Annals of Operations Research 132, 301–322, 2004  2004 Kluwer Academic Publishers. Manufactured in The Netherlands.

Locating Hybrid Fuel Cell–Turbine Power Generation Units under Uncertainty LAURA A. SCHAEFER [email protected] Mechanical Engineering Department, University of Pittsburgh, 643 Benedum Hall, Pittsburgh, PA 15261, USA ANDREW J. SCHAEFER [email protected] Department of Industrial Engineering, University of Pittsburgh, 1048 Benedum Hall, Pittsburgh, PA 15261, USA

Abstract. Hybrid gas turbine–solid oxide fuel cell power generation has the potential to create a positive economic and environmental impact. Annually, the U.S. spends over $235 billion on electricity, and electric utilities emit 550 million metric tons of carbon. The integration of distributed hybrid generation can reduce these emissions and costs through increased efficiencies. In this paper, a model is presented that minimizes the costs of distributed hybrid generation while optimally locating the units within the existing electric infrastructure. The model utilizes data from hybrid generation modules, and includes uncertainty in customer demand, weather, and fuel costs. Keywords: distributed generation, LCA, stochastic programming, integer programming

1.

Introduction

In recent years, distributed generation has received increasing attention from both the engineering and business communities. Distributed generation is defined as the placement of power generating modules of 30 MegaWatts (MW) or less near the end user (OFE, 2002). These modules can be used to entirely replace larger central power plants, or can be used for peak shaving and stand-by power. Distributed generation power modules may either be connected to the power grid or operated in isolated conditions. Many of the technologies being considered for distributed generation applications are attractive both economically and environmentally. Some common distributed generation technologies are reciprocating internal combustion engines, for applications of less than 10 MW; combustion turbines, for applications larger than 5 MW; microturbines, which can produce between 30 and 200 KiloWatts (kW); and fuel cells, which have the potential to generate power in the MW range (Kincaid, 1999). When solid oxide fuel cells (SOFCs) and turbines are combined into hybrid power generating systems, unprecedented levels of efficiency can be achieved. Within the past year, Siemens Westinghouse has produced a 220-kW hybrid power system that is capable of generating electric power at 55% efficiency. Additionally, by 2002, the EPA’s

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Environmental Science Center at Fort Meade will be powered by a 1-MW hybrid fuel cell–turbine plant. These prototypes already show great promise in increasing efficiency and lowering global warming gas emissions. It has been predicted that hybrid systems will be able to achieve efficiencies of 70–80%, and that hybrid plants will produce 50 times less nitrous oxide than current conventional gas turbines and 75% less carbon dioxide than coal-fired power plants (NETL, 2001). While it is likely that hybrid systems will indeed greatly reduce greenhouse gas emissions while producing economical power generation, the estimated savings that are commonly published do not account for the total economic impact of implementing distributed hybrid generation systems. The research outlined in this paper will focus on the analyses that must be performed in order to examine the economic feasibility and consequences of implementing hybrid fuel cell–turbine systems as distributed generation units. First, the optimal placement of the hybrid power modules must be considered so that they can be effectively interconnected with the existing electric infrastructure. Through data collection and mathematical modeling techniques, we have replicated a city, with its various industrial, commercial, and residential power needs, and its surrounding region for an urban area similar to Pittsburgh, PA. Pittsburgh is a mediumsized American city that traditionally generates much of its power using coal, but that has a well-established natural gas infrastructure. Additionally, many Pittsburgh electric customers are already exploring alternative power generation sources, including local generation and low-impact wind farms. The power needs of the various sectors (which are realistically distributed geographically) have been evaluated, and plants have been assigned a variety of generation potentials, and then placed so as to minimize costs (and, incidentally, transmission losses). Since the operation of power generation systems involves a significant amount of uncertainty, any model for optimally designing these systems must address this issue. Uncertainty can include varying supply and demand as well as factors such as the regulatory environment. The demand for different types of energy is also uncertain, and depends on factors such as economic growth, availability, and price relative to other types of energy. Other uncertain aspects include environmental impact, technological change, and available raw materials. Clearly, uncertainty is pervasive in the design and operation of power systems. Stochastic programming is an appropriate tool for many problems arising in the optimal design of power systems, including facility location and network design. Additionally, scenarios can represent various combinations of the uncertain aspects of energy problems. The model outlined above includes a variety of uncertain factors, such as power demand, raw fuel costs, and weather patterns. Energy security and network reliability issues have also been considered. As the demand for electricity grows, and as homeland security becomes a higher priority, the issue of energy security will become increasingly important. When distributed generation units fail due to either natural causes or deliberate actions, the integrity of the power

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grid must be maintained. It is possible that the use of distributed generation will be able to increase energy security without increasing electricity costs. Although the current model focuses on minimizing the economic impact, it is possible to adapt the model to also incorporate environmental factors. This adaptation could be implemented through the incorporation of a life-cycle assessment of a hybrid generation unit. This assessment must focus not only on the fuel cell and the gas turbine, but also on the complete system. During the operating life of the hybrid plant, fuel resource usage costs and waste disposal costs should also be incorporated. Whether evaluating hybrid generation solely on an economic or also on an environmental basis, the results from the model can be used to not only compare hybrid distributed generation with conventional power generation techniques, but also with “cuttingedge” technologies, such as photovoltaics and wind farms.

