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Ind. Eng. Chem. Res. 2009, 48, 4908–4916
Optimum Design of Battery-Integrated Diesel Generator Systems Incorporating Demand Uncertainty Arun P., Rangan Banerjee, and Santanu Bandyopadhyay* Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India
A diesel generator system, integrated with battery, is one of the options for decentralized power production in remote locations. The principles of process system engineering along with the concept of design space approach for system design have been employed to optimize a battery integrated diesel generator system. The design space is the set of all feasible design options, generated using time series simulation of the entire system. In this paper, a methodology is proposed to appropriately size a battery integrated diesel generator system incorporating uncertainty associated with the demand. Chance constrained programming approach is combined with the design space approach to incorporate the demand uncertainty at the design stage. With a known distribution of the demand, the proposed methodology helps in identifying a sizing curve on the diesel generator rating versus storage capacity diagram for a specified reliability level. The system reliability values given by the proposed approach are validated using sequential Monte Carlo simulation approach through illustrative examples. Sets of sizing curves may be generated for different confidence levels as required by the designer. However, under the simplified assumption of a constant coefficient of variation for the demand, these sizing curves can be combined into a single curve. The cost of energy of the system is minimized to select the optimum configuration for different reliability levels. 1. Introduction A diesel generator is a common option for supplying power in remote and isolated locations. Isolated power systems form a subset of distributed generation and typically include generators in the range of 10-250 kW.1 Battery integration in such systems has proved to be useful as a load management option ensuring the efficient operation of the generator under varying load operations.2 In many diesel generator power plants located in remote areas of Australia, Alaska, and Kenya, battery integration has been proposed and implemented, with some of the systems having supplementary photovoltaic generation.3-5 Peak load is met by the diesel generator together with the inverter operating in parallel. It allows operation of the diesel generator at or near the full load condition ensuring its efficient operation. Process design and operation, incorporating uncertainty, is an important issue in process system engineering.6,7 In isolated power systems where the diesel generator and battery storage is considered as an option, the design objective is to obtain the ratings of the generator and the battery bank required to meet a given load pattern. Depending on the load and its variation over time, the ratings of the generator-storage system are selected. The load may be treated as a stochastic variable to realistically account for the associated uncertainty. This enables the quantification of the system reliability. Load uncertainty has been accounted in power system planning by the use of probabilistic models. In this paper, the principles of process design incorporating uncertainty have been extended to design diesel generator-based power system. Filho et al.8 used probabilistic load modeling for power system expansion planning using the Brazilian North-NorthEastern system load data. It was used to model the short and long-term peak load uncertainties. It was concluded that most active and reactive daily peak load uncertainties could be modeled using normal distribution.8 Zhai et al.9 described a method for analyzing the effect of load uncertainty on unit * To whom correspondence should be addressed. E-mail: santanu@ me.iitb.ac.in. Tel.: +91-22-25767894. Fax: +91-22-25726875.
commitment risk. The unit commitment risk was defined as the probability of having insufficient committed capacity to compensate for unit failures and/or unanticipated load variation. This risk was evaluated using the Markov model for unit failures and the Gauss-Markov load model for load uncertainty. Chiang et al.10 proposed a Gauss-Markov stochastic model of load and applied it for a chronological production simulation. It was demonstrated that the load uncertainty has a significantly larger effect on cost uncertainty than the unit availability uncertainty. Valenzuela et al.11 considered the effect of uncertainty associated with the load and the generator availabilities on production costing. Heunis and Herman12 modeled the load parameter uncertainty as a bivariate distribution of mean load current and its standard deviation. It was proposed for the analysis of distribution systems where primarily residential consumers were connected. Monte Carlo simulation method has found varied application in the reliability analysis of power systems.13 In the field of isolated power systems, Billinton and Karki14 illustrated a method to evaluate the reliability indices for isolated photovoltaic-diesel systems considering the uncertainty of the atmospheric variables and the diesel system outage. Bagen and Billinton15 presented a sequential Monte Carlo simulation technique for deciding the operating strategies for photovoltaicwind-diesel systems with storage, focusing on the system reliability and possible fuel savings. In this paper, a chance constraint programming16 approach is used to incorporate uncertainty in system design. To design a battery-integrated diesel generator system with specified system reliability, the uncertainty associated with the load has to be accounted at the design stage. In this paper, a methodology is proposed to optimally size a battery-integrated diesel generator system with uncertain load. It helps in constructing a set of “sizing curves” which connect generator ratings and the corresponding minimum battery capacities. The sizing curves are plotted on a generator rating versus battery capacity diagram, for a specified reliability level. It is desirable to determine the entire range of feasible system configurations,
10.1021/ie8014236 CCC: $40.75 2009 American Chemical Society Published on Web 04/02/2009
Ind. Eng. Chem. Res., Vol. 48, No. 10, 2009
Figure 1. Schematic of a battery integrated diesel generator system.
