Optimum Design of Diesel Generator Integrated Photovoltaic-Battery

Nov 15, 2010 - locations.3 Utilizing solar resource is significant for tropical country like India; it has about 300 clear sunny days a year.4. Integr...
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Energy Fuels 2010, 24, 6565–6575 Published on Web 11/15/2010

: DOI:10.1021/ef101028p

Optimum Design of Diesel Generator Integrated Photovoltaic-Battery System P. Arun, Rangan Banerjee, and Santanu Bandyopadhyay* Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India Received August 5, 2010. Revised Manuscript Received September 14, 2010

A diesel generator integrated photovoltaic-battery system is one of the options for decentralized power generation for remote locations. A methodology for the optimum sizing of such systems incorporating the uncertainties associated with the solar insolations and electrical demand is proposed in this paper. The proposed methodology is based on the design space approach involving a time series simulation of the entire system. The chance constrained programming approach has been adopted to incorporate the resource as well as the demand uncertainties at the design stage. The set of all feasible design configurations is represented by a sizing curve. The sizing curves are represented showing the variation of the diesel generator rating and battery bank capacity for a given level of photovoltaic integration conforming to a specified reliability. Alternately, the sizing curves also represent the variation in the photovoltaic array rating and battery bank capacity for a given diesel generator rating. The methodology is validated using a sequential Monte Carlo simulation approach with illustrative examples. It is shown by means of examples that site specific conditions can be evaluated and explored using the design space approach. An illustrative example incorporating the effect of the correlation existing between the solar resource and the demand on the system sizing is presented. Selection of the optimum system configuration, based on the minimum cost of energy, is also discussed.

system performance. It is also essential to account for the uncertainty linked with the electrical demand. Hence it is important to account for the variability associated with the resource and electrical demand at the design stage. In this paper, a methodology is proposed for the optimum sizing of the diesel generator integrated photovoltaic-battery hybrid system for predefined system reliabilities. Modern hybrid systems are installed and operated in parallel topology as a result of its significant advantages over other schemes.5 The sizing of such systems is relatively more involved compared to the series connected systems. There exists simplified methods and deterministic and probabilistic simulation based approaches for the system sizing of photovoltaic-battery systems. They may be viewed as the basis for the sizing of hybrid systems. For example, the Sandia lab6 had provided simple calculations to obtain an independent design of the photovoltaic array and battery bank. A probabilistic sizing method treating storage energy variation as a random walk had been proposed for photovoltaic-battery systems.7 Deterministic simulation approaches have been applied for studying the effect of the simulation time step and the input and output power profiles on the sizing of autonomous photovoltaic systems.8 There exist different options for photovoltaic hybrid systems. It can be completely renewable energy based systems, e.g., photovoltaic-wind-battery systems.9 Another option could be photovoltaic-wind-diesel systems.10 Considering

1. Introduction Isolated hybrid systems, integrating different energy sources, are a viable means for electrifying remote locations where grid extension becomes uneconomic. Such isolated power systems meet the electricity demand of the remote location by generating power close to the load centers. There have been several successful attempts to integrate photovoltaic units and battery storage in diesel generator based isolated power plants to reduce the diesel consumption and emission. One of the largest photovoltaic-diesel generator-battery hybrid systems is the Wilpena Pound power station located in South Australia. The system integrates 100 kWp photovoltaic arrays, 400 kWh of battery storage, and 440 kW of diesel generation.1 In India, the largest hybrid system is located in Lakshadweep island comprising of 50 kWp photovoltaic array and two 75 kVA diesel generators.2 Many such hybrid plants integrating diesel generators, photovoltaic units, and battery banks are being successfully operated in various remote locations.3 Utilizing solar resource is significant for tropical country like India; it has about 300 clear sunny days a year.4 Integration of a dispatchable power system like a diesel generator and source of energy storage like a battery bank helps in overcoming the limitations associated with the solar source. The design of such hybrid power generation units involves the estimation of the capacities of the generating units and storage for satisfying a given demand pattern. For renewable energy source based isolated power systems, the randomness associated with the resource has a significant effect in the

(5) Nayar, C. V. Appl. Energy 1995, 52, 229–242. (6) Bhuiyan, M. M. H.; Asgar, A. M. Renewable Energy 2003, 28 (6), 929–938. (7) Bucciarelli, L. L., Jr. Sol. Energy 1984, 32 (2), 205–209. (8) Notton, G.; Muselli, M.; Poggi, P.; Louche, A. Renewable Energy 1996, 7, 353–369. (9) Wei, Z.; Chengzhi, L.; Zhongshi, L.; Lin, L.; Hongxing, Y. Appl. Energy 2010, 87 (2), 380–389. (10) Saheb-Koussa, D.; Haddadi, M.; Belhamel, M. Appl. Energy 2009, 86 (7-8), 1024–1030.