2.

Distributed generation and the electric industry

2.1. Deregulation The recent energy crisis in California highlights some of the problems faced by America’s existing electric infrastructure. As the electric industry has undergone deregulation and restructuring, blackouts, capacity shortages, and high prices have become more

Figure 1. Electricity deregulation by state.

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common, as have fears about power reliability. Figure 1 shows the current status of deregulation in the United States (Carner, 2001). Under deregulation, the transmission, generation, and distribution functions of the utilities are divided into separate and distinct businesses. This has the potential to create an active power trading market, and increase the attractiveness of distributed power generation (Hoffman et al., 1999). 2.2. Distributed generation When implemented in a partially or fully deregulated environment, distributed generation can be used to provide an alternative power source, reliable back-up power, or peak shaving. As an alternative power source, it is assumed that distributed hybrid generation would take the place of conventional high-emission central power plants that would otherwise be built. Distributed hybrid generation is also attractive as a source of stand-by power for companies that would incur large costs from the losses of productivity and profits that occur during blackouts. Utilizing distributed hybrid generation for peak shaving is attractive both economically and environmentally. As outlined in the previous subsection, not only do the highest electrical rates occur at peak times, but peak usage can also determine fixed charges for an entire year. Additionally, it is often the oldest and least efficient generators that must be utilized in order to meet peak power demands, thereby increasing emissions. 3.

Hybrid fuel cell–turbine power generation

As stated in the introduction, when SOFCs and turbines are combined, very high levels of efficiency can result. The properties of each of these units and the means by which they are integrated will be briefly explained. A SOFC operates at 800–1000◦ C, so it can utilize any hydrocarbon fuel (due to internal reformation) to produce electricity. A hybrid system operates by placing stacks of SOFCs on the high-pressure side of a cycle while a gas turbine operates as a bottoming engine. A simplified diagram of this interaction is given in figure 2 (Lundberg et al., 2001). Fuel is supplied to the fuel cell, a pre-fuel cell air heater, and a post-fuel cell exhaust combustor. If the fuel is natural gas, it must first pass through a desulfurizer. The turbine can directly generate AC power by means of an air-cooled generator, but the fuel cell’s DC power output must first be passed through a matrix of inverters. Although the electrochemical direct energy conversion that occurs in the fuel cell is more efficient than the combustion in the turbine section of the cycle, recent work suggests that reduced fuel utilization may have a negligible effect on the system efficiency while reducing costs (Haynes and Wepfer, 2000). Additional balance-of-plant components include the interconnecting piping, a programmable logic controller, backup fuel storage tanks, internal thermal insulation, water storage tanks, a startup boiler, an entering air filter, and an inert gas (usually nitrogen) safety system. The air filter is used to clean the air before it enters the compressor, and

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Figure 2. A hybrid power system.

the startup boiler is used to bring the fuel cell subsystem up to its operating temperature. For an accurate analysis, all of these components must be included with the fuel cell stack and turbine to determine the total system cost. Currently, directed research initiatives such as the Solid State Energy Conversion Alliance (SECA) have focused on the development of solid-state fuel cells. Through application flexibility and mass production, SECA aims to lower the cost of SOFCs to $400/kW within the next decade (Williams and Surdoval, 2000). Research is also being conducted through other programs into advanced turbine technology. 4.

Optimal power network design

Several issues regarding distributed power generation remain unexamined. One is the optimal location of the power plants. Power plants should be located near customers to minimize transmission losses. Transmission losses are proportional to the distance the power is transmitted. If power generators are close to customers, transmission losses will be smaller. Certain locations may not be permitted due to zoning restrictions, or may be prohibitively expensive. Closely related to the question of locating power plants is determining where each type of plant should be located. Different types of plants have different power generation capacities, different environmental impacts, and so on. For instance, a small generator may be desirable for a residential neighborhood, whereas a larger generator might be appropriate adjacent to an industrial park.