called design space, to meet a given load. Generation of design space helps in optimizing and selecting an appropriate system configuration. The concept of design space was proposed by Poddar and Polley17 for designing heat exchangers. The concept has been applied to design and optimize single equipment such as heat exchanger18,19 and distillation column.20 Kulkarni et al.21 extended the concept of design space to design and optimize a solar thermal system. Kulkarni et al.21-23 introduced the concept of design-space for optimum sizing of solar hot water systems. A methodology for generating the design space for battery integrated diesel generator system following a deterministic approach was illustrated by Arun et al.2 The concept of design space has been applied and demonstrated for other options for isolated power generation like photovoltaic-battery and wind-battery systems following a deterministic approach.24,25 In this paper, the design space approach is extended to incorporate the demand uncertainty on isolated power systems involving diesel generator and battery bank. The method helps in incorporating the demand uncertainty at the design stage to arrive at the complete set of feasible design options satisfying a specified reliability. The sizing curve depends on the load. Analyzing the effect of uncertainty in load would result in an improved system design. After establishing the design space, the overall system is optimized on the basis of the minimum cost of energy. For a typical remote location, the optimal cost of energy is obtained as 0.35 US$/kWh with a system consisting 8.3 kW diesel generator and 65.6 kWh of battery storage to meet the system confidence level of 0.5. On the other hand, to satisfy the confidence level of 0.99 for the same location, the optimum system consists of 10.2 kW diesel generator along with 80.9 kWh of battery storage and the associated cost of energy is increased to 0.37 US$/kWh. 2. Description of the Battery-Integrated Diesel Generator System Battery integration is useful in diesel generator-based isolated power systems when a varying demand profile has to be met.2 For systems with only diesel generators, generators are sized on the basis of the expected peak demand. In such a system, the generator operates at part load condition for a large duration of time. With battery banks integrated, the overall system efficiency is improved. Hybrid systems incorporating renewable energy-based units such as photovoltaic panels, wind generators, etc. may also be integrated in to the diesel grids with battery banks. The schematic of a diesel generator-battery system configuration is shown in Figure 1. The components include diesel generator set, battery bank, and bidirectional converter. A generic system may also include other renewable energy-based power generators such as photovoltaic, wind generators, etc.
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During the system operation, the diesel generator and the converter can operate in parallel. The operational characteristic of this power system has a major implication in the sizing of the diesel generator set and the battery bank. The load may be shared by the diesel generator and the inverter together. Thus, it is possible to have the rating of the diesel generator lower than the expected peak demand. This improves the capacity factor of the diesel generator as the generator operates at considerably higher load factors compared to the diesel-only mode. The generator can go off-line when the load goes below a preset value depending on the allowable minimum loading of the generator. The system operates in one of following three modes based on the generator dispatch strategy: (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) when the generator is allowed to be shut down. (ii) Diesel generator only operation: At certain periods, the system operation would be such that the generator simultaneously meets the load and charges the battery bank enabling it to attain its full capacity. (iii) Parallel operation: This corresponds to the peak load periods when both diesel generator and battery operate together to serve the load. Nayar4 has discussed the following advantages of the parallel configuration over other system topologies: • System load can be met optimally, as whenever the generator is operating, it is operating at full load or near full load conditions corresponding to higher efficiencies. • Diesel generator maintenance can be minimized by reducing the part load operation of the diesel engine. • Reduction in the rated capacities of the diesel generator and the battery bank while meeting the peak load can be achieved. 3. System Modeling and Generation of the Design Space A methodology to obtain the system sizing for the diesel generator-battery system for a given random load profile is discussed in this section. The hourly load is treated as a stochastic variable and the system sizing corresponds to a specified system reliability level. The minimum battery capacity for a given generator rating is determined conforming to a specified reliability value. To solve this optimization problem under uncertainty, a time series simulation approach, based on the energy balance of the overall system, is employed. The net power flow across the storage is accounted considering the efficiencies for the power conversion during charging and discharging processes. The rate of change of energy stored (dQB/ dt) in the battery bank is proportional to the net power generated. The net power generated is the difference between the power generated by the diesel generator (P(t)) and the power required by the random load (D(t)) at any point of time t. dQB ) (P(t) - D(t)) f(t) dt
(1)
where f(t) represents the efficiencies associated with the charging and discharging processes (including the efficiency of the bidirectional converter) at the time step t: f(t) ) ηc whenever P(t) g D(t) 1 f(t) ) whenever P(t) < D(t) ηd
(2)
The above equations relate the rate of change of stored energy with the input and demand power as well as the power