*To whom correspondence should be addressed. Telephone: þ91-2225767894. Fax: þ91-22-25726875. E-mail: [email protected]. (1) http://www.pvresources.com/en/hybrid.php, December 2009. (2) Bharat Heavy Electricals Ltd. www.bhel.com/images/pdf/ENJune2006.pdf, December 2009. (3) Ashari, M.; Nayar, C. V. Sol. Energy 1999, 66 (1), 1–9. (4) Banerjee, R. Energy Policy 2006, 34 (1), 101–111. r 2010 American Chemical Society

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diesel generator integrated photovoltaic-battery systems, a case study illustrating the sizing of the hybrid system involving a photovoltaic system with a backup generator was presented by Notton et al.11 Wichert12 provided a detailed review of photovoltaic-diesel hybrid systems highlighting their merits and limitations. Seeling-Hochmuth13 suggested system optimization for design and operation strategy for hybrid photovoltaic energy systems using genetic algorithms. Muselli et al.14 discussed an optimal sizing procedure for photovoltaic-diesel systems by minimizing the total energy cost. Ashari and Nayar3 analyzed dispatch strategies for the operation of a solar photovoltaic-diesel-battery hybrid power system based on “set points”. Lopez and Agustin15 reported the design and control strategies of photovoltaic-diesel systems using genetic algorithms. The multiobjective design of isolated hybrid systems simultaneously minimizes the total cost, the pollutant emissions, and the unmet load has been presented by Lopez and Agustin.16 Few software tools are also available for the design of isolated power systems. RETScreen17 is useful for preliminary system sizing of isolated power systems. Hybrid218 developed by the National Renewable Energy Laboratory is capable of performing the detailed time series simulation of hybrid power systems. The hybrid optimization model for electric renewables (HOMER)19 uses hourly simulations for arriving at optimal system sizing of isolated power systems. However, most of the available software tools identify and simulate a single design option. They generally do not generate and evaluate the different design options. The utility and applicability of the software tools get significantly enhanced if it is possible to explore and represent various feasible design configurations. Such a methodology will be useful for system optimization considering a different type of objective functions (e.g., minimum cost of energy, minimum carbon dioxide emissions, maximum reliability level, etc.) and/or configuring the overall system with multiple-objective functions. Also, the effect of nonlinearity in the system modeling and the randomness associated with the design variables needs to be accounted in such tools so that they can provide system configurations with predefined reliability levels. For the system planner, it will be beneficial to obtain the set of all feasible system configurations or the design space capable of meeting the expected demand and conforming to a desired reliability. Kulkarni et al.20-22 introduced the concept of design space for optimum sizing of solar hot water systems. A methodology for generating the design space for a battery integrated diesel generator system following a deter-

Figure 1. Schematic of an integrated photovoltaic- diesel generatorbattery system.

ministic approach was illustrated by Arun et al.23 The concept of design space has been extended and demonstrated for other options for isolated power generation like photovoltaicbattery systems and wind-battery systems following a deterministic approach.24-26 The design space methodology has also been extended to incorporate the solar resource uncertainty for photovoltaic-battery systems27 and wind-battery systems28 and demand uncertainty for battery integrated diesel generator systems.29 Existing methodologies and tools do not include various uncertainties associated with the resource and the demand at the design stage for hybrid energy systems. This paper extends the design space approach for isolated systems involving diesel generators and photovoltaic units, integrated with battery storage. The methodology is presented as an extension from the deterministic framework to incorporate the uncertainties associated with both the solar insolation and the electrical load demand using the chance constrained programming approach. It enables the construction of a set of sizing curves which graphically represent the set of feasible design configurations meeting a desired reliability level. This helps in identifying the entire range of feasible system configurations or the design space conforming to a specified reliability level. Generation of the design space helps in selecting and optimizing an appropriate system configuration based on the desired objective. 2. Photovoltaic-Diesel -Battery System The schematic of a parallel hybrid system configuration is given in Figure 1. The system components include diesel generator set, photovoltaic array, battery bank, and bidirectional converter. A controller is also needed to supervise the operation of the system in selecting the most appropriate operational mode to supply the load. Parallel mode of operation enables the photovoltaic array and the diesel generator to supply a portion of the load directly. The rating of the diesel generator can be lower than the expected peak demand.

(11) Notton, G.; Muselli, M.; Louche, A. Renewable Energy 1996, 7 (4), 371–391. (12) Wichert, B. Renewable Sustainable Energy Rev. 1997, 1 (3), 209– 228. (13) Seeling-Hochmuth, G. C. Solar Energy 1997, 64 (2), 77–87. (14) Muselli, M.; Notton, G.; Louche, A. Solar Energy 1999, 65 (3), 143–157. (15) Lopez, R. D.; Agustin, J. L. B. Solar Energy 2005, 79, 33–46. (16) Lopez, R. D.; Agustin, J. L. B. Renewable Energy 2008, 33 (12), 2559–2572. (17) RETScreen International Clean Energy Project Analysis Software, CANMET Energy Technology Centre Varennes, http://www. retscreen.net/ang/t_software.php, December 2009. (18) Hybrid2, University of Massachusetts Amherst, http://www. ceere.org/rerl/rerl_hybridpower.html, December 2009. (19) HOMER, http://www.homerenergy.com, December 2009. (20) Kulkarni, G. N.; Kedare, S. B.; Bandyopadhyay, S. Solar Energy 2007, 81 (8), 958–968. (21) Kulkarni, G. N.; Kedare, S. B.; Bandyopadhyay, S. Solar Energy 2008, 82, 686–699. (22) Kulkarni, G. N.; Kedare, S. B.; Bandyopadhyay, S. Energy Convers. Manage. 2009, 50, 837–846.

