 0.95 x 0.90 = 0.071
and, from Eqn. (G.3)
A = 5000/(4300 x 0.071 x 0.72)
= 22.8 m2
or, in terms of the manufacturer’s ratings (at 25°C)
array rating = 1000 W/m2 x 0.10 x 22.8 m2 = 2.28 kWp.
This compares to the 3.0 kWp rating required without the MPPT. A trade off obviously exists between cost, system complexity, system efficiency and reliability.
These calculations have been done on the basis of design 2 and can be simply repeated for designs 1, 3 and 4 by using the appropriate nomograms, which will give different values for POA and S. This is shown in Table G.2.
Table G.2. Planeofarray design insolations as a function of storage (S). ______________ planeof array insolation (kWh/m2/day)___________________________

From Table G.2, the optimum tilt angle can be seen to be latitude + 10°. This then leaves four potential combinations of array size and storage capacity, each of which provides the required LOLP. Selection is then made on the basis of least cost. Costing may be done purely on an initial cost basis, or else a lifetime cost basis, depending on consumer preference. For the latter, battery life, which varies significantly with temperature and depthofdischarge (DOD), must be considered.
The battery life for flooded lead acid batteries can be estimated from
CL = (89.59 – 194.29T )exp(1.75 x DOD) (G.7)
where CL is the battery life (in cycles), T is the battery temperature and DOD is the depthofdischarge.
In a PVstorage system, the depthofdischarge varies from cycle to cycle. We define each cycle as one day, and DOD as the maximum depthofdischarge for that day.
It has been shown statistically that the distribution of DODs for all battery cycles can be generalised as a function of the LOLP and the days of storage, thus enabling Eqn. (G.7) to be used to give a close estimate of actual battery life.
Using a temperature of 25°C (constant) and an LOLP of 0.001, the anticipated battery life for designs 14 are 10.7, 10.9, 10.9 and 10.9 years, respectively. The lives are quite long because, for an LOLP of 0.001, the batteries remain well charged most of the time.
REFERENCES
Mack, M. (1979), ‘Solar power for telecommunications’, The Telecommunication Journal of Australia, 29(1), pp. 2044.
Chapman, R. N. (1987), ‘A simplified technique for designing least cost standalone PV/storage systems’, Proc. 19th IEEE Photovoltaic Specialists Conference, New Orleans, pp. 11171121.
Leave a reply