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conversion efficiencies during charging/discharging. The mean (µD(t)) and standard deviation (σD(t)) of the load is assumed to be known for the time step considered. For a deterministic demand, from the energy balance, the stored energy QB over a time period of ∆t may be expressed as QB(t + ∆t) ) QB(t) +
∫
t+∆t
t
(P(t) - D(t)) f(t) dt
(3)
may be daily, seasonal, or yearly load curve), the nominal charging and discharging efficiencies of the battery bankconverter system, and the desired confidence level. In the generalized methodology, to obtain the minimum generator rating, a numerical search is performed that satisfies the energy balance eq 9 and the following constraints: QB(t) g 0
∀t
(10)
For a relatively small time period, eq 3 may be approximated as
QB(t ) 0) ) QB(t ) Tmax)
QB(t + ∆t) ) QB(t) + (P(t) - D(t)) f(t) ∆t
Equation 10 ensures that the battery energy level is always positive, while eq 11 represents the repeatability of the battery state of energy over the time horizon of analysis. The repeatability condition implies that there is no net energy supplied to or drawn from the battery bank over the time horizon of analysis. It is assumed that the load is recurring in the same pattern after time Tmax. The required battery bank capacity (Br) is obtained as
(4)
To incorporate the demand uncertainty in the system sizing, the electrical load is treated as a random variable. The constraint16 related to the power production level at any time t will have a probability associated with it, due to the load randomness. The power generation, limited by the rated capacity of the generator, follows a random process as it has to meet a randomly varying load. Hence, the constraint related to the power production of the generator may be expressed as a chance constraint: prob[P(t) e Pr] g R
∀t
(5)
where Pr is the rated power of the generator and R the specified reliability level (0 e R e 1). It indicates that the power delivered by the system to meet the random load depends on the confidence level (R). On the basis of the system energy balance 4, the chance constraint 5 may be written as
[
prob D(t) -
]
QB(t + ∆t) QB(t) + e Pr g R f(t) ∆t f(t) ∆t
(6)
By rearranging eq 6, the random demand is brought to the right side of the term inside the square brackets: prob
[
]
QB(t + ∆t) QB(t) + Pr g D(t) g R f(t) ∆t f(t)∆t
(7)
Equation 7 is a chance constraint with the right side of the term inside the square bracket being a random variable. The corresponding deterministic equivalent is expressed relating the battery energy, input power, and random demand power as follows: QB(t + ∆t) e QB(t) + [Pr - µD(t) - zRσD(t)]f(t) ∆t
(8)
The battery energy over the time ∆t considering the net power of the generator P*(t) (including its part load operation) may be expressed as QB(t + ∆t) ) QB(t) + [P*(t) - µD(t) - zRσD(t)]f(t) ∆t (9) where zR is the value of standard normal variate with a cumulative probability of R. It may be noted that in deriving eqs 8 and 9, the load is assumed to follow a normal distribution with mean µD(t) and standard deviation σD(t). The assumption of normal distribution for the hourly load may be justified on the basis of the central limit theorem as the load to be met by the system arises from the cumulative effect of various independent random variables with finite mean and variance.8 The minimum generator rating required and the corresponding storage capacity for meeting the specified load and reliability level may be obtained by solving eq 9 over the entire duration. To solve eq 9, required inputs are the mean and standard deviation values of the load over a specific time interval (which
Br )
max{QB(t)} DOD
(11)
(12)
where DOD is the allowable depth of discharge of the battery bank. The proposed procedure provides the value of the minimum diesel generator capacity (Pr ) Pmin) and the corresponding capacity of the battery bank (B) for a desired confidence level. Any diesel generator rating, higher than the minimum, is capable of supplying the load for the given reliability level. However, the capacity of the battery bank would decrease due to the additional power generation from the generator. It is important from the designers’ perspective to identify all feasible combinations of the generator rating and the corresponding storage capacity. As the diesel generator rating is higher than the minimum possible, the system operation would be such that at certain periods there would be part load operation of the generator. Simulations to obtain the minimum storage capacity are carried out for different values of the diesel generator ratings (Pr > Pmin). For each value of Pr considered, the corresponding minimum battery bank capacity is obtained by minimizing the required storage capacity 12. 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 are plotted on a generator rating versus battery bank capacity diagram. This defines the sizing curve of the system for a given confidence level (R). For a load curve with time step ∆t and time horizon of Tmax, the confidence level (R) and the loss of load expectation (LOLE) is approximately related by the following expression: LOLE ≈
(1 - R)∆t Tmax
(13)
The above relation is valid for a load curve with a dominant peak demand. However, for a constant load curve, ∆t should be approximated by Tmax to calculate the LOLE of the system. The sizing curve represents the minimum storage capacity required for a given diesel generator rating. It divides the entire space into a feasible region and an infeasible region. The region above the sizing curve represents the feasible region as any combination of diesel generator rating and battery capacity represents a feasible design option satisfying the specified reliability. The entire feasible region including the sizing curve is the design space for a given problem. A typical sizing curve
Ind. Eng. Chem. Res., Vol. 48, No. 10, 2009
Figure 2. Typical sizing curve and the design space for a diesel generator-battery system for a given reliability level.