(23) Arun, P.; Banerjee, R.; Bandyopadhyay, S. Energy 2008, 33 (7), 1155–1168. (24) Arun, P.; Banerjee, R.; Bandyopadhyay, S. Energy Sustainable Dev. 2007, 11 (4), 21–28. (25) Roy, A.; Arun, P.; Bandyopadhyay, S. Sol. Energy Soc. India J. 2007, 17 (1-2), 54–69. (26) Roy, A.; Kedare, S. B.; Bandyopadhyay, S. Appl. Energy 2009, 86 (12), 2690–2703. (27) Arun, P.; Banerjee, R.; Bandyopadhyay, S. Solar Energy 2009, 83 (7), 1013–1025. (28) Roy, A.; Kedare, S. B.; Bandyopadhyay, S. Appl. Energy 2010, 87 (8), 2712–2727. (29) Arun, P.; Banerjee, R.; Bandyopadhyay, S. Ind. Eng. Chem. Res. 2009, 48 (10), 4908–4916.

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converted into electrical energy. It is assumed that the charging and discharging efficiencies of the battery bank and converter remains constant over time. It is also assumed that the self-discharge loss for the battery is negligible. The minimum generator ratings required and the corresponding storage capacity for meeting the specified load may be obtained by solving eq 3 over the entire duration. In the generalized methodology, to obtain the minimum generator ratings, a numerical search is performed that satisfies the energy balance and the following conditions:

This permits the generator operation at considerably higher load factors compared to the diesel generator-only mode. The presence of the photovoltaic array acts as a supplementary power source and reduces the fuel consumption of the generator. The system may operate in one of three modes based on the generator dispatch: (i) converter only operation: At lower loads, the system would operate in the inverter mode (depending on the available state of charge of the battery, with the battery discharging). The generator may be shut down during this period. (ii) Charging operation: At certain periods, the system operation would be such that the generator meets the load and charges the battery bank enabling it to attain its full capacity. Depending on the availability of solar insolation, the photovoltaic system operates to feed the power for charging the battery bank. (iii) Parallel operation: This corresponds to the peak load periods when both diesel generator and battery operate together to serve the load.

QB ðtÞg0 "t

ð4Þ

QB ðt ¼ 0Þ ¼ QB ðt ¼ TÞ

ð5Þ

Equation 4 ensures that the battery energy level is always nonnegative, while eq 5 represents the repeatability of the battery state of energy over the time horizon. The required battery bank capacity (Br) is obtained as

3. System Modeling and Generation of the Design Space A methodology to obtain the system sizing of the integrated system is discussed in this section. The proposed approach employs a time series simulation based on the energy balance of the overall system. The net power generated is represented as the difference between the total power generated (from the diesel generator set (P) and the photovoltaic array (PP)) and the power required by the load (D) as: 8 > ¼ ðPP ðtÞ þ PðtÞηconv Þ f ðtÞ > > > > > > and f ðtÞ ¼ ηc > whenever PP ðtÞ þ PðtÞg0 dQB <   PðtÞ dt > f ðtÞ ¼ PP ðtÞ þ > > ηconv > > > 1 > > : whenever PP ðtÞ þ PðtÞ < 0 and f ðtÞ ¼ ηd ð1Þ

Br ¼

maxfQB ðtÞg DOD

ð6Þ

where DOD is the allowable depth of discharge of the battery. The proposed procedure provides the value of the minimum diesel generator capacity (P = Pmin) and the corresponding capacity of the battery bank (B) for a specified value of the photovoltaic array rating. Alternately the minimum photovoltaic array rating or area (A = Amin) and the corresponding capacity of the battery bank (B) for a specified value of the diesel generator rating may be determined by system simulation. Any generator rating, higher than the minimum, is capable of supplying the load. However, the battery bank capacity is expected to be lower for higher generator ratings. It is important from the designers’ perspective to identify all the feasible combinations for the generator rating and the corresponding storage capacity. Simulations to obtain the minimum storage capacity may be carried out for different values of generator ratings. For each value of generator rating considered, the corresponding minimum battery bank capacity is obtained by minimizing the required storage capacity in eq 6. The optimization variables are the initial battery energy, QB (t = 0) and the net power delivered by the generator P*(t). The combinations of the different generator ratings and the corresponding minimum storage requirements may be plotted on a generator rating vs battery bank capacity diagram which may be called the sizing curve of the system. The sizing curve represents the minimum storage capacity required for a given generator rating. The sizing curve divides the entire space into regions, a feasible and an infeasible region. The region above the sizing curve represents the feasible region as any combination of generator rating and battery capacity represents a feasible design option. The entire feasible region including the sizing curve is the design-space for a given problem. For simplicity of representation in a two-dimensional plane, the design space may be generated and represented for a specified value of the diesel generator rating or photovoltaic array rating. For systems with two generation units, the sizing curves then represent the variation in storage for different values of one of the generator units maintaining the other constant. This is illustrated for the diesel generatorphotovoltaic-battery bank system. Starting from a diesel generator-battery system, the effect of integrating photovoltaic