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tions from the design space are compared with those produced 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 diesel generator-battery bank configurations are selected from the design space. The selected system is simulated using the battery energy balance (eq 9). The hourly demand is considered as a random variable in the simulation. The random demand values used in the simulation are sampled from a normal distribution with specified mean and standard deviation for the time step. For comparison, data set is used such that the mean and standard deviation data of the hourly demand correspond to the same as used in the chance constrained model for deriving the sizing curve. The initial battery level is assumed to follow a uniform distribution based on the minimum and maximum energy capacity of the battery. The hourly simulations are carried out for the span of a year (using 8760 hly data points), 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 Rh ≈ 1 -
∑h H
(14)
where ∑h 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 horizon would correspond to the system confidence level (R). Further, the system LOLE is estimated as LOLE ≈
Figure 3. Flowchart for Monte Carlo simulation of the generic system for reliability estimation.
and associated design space are shown in Figure 2. Sets of sizing curves may be generated for different confidence levels as required by the designer. However, under the simplified assumption of a constant coefficient of variation for the demand, these sizing curves are coalesced into a single curve. This is explained in a subsequent section. 3.1. Monte Carlo Simulation Approach for Validating the System Reliability. The sequential Monte Carlo method helps in simulating the occurrence of random events over time, recognizing the underlying statistical properties of the system. Using this approach it is possible to model the time dependence of the random variables involved in the analysis of a system. Monte Carlo simulation method is used for validating the results obtained with the chance constrained model for system sizing. The methodology of random simulation is represented in Figure 3. The estimated values of confidence level and LOLE obtained from the Monte Carlo simulation for different system configura-
∑t Tmax
(15)
where ∑t corresponds to the total duration when there is loss of load over the entire time frame (i.e., QB(t) < Bmin) and Tmax is the total time frame of simulation (8760 h). The LOLE is computed at each iteration by randomly varying the hourly demand and the initial battery level. The simulation is terminated as the estimated LOLE value for a system achieves a specified degree of confidence. This provides a trade off between the accuracy level and the computation time after several iterations. The system reliability is estimated as the mean of the results obtained over the repeated simulations. The coefficient of variation (ε) of LOLE is used as the parameter for deciding the stopping criterion: ε)
σπ µπ
(16)
where µπ is the estimated mean of the LOLE and σπ is its standard deviation. The simulation is stopped when the value of coefficient of variation attains a reasonably steady value over different iterations. 3.2. Example 1: System Sizing for Specified Reliability Level. The methodology of system sizing and generation of the design space for different reliability levels is illustrated with an example in this section. The load data reported by Valenzuela et al.11 is used for the analysis. The hourly load and the standard deviation data have been appropriately scaled down to make it
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Figure 4. The daily load curve along with the hourly standard deviation for example 1.
Figure 6. Variations in load, generator power, and battery energy over the day for a typical system configuration having confidence level of 0.8 (Pr ) 16.7 kW, Br ) 36.6 kWh, LOLE ≈ 0.01). Table 2. Comparison of System Confidence Level and LOLE with Monte Carlo Simulation for Representative Configurations from the Design Space confidence LOLE level (analytical) 0.5 0.8 0.9 0.95 0.99
Figure 5. Sizing curve and design space for example 1 for different values of confidence levels. Table 1. Input Parameters Used in the System Sizing and Optimization net charging efficiency, % net discharging efficiency, % depth of discharge, % diesel generator fuel curve coefficients a, L/kWh b, L/kWh
85 85 50 0.08415 0.246
applicable for an isolated location. The daily load curve along with the hourly standard deviation values of the load are shown in Figure 4. Following the proposed methodology, the system sizing curves for different confidence levels are obtained and illustrated in Figure 5. The charging/discharging efficiencies and depth of discharge considered in the analysis are given in Table 1. The hourly load is assumed to follow a normal distribution. The system sizing, corresponding to R ) 0.5, matches the design when the load is taken as deterministic (based on the mean demand). For this case, the minimum generator rating is 15.5 kW and corresponding battery bank capacity is 33 kWh. The generator rating without any storage requirement is 17.5 kW (Figure 5). The generator rating without any storage corresponds to the maximum demand. Any system configuration located on the sizing curve or inside the design space will always meet the load balance conditions ensuring the specified confidence level, while any configuration in the infeasible region (i.e., left of the sizing curve in Figure 5) will not. It is observed that as the reliability levels are increased the sizing curve shifts toward higher system capacities. This may be explained from eq 9. As the system confidence level increases, the probability of the demand fluctuation increases the demand and hence, higher system capacities are required. To meet the reliability requirements subject to the uncertainty in demand, higher capacities of generator rating and battery bank are required as compared to the deterministic assumption of the demand. For example,
0.021 0.010 0.004 0.002 0.0004
generator battery rating capacity confidence level LOLE (kW) (kWh) (Monte Carlo) (Monte Carlo) 16.5 17.2 17.4 17.9 18.9
11.1 26.3 38.7 40.4 43.7
0.5726 0.9863 1 1 1
0.12 0.0004 0 0 0