where P*(t) = P(t) - D(t). In the above equations f(t) represents the efficiencies in charging and discharging processes of the battery bank,and ηconv represents the converter efficiency. The power generated by the photovoltaic array at any given time t is given by ð2Þ PP ðtÞ ¼ η0 AIT ðtÞ where η0 is the photovoltaic system efficiency, A is the total array area (m2), and IT is the total insolation incident on the array (watt/square meter) at that time step. For a relatively small time period Δt, the stored energy QB may be expressed as follows for charging: 8 > ¼ QB ðtÞ þ ðPP ðtÞ þ PðtÞηconv Þf ðtÞΔt > > > > > > PP ðtÞ þ PðtÞg0 < whenever   ð3Þ QB ðt þ ΔtÞ PðtÞ > > f ðtÞΔt ¼ Q ðtÞ þ P ðtÞ þ > B P > ηconv > > > : whenever PP ðtÞ þ PðtÞ < 0 During the system operation over the time period Δt, whenever the total energy supplied by the diesel generator and the photovoltaic array is greater than the demand, the energy surplus is used for charging the battery. If the generator dispatch is such that the total energy delivered by the generators is lower than the load, then the battery supplements the deficit. The load is met if the battery has not reached its depth of discharge, and in that case, the stored chemical energy is 6567

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Figure 2. System sizing for various levels of diesel generator integration.

Figure 4. System sizing for various levels of photovoltaic array integration.

With addition in array capacity, the system configuration reaches the photovoltaic-battery system as one of the limiting cases. The steps for generating the design space are presented in Figure 2. The sizing curve and associated design space showing the variation in diesel generator-battery combinations for a fixed photovoltaic array rating are given in Figure 3. Alternately, the system sizing may be carried out with photovoltaic-battery systems as the starting point and obtaining the system capacities for various levels of diesel generator integration. With an increase in diesel capacity, the limiting case of a diesel generator-battery is attained. The steps for generating the design space in this case are presented in Figure 4. The sizing curve and associated design space showing the variation in photovoltaic array-battery combinations for a fixed diesel generator rating is given in Figure 5. It may be noted that the actual design space is three-dimensional in nature. A threedimensional design space is shown later with an illustrative example. The complete set of feasible configurations of the diesel generator, photovoltaic array, and battery bank may be represented in a three-dimensional space. However, for simplicity of representation, the sizing curves showing the variation of generator ratings with battery capacity is plotted

Figure 3. Typical sizing curve and design space for the hybrid system for a specified diesel generator level.

units in steps is obtained. The minimum diesel generatorbattery combination for a specified photovoltaic array rating is identified and the sizing curve is generated for different values of diesel ratings greater than the identified minimum. 6568

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Arun et al. Table 1. Input Parameters Used in the System Sizing and Optimization photovoltaic system efficiency, % net charging efficiency, % net discharging efficiency, % inverter efficiency depth of discharge, %

10 85 85 90 70

Figure 5. Typical sizing curve and design space for the hybrid system for a specified photovoltaic array level.

Figure 8. Sizing curves for the integrated system starting from the photovoltaic-battery configuration.

sizing are listed in Table 1. Nominal system efficiency values are assumed for the battery charging/discharging and for the converter, for simplicity. Temperature effect on the photovoltaic module is not explicitly considered which is, however, incorporated through the overall efficiency of the module. The set of sizing curves are generated for the integrated system based on the procedure described in the previous section. The system sizing has been illustrated considering an averaged day profile for the solar insolation and electrical demand. The seasonality can be captured, and accurate system sizing can be obtained if the hour by hour variation in the resource and the demand are considered on a yearly basis. The data for such an extended time frame is not generally available for remote locations. However it may be noted that the proposed methodology is not limited by the time frame and time step of resource and demand. This has been systematically illustrated for the case of system sizing of the photovoltaic battery system using the design space approach by using hourly data for a complete year.27 It has been observed that for the case of photovoltaic-battery systems, adopting monthly average daily data for the solar insolation would lead to system oversizing.31 Considering this observation, hourly data for a day has been used for illustration of the method. For representing the feasible systems on a two-dimensional space, the system configurations corresponding to a specified constant level of the power generating system (diesel generator rating or photovoltaic array rating) are illustrated. The sizing curves and associated design space are represented for a given value of diesel generator rating showing the corresponding photovoltaic array-battery combinations (Figure 8). The set of sizing curves are plotted for a no diesel generator system (corresponding to photovoltaic-battery systems), representative diesel generator ratings of 3, 6, 7.7 (minimum load), 9, 12, and 13.3 kW (mean load). At a generator rating of 13.8 kW and a battery capacity of 50 kWh, it is observed that the load

Figure 6. Load curve for the rural location in Kerala, India.

Figure 7. Variation in the mean (bold curve) and standard deviation (thin curve) of the hourly solar insolation for a representative day for the rural location in Kerala, India.

on a two-dimensional plane. For the system optimization, the entire feasible design space is searched (i.e., simultaneous variations of all the design variables) to obtain the global optimum value of the objective function. The methodology of system sizing is illustrated through examples in the following sections. 3.1. Illustrative Example: Deterministic System Sizing. An illustrative example for the sizing of the integrated system is presented in this section. The solar resource data and electrical demand data for a rural community in Kerala, India have been considered.30 For a representative day, the hourly variation in the electrical demand for the rural community is represented in Figure 6. The hourly variation in the mean and standard deviation of the solar insolation is presented in Figure 7. Other input parameters used for the hybrid system

(31) Arun, P. Optimal Design of Isolated Power Systems. Ph.D. Thesis, IIT Bombay, India, 2009.

(30) Ashok, S. Renewable Energy 2007, 32, 1155–1164.

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Figure 11. Sizing curves for the integrated system starting from the diesel-battery configuration.

Figure 9. Sizing curve without photovoltaic integration.

Figure 12. Variation in battery energy value for representative system configurations over a representative day.

For all the systems, the battery discharges through the converter for the morning hours to meet the peak load. There is an increase in the battery energy value for the rest of the hours through the photovoltaic array operation and the diesel generation until the evening peak. Further, the battery bank is discharged in the evening hours for all the systems in order to meet the prevailing demand.

Figure 10. Three dimensional design space representing the feasible system configurations.

requirements are met without any contribution from the photovoltaic system. Further, the load may be met by the integration of the diesel generator and the battery bank. The corresponding diesel generator-battery bank configurations are represented in Figure 9. It may be noted that at the maximum load of 20.2 kW, the storage component reduces to zero reaching the diesel generator only system (Figure 9). The three-dimensional design space showing the variation of the diesel generator rating, photovoltaic array rating, and battery capacity is illustrated in Figure 10. The set of sizing curves plotted in Figure 11 illustrate the effect of discrete integration of photovoltaic arrays starting from the set of diesel generator battery-bank configurations. As expected, it is observed that the requirement of the minimum diesel generator drops with addition of photovoltaic units with an increase in the corresponding storage capacity. Storage forms an important component in the various configurations of isolated systems considered. For a photovoltaic battery system, the storage component is of relatively higher magnitude to meet the periods of low sunshine. The amount of storage required is lower for a battery integrated diesel generator system and an integrated photovoltaic-diesel generator battery system due to the presence of the diesel generator. The variation in battery energy value QB(t) for representative configurations is presented in Figure 12.

4. Probabilistic System Sizing In reality, the electrical demands as well as the solar insolation have inherent variability. Uncertainties associated with these variables are to be incorporated at the design stage to reduce frequent failure of a hybrid energy system. A probabilistic approach is essential to design such a hybrid energy system to satisfy a specified reliability criterion. The extension of the design space methodology for sizing the hybrid isolated power system for a given random load profile and random solar insolation profile is discussed in this section. The hourly load and solar insolation are treated as stochastic variables, and the system sizing corresponds to a specified system reliability level. The system sizing problem is solved as an optimization problem under uncertainty. It is developed as a systematic extension of the deterministic time series simulation of the system incorporating the random source and demand variables by means of the chance constrained programming approach.32 (32) Charnes, A.; Cooper, W. W. Manage. Sci. 1959, 5, 73–79.

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For simplicity of analysis, it is assumed that the hourly solar insolation and demand are independent normally distributed random variables. However, the proposed methodology has no restriction as far as the selection of the probability distribution is concerned. In the present case it is also possible to evaluate the error under such an assumption. The constraint defining power production of the generator may be expressed as a chance constraint as prob½PðtÞePr gR

"t

erator-battery bank configurations are selected from the design space. The selected system is simulated using the battery energy balance (eq 3). The hourly demand and solar insolation are considered as random variables in the simulation. The random demand and solar insolation values used in the simulation are sampled from respective normal distributions with specified mean and standard deviation for the time step. For comparison, a data set is used such that the mean and standard deviation data of the hourly demand and solar insolation correspond to the same as used in the chance constrained model for deriving the sizing curve. The hourly simulations are carried out for the span of a year (using 8760 hly data points), and the battery energy level (QB) is checked with the minimum permissible battery energy level (Bmin) at each hour. The system confidence level (R) is estimated for the configuration based on the hourly distribution of the durations when the demand is not met. It is estimated as P h ð11Þ Rh  1 H

ð7Þ

where R represents the specified reliability of compliance of the constraint. Relation 7 is a chance constraint with the righthand side of the term inside the square brackets and represents a random variable. For simplicity it is assumed that the converter operates at maximum efficiency, and hence, only the battery charging/discharging efficiencies are considered. On the basis of the energy balance of the system, the chance constraint may be expressed as   QB ðt þ ΔtÞ QB ðtÞ ð8Þ - Pr ePP ðtÞ - DðtÞ gR prob f ðtÞΔt f ðtÞΔt

where Σh corresponds to the total duration when there is a loss of load for the specified hourly time band, say 0500-0600 h or 2300-2400 h during the day (i.e., QB(t) < Bmin). H is the total hours considered in that interval (365 h) for yearly simulation. The minimum value of (Rh) obtained for each of the 24 h time horizons would correspond to the system confidence level (R). Further, the system LOLE is estimated as P t ð12Þ LOLE  Tmax

It may be noted that the quantity PP(t) - D(t) also follows normal distribution since it is the difference of two normal random variables. The deterministic equivalent31,33 for the energy balance in terms of the battery energy value, mean, and standard deviation of the random variables is obtained as QB ðt þ ΔtÞeQB ðtÞ þ ºPr þ μPP ðtÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! - μDðtÞ þ zR σPP ðtÞ 2 þ σDðtÞ 2 ß f ðtÞΔt