with a confidence level of 0.99, the minimum generator requirement is 18.9 kW and the corresponding battery capacity is 44 kWh. The minimum generator rating required in this case without any storage is increased to 22 kW. Variation in load, generator power, and battery energy over the day for one of the feasible systems from the design space is illustrated in Figure 6. The generator power and the battery energy have been determined by simulating the system based on the load balance 4 considering the mean load and the equipment constraints. The load and generator power are expressed as a percentage of the rated capacity of the generator and the battery energy is expressed as a percentage of the rated capacity of the battery bank. The peak load occurring in the afternoon and the evening is met by the generator and the inverter together by parallel operation. The generator power and the battery energy shown correspond to the expected values of the hourly demand. But due to the variation in the load over the time step, the extra capacity in terms of generation and storage are required to meet the uncertain demand. This is provided by the system sizing. Thus, the system configurations in the design space are capable of meeting the random load and satisfy the required system reliability as well. The reliability levels given by the chance constrained model are validated through the sequential Monte Carlo approach. Specific configurations selected from the sizing curve are simulated to study its performance. The estimated values of confidence level and LOLE for the configurations selected from the sizing curve with the Monte Carlo simulation are given in Table 2 for specified reliability levels 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. This indicates that the proposed model offers a conservative design for the system. 4. System Optimization The sizing curves identify the design space that helps the designer in choosing an optimum system configuration based
Ind. Eng. Chem. Res., Vol. 48, No. 10, 2009 Table 3. Economic Parameters Considered for System Optimization discount rate, d % diesel generator life, years battery bank life, years converter life, years charge controller life, years cost of diesel generator, US$/kW cost of battery bank, US$/kWh cost of converter, US$/kW cost of charge controller, US$/kWh fuel cost, Cf, US$/L
10 10 5 10 10 555.6 88.9 400 7.78 0.84
0.5 0.8 0.9 0.95 0.99
24
11.1 26.3 38.7 40.4 43.7
103.1 103.9 104.1 105.0 106.7
ACC + AOM + AFC Egen
102.4 104.4 104.6 105.3 106.8
(17)
ACC is the annualized capital cost. 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 optimum dispatch of the generator, and Egen is the total annual energy to be delivered based on the expected demand. The annualized capital cost (ACC) of the system is calculated as ACC )
∑C
× CRFi
(18)
d(1 + d)ni (1 + d)ni - 1
(19)
i
0i
where CRFi )
(20)
where the hourly fuel consumption (L/h) m(t) is related to the part load characteristic of the diesel generator.
on the intended objective function for different reliability levels. In this section, the procedure for selecting an optimum system configuration is discussed. The cost of energy (COE), which accounts the capital as well as the operating costs associated with the system, is chosen as an appropriate economic parameter to evaluate and to optimize the system configuration. The COE is calculated as COE )
∑ m(t) t)1
generator battery operating cost operating cost rating capacity (based on average (based on Monte Carlo (kW) (kWh) load, US$/day) simulation, US$/day) 16.5 17.2 17.4 17.9 18.9
for the economic analysis are given in Table 3. The system operating cost (the fuel cost for the generator operation) is determined for the optimal power dispatch of the generator where the generator is operational throughout the time period of analysis and meets the expected demand. The fuel cost (FC) is calculated for a day as FC ) Cf
Table 4. Comparison of the Operating Cost with Monte Carlo Simulation confidence level
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C0i is the capital cost of the ith system component (corresponding to the diesel generator, battery bank, 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). To demonstrate the optimization procedure, example 1 has been considered. The cost data considered
m(t) ) aPr + bP(t)
(21)
In the above equation, a and b are constants (given in Table 1), Pr is the rated power of the generator, and P(t) denotes the actual power generated by the diesel generator.26 The objective is to minimize the total fuel cost for a given diesel generator sizing. The optimum dispatch P(t) has been determined satisfying the load balance conditions 4 and battery charge constraints (10-11). It is further assumed that the generator has to operate between the rated and a specified minimum value. In example 1, the minimum loading on the generator (Pmin) is taken as 30% of the rated power. Pmin e P(t) e Pr
(22)
The battery energy has to be between the rated and minimum values depending on the allowable depth of discharge. Bmin e QB(t) e Br
(23)
The initial battery energy (at time t ) 0) is also taken as an optimization variable. The system sizing is performed on the basis of the modified energy balance that incorporates the load uncertainty. The operating cost is evaluated on the basis of the expected demand. To evaluate and compare the actual variation in the operating cost with demand uncertainty, Monte Carlo simulation of the entire system has been performed. The comparison of the operating cost with normally distributed random demand and that obtained with the mean demand is given in Table 4. It shows that the assumption of expected demand for the operating cost is valid, as the difference between the costs is not significant. The optimum cost of energy for different reliability levels are evaluated. It is seen that overall minimum cost of energy is obtained for the diesel only system for all the reliability levels for this case. Variation in the optimum cost of energy with LOLE is shown in Figure 7 illustrating the increase in cost of generation with increase in reliability requirements. 5. Generalized Sizing Curve for Constant Coefficient of Variation of Demand In many locations the coefficient of variation of the hourly demand may not significantly vary with time. Under such a condition, different sizing curves corresponding to different confidence levels may be combined into a single sizing curve, and this may be utilized to arrive at different combinations of generator rating and battery capacities for various reliability levels. The coefficient of variation for the demand (xD) is given by
Figure 7. Variation of optimum cost of energy with LOLE for example 1.