ð9Þ

where Σt corresponds to the total duration when there is loss of load over the entire time frame (i.e., QB(t) < Bmin) and Tmax the total time frame of simulation (8760 h). The simulations are repeated using the random samples of the hourly values of the solar insolation and the electrical demand for a complete year. At each iteration (i.e., at the end of 1 year) of the random simulation, the system LOLE is estimated. Hence the hourly sampling of the solar resource and demand are replicated for each simulation run, ensuring reasonable estimates of the system reliability in terms of the system LOLE. The simulation is terminated as the estimated LOLE value for a system achieves a specified degree of confidence. The coefficient of variation (ε) of LOLE is used as the parameter for deciding the stopping criterion: σπ ε ¼ ð13Þ μπ

Considering the net power from the generator, P*(t) (including its part load operation), the battery energy balance may be expressed as QB ðt þ ΔtÞ ¼ QB ðtÞ þ ºPðtÞ þ μPP ðtÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! - μDðtÞ þ zR σPP ðtÞ 2 þ σDðtÞ 2 ß f ðtÞΔt

ð10Þ

The battery capacity for a given array area and generator rating for meeting the load with a specified reliability may be obtained by solving eq 10 over the entire duration. The mean and standard deviation of the hourly power output is calculated using the mean and standard deviation values of the hourly solar insolation. The required battery capacity for a given array size and diesel generator rating may be obtained following the procedure described in section 3 using the modified energy balance. 4.1. Monte Carlo Simulation Approach for Validating the System Reliability. The sequential Monte Carlo simulation method is used for validating the results obtained with the chance constrained model for system sizing of the integrated system. The estimated values of confidence level and LOLE obtained from the Monte Carlo simulation for different system configurations from the design space are compared with those obtained by the chance constrained model. The sizing curve and design space are generated following the procedure detailed in the previous section for specified confidence levels. For checking the system reliability predicted by the design space, specific photovoltaic-diesel gen-

where μπ is the estimated mean of the LOLE and σπ being its standard deviation. The simulation is stopped when the value of the coefficient of variation attains a reasonably steady value over different iterations. The methodology of sequential Monte Carlo simulation is represented in Figure 13. 4.2. Illustrative Example: System Sizing for Specified Reliability Level. An example illustrating the influence of the load and demand uncertainty on the sizing of hybrid systems is presented in this section. The input data set corresponds to the same values as considered in example 1. The additional data required is the standard deviation of the hourly solar insolation (Figure 7) and demand. The coefficient of variation of the hourly demand is assumed to be 0.1. The sizing curves for the integrated system incorporating resource and demand uncertainty have been illustrated for specified values of photovoltaic array rating and diesel

(33) Sreeraj, E. S.; Chatterjee, K.; Bandyopadhyay, S. Solar Energy 2010, 84 (7), 1124–1136.

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Figure 15. Sizing curves for the integrated system for a generator rating of 7.7 kW.

given by the chance constrained model. It shows that the random simulation of the system predicts a higher reliability for the system compared to the chance constrained model for a majority of the cases. This indicates that the proposed model offers a conservative design for the system. 5. System Sizing with Correlated Solar Insolation and Load Demand In generation of the design space for isolated systems, it is of interest to account for the correlation existing between the resource and the demand. The strength and direction of the relationship between the two random variables, solar insolation, and the electrical demand can be expressed using the correlation coefficient (FI,D). It may be noted that the correlation coefficient depends upon the nature of the load profile prevailing in the location. For a system serving predominantly a lighting demand prevailing in the night, the correlation coefficient is expected to be negative. Similarly a strong positive correlation is expected between the solar insolation and the demand when the load also peaks with the resource. A typical example would be isolated systems serving predominantly cooling or air conditioning loads prevailing during the day time. In this section, the system sizing incorporating the effect of the correlation is illustrated. The battery energy balance (eq 10) incorporating the solar and demand uncertainty is modified accounting for the correlation coefficient (FI,D) as31,33 QB ðt þ ΔtÞ ¼ QB ðtÞ þ ºPðtÞ þ μP ðtÞ

Figure 13. Flowchart for Monte Carlo simulation of the generic system for reliability estimation.

Figure 14. Sizing curves for the integrated system for an array rating of 10 kWp.

P

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! - μDðtÞ þ zR σPP ðtÞ 2 þ σDðtÞ 2 - 2FI, D σ PP ðtÞ σDðtÞ ß f ðtÞΔt

generator rating. For the integrated system, Figure 14 represents the sizing curves for different confidence levels showing the variation of diesel rating and battery capacity for a fixed photovoltaic array rating of 10 kWp. As expected, owing to the source and demand randomness, system capacities need to increase to meet the generation requirements meeting a specified reliability. The sizing curves are also illustrated for a specified fixed diesel generator rating of 7.7 kW (minimum load) representing the variation in the photovoltaic array rating and the battery capacities (Figure 15) The reliability levels predicted by the chance constrained model are validated through the sequential Monte Carlo approach. Specific representative configurations from the design space are simulated to study its performance. The estimated values of confidence level and LOLE for these configurations selected from the sizing curve with the Monte Carlo simulation are given in Table 2 for specified reliability levels

ð14Þ An example illustrating the influence of the correlation on the sizing of hybrid systems is presented in this section. The input data set corresponds to the same values as considered in the illustrative example. The additional data required is the correlation coefficient between the hourly solar insolation and demand. For the location considered in the illustration, the value of the correlation coefficient is estimated to be -0.46. The load profile for the site is such that high load prevails during the nonsunshine hours. The sizing curves for the integrated system incorporating resource and demand uncertainty have been illustrated for specified values of the photovoltaic array rating and diesel generator rating incorporating the correlation effect. For the integrated system, Figure 16 represents the sizing curves for different confidence 6572

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Table 2. Comparison of System Confidence Level and LOLE with Monte Carlo Simulation for Representative Configurations from the Design Space confidence level

LOLE (analytical)

generator rating (kW)

photovoltaic array rating (kWp)

battery capacity (kWh)

confidence level (Monte Carlo)

LOLE (Monte Carlo)

0.5 0.5 0.8 0.8 0.95 0.95

0.021 0.021 0.010 0.010 0.002 0.002

7.7 12 7.7 16 7.7 18

40 10 100 10 140 10

159.59 65.21 194.10 37.11 240.98 33.49

0.6630 0.4767 0.9973 1 0.9973 1

0.0212 0.0345 0.0003 0 0.0001 0

Table 3. Input Parameters Used for System Optimization discount rate, d % diesel generator life, years photovoltaic system life, years battery bank life, years converter life, years cost of diesel generator, U.S.$/kW cost of photovoltaic system, U.S.$/kW cost of battery bank, U.S.$/kWh cost of converter, U.S.$/kW balance of system cost as a fraction of total capital cost operation and maintenance cost as a fraction of total capital cost diesel generator fuel curve coefficient a, liters/kWh diesel generator fuel curve coefficient b, liters/kWh

10 10 20 5 10 500 3 000 80 360 0.10 0.01 0.084 15 0.246

Figure 16. Sizing curves for the integrated system for a generator rating of 7.7 kW considering correlation between solar insolation and load (sizing curve represented by the dotted line account for the correlation effect).

levels for a specified fixed diesel generator rating of 7.7 kW (minimum load) representing the variation in photovoltaic array rating and the battery capacities. It may be noted that for lower array ratings and higher confidence levels, the effect of correlation is predominant. 6. System Optimization The identification of the design space helps the designer in choosing an optimum system configuration based on the desired objective. In this section, the selection of an optimum system configuration is discussed. The cost of energy (COE), which accounts for the capital cost as well as the operating cost associated with the system, is chosen as an appropriate economic parameter to evaluate and to optimize the system configuration. It is calculated as ACC þ AOM þ AFC ð15Þ COE ¼ Edem The annualized capital cost (ACC) is calculated as X ðC0i ÞðCRFi Þ ACC ¼

Figure 17. Variation in fuel cost for an integrated system with an addition in array capacity.

ð16Þ

annual energy delivered. To demonstrate numerically, example 1 has been considered. The cost data considered for the economic analysis are given in Table 3.3,34,35The system operating cost (the fuel cost for the diesel generator operation) is evaluated based on the optimal power dispatch of the generator. The fuel cost (FC) is calculated for a day as 24 X mf ðtÞ ð18Þ FC ¼ Cf

ð17Þ

where the hourly fuel consumption (liters/hour), mf(t) is related to the part load characteristic of the diesel generator. ð19Þ mf ðtÞ ¼ aPr þ bPðtÞ

t¼1

i

where ni

CRFi ¼

dð1þdÞ ð1þdÞni - 1

In the above equation, a and b are constants (given in Table 3), Pr is the rated power of the generator, and P(t) denotes the actual power generated by the diesel generator.36 In the proposed methodology, the generator dispatch is optimized

C0i is the capital cost of the ith system component (corresponding to the diesel generator, photovoltaic array, battery bank, balance of system, converter, and charge controller). CRFi is the capital recovery factor for the ith component, and it is a function of the discount rate (d) and life of the component (ni). The annual operating and maintenance cost of the system (AOM) has been taken as 2.5% of the capital cost. AFC is the annual fuel cost estimated based on the optimum dispatch of the generator, and Edem is the total

(34) Kolhe, M.; Kolhe, S.; Joshi, J. C. Energy Economics 2002, 24 (2), 155–65. (35) NREL. A Review of PV inverter technology cost and performance projections, www.nrel.gov/pv/pdfs/38771.pdf, December 2009. (36) Skarstein, O.; Uhlen, K. Wind Eng. 1989, 13 (2), 72–87.

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Table 4. Minimum Cost of Energy Configurations from the Design Space array rating (kWp) 83 9.3

generator rating kW 12 13.8 20.2

battery capacity kWh

cost of energy (U.S.$/kWh)

CO2 emissions (tonnes/year)