xD )
σD(t) µD(t)
(24)
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Figure 8. Load curve for example 2.
Whenever the coefficient of variation of the demand is constant, eq 9 may be rewritten as QB′ (t + ∆t) ) QB′ (t) + [P′(t) - µD(t)]f(t) ∆t
Figure 9. Generalized sizing curve for system sizing under constant coefficient of variation of demand (example 2).
(25)
where QB′ )
QB 1 + x D zR
(26)
P′ )
P* 1 + xDzR
(27)
and
A single curve may be plotted between the variables Pr′ and B′ to represent the single sizing curve. This may be generated using the deterministic load. The system sizing for different reliability levels may be quickly determined from such a plot. For a given generator rating and reliability level, the corresponding values of Pr′ may be found from eq 28. Value of B′ may be obtained from the generalized curve. The required battery capacity may be calculated using the relation Br ) B′(1 + xDzR)
Figure 10. Sizing curve and design space for example 2 for different values of confidence levels and the optimum system configurations.
(28)
This is illustrated for by means of an example in the following subsection. 5.1. Example 2: Generalized Sizing Curve. The load curve for a typical remote location is represented in Figure 8.27 For this location, a constant coefficient of variation value of 0.1 is assumed for the hourly electrical demand. The generalized sizing curve on the B′ versus P′r diagram is shown in Figure 9. From this curve the battery capacity for any generator rating and LOLE value may be obtained. For example, for a generator rating of 14 kW and a confidence level R ) 0.99 (LOLE ≈ 0.0004) the required battery capacity may be determined from this plot. The value of P′r is obtained from eq 27 and is represented on the generalized sizing curve. The value of B′ is obtained corresponding to this value of P′.r The required battery capacity is found to be 38 kWh using eq 28. The optimum system configurations are also identified for the location based on the minimum cost of energy for different reliability levels. The optimum system configurations are represented in the design space (Figure 10). It is found that for this case, the optimum systems are the diesel generator-battery combinations corresponding to the minimum generator rating for all the reliability levels. The variation of the optimum cost of energy with LOLE is given in Figure 11. With the requirement of higher capital and
Figure 11. Variation of optimum cost of energy with LOLE for example 2.
operating costs for systems providing higher reliability levels, the cost of energy is expected to be higher. The exploration of the design space aids in the selection of an appropriate configuration considering the economic factors and the reliability requirements. 6. Conclusions Load uncertainty is an important issue in the design of isolated power systems. The concept of design space approach for the optimum system sizing of diesel generator-battery bank systems incorporating load uncertainty at the design stage is presented in this paper. Chance constrained programming approach has been adopted for system sizing under demand uncertainty. For given random demand profile and system characteristics, set of sizing curves may be plotted on the diesel generator rating versus storage capacity diagram for specified reliability levels for identifying the set of all feasible design configurations or design space. It offers flexibility in the overall design process as the complete set of feasible configurations satisfying the reliability require-
Ind. Eng. Chem. Res., Vol. 48, No. 10, 2009
ments is generated. The methodology of sizing and optimization are illustrated through representative examples. The approach is presented by assuming the hourly demand to follow a normal distribution. This assumption is based on the central limit theorem. However, the methodology is not limited by this assumption. The reliability levels given by the design space approach are found to be in agreement with that obtained through the Monte Carlo simulation approach. Comparison of the system reliability based on the values of system confidence level and LOLE are presented. The sizing curve depends on the load. Analyzing the effect of uncertainty in load has been considered to improve the system design. It has been demonstrated that to supply cost optimal power for a typical remote location, compared to the deterministic approach, the size of the diesel generator as well as the required battery requirement both increase by 23% with the confidence level of 0.99. Selection of optimum systems based on minimum cost of energy is illustrated. It is shown that for the condition of constant coefficient of variation of demand a generalized sizing curve may be generated. The generalized sizing curve enables the identification of system configurations