376 65 50

0.43 0.28 0.29 0.30

0 73.48 84.50 81.69

so that the system operates under the most energy efficient conditions. The objective is to minimize the total fuel cost for a given diesel generator rating. The controller decisions are based on the energy balance and the operating limits of the various components of the system. The optimum dispatch, P(t) has been determined satisfying the load balance conditions (eq 3) and charge constraints, eqs 4 and 5. It is further assumed that the generator has to operate continuously between the rated and a specified minimum value. The minimum loading on the generator is taken as 30% of the rated power. ð20Þ Pmin ePðtÞePr

uncertainty at the design stage is presented in this paper. The chance constrained programming approach has been adopted for system sizing under demand and resource uncertainty. For given random demand profile, resource profile, and system characteristics, a set of sizing curves may be plotted on the diesel generator rating vs storage capacity diagram for a given photovoltaic array rating meeting a specified reliability level. For a specified reliability level, a set of sizing curves may also be plotted on the photovoltaic array rating vs storage capacity diagram for a given diesel generator rating. The reliability levels given by the design space approach are found to be in agreement with that obtained through the Monte Carlo simulation approach. The methodology has been extended to incorporate the effect of correlation existing between the solar resource and demand. With an illustrative example, it is observed that for lower array ratings and higher confidence levels, the effect of correlation is predominant. The optimum system selection is considered based on the minimum cost of energy. The system sizing and optimization is carried out based on the solar resource and demand profile for an averaged day for the location. On the basis of this assumption for the remote location in south India, it is found that the optimum cost of energy of 0.28 U.S.$/kWh corresponds to a system comprising a photovoltaic array rating of 9.3 kWp, diesel generator rating of 12 kW, and battery capacity of 65 kWh. A recent study has estimated that the delivered cost of electricity (generated in a coal thermal power plant) in remote areas, located in the distance range of 5-25 km, is found to vary in a wide range from 0.06 to 4.62 U.S.$/kWh depending on the peak electrical load and load factor. For the location considered in this study, the cost of energy through grid extension would lie in the range of 0.07 to 0.55 U.S.$/kWh based on the distance from the existing grid and the expected plant load factor.37 Hence, the proposed hybrid system may be considered as an economically viable option.

The battery energy has to be between the rated and minimum values depending on the allowable depth of discharge. ð21Þ Bmin eQB ðtÞeBr The initial battery energy (at time t = 0) is also taken as an optimization variable. To obtain the economic optimum, the cost of energy is evaluated over the various configurations in the design space based on the expected values of the resource and demand. The selection of the optimum system is based on the minimum cost of energy. For a given generator rating, with an increase in the photovoltaic integration, the operating cost decreases owing to the reduction in diesel generator fuel consumption. As an example, for a generator rating of 3 kW, this variation is shown in Figure 17. It may be observed that the operating cost initially decreases rapidly and thereafter stays constant. It is observed that the minimum cost of energy for a given diesel generator rating corresponds to the system with a minimum array area and battery capacity. The minimum costs of energy configurations for different options are summarized in Table 4. It is also important to estimate the CO2 emissions for these options along with the cost to account the impact on the environment. Though the photovoltaicbattery system is the costliest option, it is a clean and sustainable means of power production. It may be noted that the cost factor does not always dictate the decision for the system selection. From Table 4, it may be observed that about a 10% reduction in CO2 is possible by opting for photovoltaic integration as compared to the diesel generator only system. The hardware costs associated with the components in the isolated system are not expected to change significantly with time but the fuel costs are subject to market uncertainties. For the present example, a 38% increase in the fuel cost will make the photovoltaic-battery system cost competitive with the diesel generator only system.

Acknowledgment. The first author is grateful to the Ministry of New and Renewable Energy, Government of India, for providing the financial support for the research work.

Nomenclature a = fuel curve coefficient, L/kWh b = fuel curve coefficient, L/kWh, B = battery bank capacity, Wh Bmin = minimum battery energy value, Wh Br = maximum battery energy value, Wh C0 = capital cost, U.S.$ Cf = fuel cost, U.S.$/L d = discount rate Edem = energy delivered, Wh f = net charging/discharging efficiency FC = daily fuel cost, U.S.$ h = loss of load duration in the estimation of hourly confidence level

7. Conclusions A systematic methodology for the sizing of integrated isolated systems involving photovoltaic arrays, a diesel generator, and a battery bank is presented in this paper. The concept of design space approach for the optimum system sizing of integrated photovoltaic -diesel generator-battery bank systems incorporating load uncertainty and solar resource

(37) Nouni, M. R.; Mullick, S. C.; Kandpal, T. C. Renewable Sustainable Energy Rev. 2008, 12 (5), 1187–1220.

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ηc = charging efficiency ηconv = converter efficiency ηd = discharging efficiency μD = mean demand, W μPP(t) = mean photovoltaic power, W μπ = mean of loss of load expectation σD = standard deviation of demand, W σπ = standard deviation of Loss of Load Expectation σPP(t) = standard deviation of photovoltaic power, W ξ = coefficient of variation of loss of load expectation

H = total duration considered in the estimation of hourly confidence level IT = solar insolation intensity on tilted surface, W/m2 mf = fuel flow rate, L/h n = life, years P = power generated by the diesel generator, W P* = net power generated by the diesel generator, W Pmin = minimum diesel generator power, W PP = power generated by the photovoltaic array, W Pr = rated diesel generator power, W QB = battery energy, Wh t = loss of load duration in the estimation of LOLE T = index of time in the load time series Tmax = maximum time considered in the expression for LOLE zR = standard normal variate with a cumulative probability of R

Abbreviations ACC = annualized capital cost, U.S.$/year AFC = annual fuel cost, U.S.$/year AOM = annual operation and maintenance cost, U.S.$/ year COE = cost of energy, U.S.$/kWh CRF = capital recovery factor DOD = depth of discharge LOLE = loss of load expectation

Greek Symbols η0 = photovoltaic system efficiency

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