corresponding to any desired reliability level from the single plot. A large number of engineering problems require that decisions to be made in the presence of uncertainty. Other than chance constraint programming approach, other methodologies such as stochastic programming,28-30 robust stochastic programming,31 fuzzy programming,32 possibility theory,33 stochastic dynamic programming,34 etc. have also been developed to addresses different optimization problems with uncertainty. Sahinidis35 has recently reviewed different optimization techniques with uncertainty. In this paper, the principles of process system engineering along with the concept of design space approach for system design have been employed to optimize a battery integrated diesel generator system incorporating chance constraint programming approach. In the proposed approach, the energy demand is the only random variable. The proposed methodology may be suitably generalized and extended to address system level design and optimization of different processes and with multiple uncertainties. The major problem associated with chance constraint programming with multiple uncertainties is the nonconvexity associated with the chance constraint.35,36 Present research is directed toward development of methodology to integrate the concept of design space with chance constraint programming having multiple uncertainties. Acknowledgment The first author is grateful to the Ministry of New and Renewable Energy, Government of India, for providing financial support for the research work. Nomenclature a ) fuel curve coefficient, L/kWh b ) fuel curve coefficient, L/kWh B ) battery bank capacity, Wh B′ ) transformed battery capacity variable used in the equation for generalized sizing curve, Wh Bmin ) minimum battery energy value, Wh Br ) maximum battery energy value, Wh C0 ) capital cost, US$ Cf ) fuel cost, US$/L d ) discount rate
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Egen ) energy delivered, Wh f ) net charging/discharging efficiency FC ) daily fuel cost, US$ h ) loss of load duration in the estimation of hourly confidence level H ) total duration considered in the estimation of hourly confidence level mf ) fuel flow rate, L/h n ) life, years P ) power generated by the diesel generator, W P′ ) transformed generator power variable used in the equation for generalized sizing curve, W P* ) net power generated by the diesel generator, W Pmin ) minimum diesel generator power, W Pr ) rated diesel generator power, W QB ) battery energy, Wh tj ) time interval of loss load in the expression for LOLE T ) index of time in the load time series Tmax ) maximum time considered in the expression for LOLE xD ) coefficient of variation of hourly demand zR ) standard normal variate with a cumulative probability of R Greek Symbols R ) confidence level Rh ) hourly confidence level ηc ) charging efficiency ηd ) discharging efficiency µD ) mean demand, W µπ ) mean of loss of load expectation σD ) standard deviation of demand, W σπ ) standard deviation of loss of load expectation ξ ) coefficient of variation of loss of load expectation Acronyms ACC ) annualized capital cost, US$/y AFC ) annual fuel cost, US$/L AOM ) annual operation and maintenance cost, US$/y COE ) cost of energy, US$/kWh CRF ) capital recovery factor DOD ) depth of discharge LOLE ) loss of load expectation
Literature Cited (1) Willis, H. L.; Scott, W. G. Distributed Power Generation Planning and EValuation; Marcel Dekker: New York, 2000. (2) Arun, P.; Banerjee, R.; Bandyopadhyay, S. Optimum Sizing of Battery Integrated Diesel Generator for Remote Electrification through Design-Space Approach. Energy 2008, 33, 1155. (3) Durand, S. J.; Bower, W.; Chapman, R.; Smith, G. The Alaska Energy Authority PV-Diesel Hybrid Assessment and Design Program. Proceedings of the 22nd IEEE Photovoltaic Specialists Conference, 1991, 646. (4) Nayar, C. V. Recent Developments in Decentralized Mini-Grid Diesel Power Systems in Australia. Appl. Energy 1995, 52, 229. (5) Sebitosi, A. B.; Pillay, P.; Khan, M. A. An Analysis of off-Grid Electrical Systems in Rural Sub-Saharan Africa. Energy ConVers. Manage. 2006, 47, 1113. (6) Zhang, L.; Xue, C.; Malcolm, A.; Kulkarni, K.; Linninger, A. A. Distributed System Design under Uncertainty. Ind. Eng. Chem. Res. 2006, 45, 8352. (7) Ricardez Sandoval, L. A.; Budman, H. M.; Douglas, P. L. Simultaneous Design and Control of Processes under Uncertainty: A Robust Modelling Approach. J. Process Control 2008, 18, 735. (8) Filho, M. B. D. C.; Leite Da Silva, A. M.; Arienti, V. L.; Ribeiro, S. M. P. Probabilistic Load Modeling for Power System Expansion Planning. Proc. 3rd Int. Conf. Prob. Methods Appl. Electr. Power Syst. 1991, 203. (9) Zhai, D.; Breipohl, A. M.; Lee, F. N.; Adapa, R. The Effect of Load Uncertainty on Unit Commitment Risk. IEEE Trans. Power Syst. 1994, 9, 510.
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Ind. Eng. Chem. Res., Vol. 48, No. 10, 2009
(10) Chiang, J.; Breipohl, A. M.; Lee, F. N.; Adapa, R. Estimating the Variance of Production Cost Using a Stochastic Load Model. IEEE Trans. Power Syst. 2000, 15, 1212. (11) Valenzuela, J.; Mazumdar, M.; Kapoor, A. Influence of Temperature and Load Forecast Uncertainty on Estimates of Power Generation Production Costs. IEEE Trans. Power Syst. 2000, 15, 668. (12) Heunis, S. W.; Herman, R. A Probabilistic Model for Residential Consumer Loads. IEEE Trans. Power Syst. 2002, 17, 621. (13) Billinton, R.; Li, W. Reliability Assessment of Electric Power Systems using Monte Carlo Methods; Plenum Press: New York, 1994. (14) Billinton, R.; Karki, R. Reliability/Cost Implications of Utilizing Photovoltaics in Small Isolated Power Systems. Reliab. Eng. Syst. Safety 2003, 79, 11. (15) Bagen; Billinton, R. Evaluation of Different Operating Strategies in Small Stand-Alone Power Systems. IEEE Trans. Energy ConVers. 2005, 20, 654. (16) Wendt, M.; Li, P.; Wozny, G. Nonlinear Chance-Constrained Process Optimization under Uncertainty. Ind. Eng. Chem. Res. 2002, 41, 3621. (17) Poddar, T. K.; Polley, G. T. Heat Exchanger Design through Parameter Plotting. Chem. Eng. Res. Des. 1996, 74, 849. (18) Muralikrishna, K.; Shenoy, U. V. Heat Exchanger Design Targets for Minimum Area and Cost. Chem. Eng. Res. Des. 2000, 78, 161. (19) Serna-Gonzalez, M.; Ponce-Ortega, J. M.; Castro-Montoya, A. J.; Jimenez-Gutierrez, A. Feasible Design Space for Shell-and-Tube Heat Exchangers Using the Bell-Delaware Method. Ind. Eng. Chem. Res. 2007, 46, 143. (20) Muralikrishna, K.; Madhavan, K. P.; Shah, S. S. Development of Dividing Wall Distillation Column Design Space for a Specified Separation. Chem. Eng. Res. Des. 2002, 80, 155. (21) Kulkarni, G. N.; Kedare, S. B.; Bandyopadhyay, S. Determination of Design Space and Optimization of Solar Water Heating Systems. Solar Energy 2007, 81, 958. (22) Kulkarni, G. N.; Kedare, S. B.; Bandyopadhyay, S. Design of Solar Thermal Systems Utilizing Pressurized Hot Water Storage for Industrial Applications. Solar Energy 2008, 82, 686.
(23) Kulkarni, G. N.; Kedare, S. B.; Bandyopadhyay, S. Optimization of Solar Water Heating Systems through Water Replenishment. Energy ConVers. Manage. 2008, in press. (24) Arun, P.; Banerjee, R.; Bandyopadhyay, S. Sizing Curve for Design of Isolated Power Systems. Energy Sustain. DeV. 2007, 11, 21. (25) Roy, A.; Arun, P.; Bandyopadhyay, S. Design and Optimization of Renewable Energy Based Isolated Power Systems. SESI J. 2007, 17, 54. (26) Skarstein, O.; Uhlen, K. Design Considerations with Respect to Long Term Diesel Saving in Wind/Diesel Plants. Wind Eng. 1989, 13, 72. (27) Fung, C. C.; Ho, S. C. Y.; Nayar, C. V. Optimization of a Hybrid Eenergy System Using Simulated Annealing Technique. Proc. IEEE TENCON 1993, 235. (28) Kall, P.; Wallace, S. W. Stochastic Programming; Wiley: New York, 1994. (29) Birge, J. R.; Louveaux, F. V. Introduction to Stochastic Programming; Springer: New York, 1997. (30) Sherali, H. D.; Fraticelli, B. M. P. A Modified Benders’ Partitioning Approach for Discrete Subproblems: An Approach for Stochastic Programs with Integer Recourse. J. Global Optim. 2002, 22, 319. (31) Mulvey, J. M.; Vanderbei, R. J.; Zenios, S. A. Robust Optimization of Large-Scale Systems. Oper. Res. 1995, 43, 264. (32) Delgado, M.; Herrera, F.; Verdegay, J. L.; Vila, M. A. Postoptimality Analysis on the Membership Function of a Fuzzy Linear Programming Problem. Fuzzy Sets Syst. 1993, 53, 289. (33) Tong, S. Interval Number and Fuzzy Number Linear Programming. Fuzzy Sets Syst. 1994, 66, 301. (34) Bertsekas, D. P.; Tsitsiklis, J. N. Neuro-dynamic Programming; Athena Scientific: Belmont, MA, 1996. (35) Sahinidis, N. V. Optimization under Uncertainty: State-of-the-Art and Opportunities. Comput. Chem. Eng. 2003, 28, 971. (36) Sen, S.; Higle, J. An Introductory Tutorial on Stochastic Linear Programming. Interfaces 1999, 29, 33.
ReceiVed for reView September 22, 2008 ReVised manuscript receiVed November 11, 2008 Accepted March 11, 2009 IE8014236