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Another issue is energy security and reliability. Distributed power systems should be at least as reliable as the existing power infrastructure. The goal is to minimize the fixed cost of constructing power generators plus the expected cost of delivering power to customers. This methodology could also consider the possibility of multiple generators failing. The result is a network design that can recover from individual power failures without a corresponding power disruption. The model presented in this paper can easily be extended to consider this case. Customer demand varies over time. Demand for power varies by time of day, day of the week, and time of year. In addition, random factors such as weather contribute uncertainty. The methodology we are proposing can consider uncertain and dynamic customer demand, minimizing the expected cost of generating and distributing electricity. Given the enormous costs of new power plants, the need for methods that can optimally determine a distributed power network and the best way to deliver power to customers is clear. One goal of this project is to demonstrate the advantages of distributed power generation. Using fuel cell technology, the efficiency of hybrid power systems increases dramatically. By making distributed power more reliable and efficient, this research will yield environmental benefits as well. We believe that by considering lower transmission losses and reliability the benefits of distributed power may be even greater than current expectations. We will demonstrate that it is possible to design reliable power networks that can meet uncertain and changing demands for power.

4.1. Deterministic power network design We first consider the case where customer demand is known with certainty, and each power generator is always available. Let I = {1, . . . , |I |} be the set of all possible locations for power plants, let J = {1, . . . , |J |} be a set of customers, and let K = {1, . . . , |K|} be a set of possible power plant configurations. Let xik be 1 if a plant of type k is located at i, and let it be 0, otherwise, and let cik be the cost of building a plant of type k at location i. This cost will include the initial construction cost, and the present value of any fixed operating costs. We assume that only one power plant type is allowed at any location. Let yij be the amount of power supplied from the plant at location i to customer j . Let dij be the per-unit cost of supplying power from a plant at i to customer j . This cost includes the present value of any variable operating costs and transmission losses. Let gk be the power capacity of a plant of type k, and let hj be the demand of customer j . The goal is to minimize the sum of the fixed costs plus the variable costs, i.e., min



i∈I k∈K

cik xik +



i∈I j ∈J

dij yij .

(1)

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This objective function is to be minimized subject to several sets of constraints. Each customer demand must be satisfied, so


yij = hj ,

∀j ∈ J.

(2)

i∈I

Only one type of power plant may be built at any location, so


xik
1,

∀i ∈ I.

(3)

k∈K

A power plant may only supply power if it has been built. Furthermore, it may not supply more power than its total capacity, so
j ∈J

yij





gk xik ,

∀i ∈ I.

(4)

k∈K

The variables yij must be nonnegative. This is a capacitated facility location problem, which is well studied in the optimization literature. The major difficulty is that the x variables are required to be either 0 or 1. However, it is possible to solve large-scale instances to optimality using integer programming techniques. In addition, many heuristics have been developed for capacitated facility location problems (Bramel and Simchi-Levi, 1997; Mirchandani and Francis, 1990).

4.2. Considering uncertain power demand The demand for power is not constant, and it is not known with certainty. Residential power demand is usually higher in the summer than in the spring. Industrial power demand may depend on regional economic growth. However, the vast majority of power network design models do not consider this inherent uncertainty. We believe that deterministic planning models are insufficient for long-range power planning problems. In such cases, the objective should be to minimize the current cost of constructing power facilities, plus the expected cost of delivering power to customers. Let ξ1 , . . . , ξr be a set of possible scenarios. A scenario is a complete description of a possible outcome of the randomness in the system. Initially we consider the case where a scenario describes each customer’s power demand. For any scenario ξ , with 1

r, let p be the probability that scenario ξ occurs. We assume that these scenarios are mutually exclusive, and together describe all possible states, so that sum of the p probabilities is 1.

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For any scenario ξ , let yij (ξ ) be the amount of power supplied from the power plant located at i to the customer located at j , let dij (ξ ) be the per-unit cost, and let hj (ξ ) be the demand of customer j under scenario ξ . The model then becomes min





cik xik +

i∈I k∈K

r


p





dij (ξ )yij (ξ ),

(5)

i∈I j ∈J

=1

subject to


yij (ξ ) = hj (ξ ),

i∈I




yij (ξ )


j ∈J







gk xik ,

∀j ∈ J, 1

r,

(6)

∀i ∈ I, 1

r,

(7)

∀i ∈ I,

(8)

k∈K

xik
1,

k∈K

xik ∈ {0, 1}, yij (ξ )  0,

∀i ∈ I, ∀k ∈ K, ∀i ∈ I, ∀j ∈ J, 1

r.

(9) (10)

This model minimizes the fixed cost plus the expected power distribution cost subject to several constraints. Constraints (6) require that each customer receive enough power to satisfy demand. Constraints (7) allow power to be generated by a generator of type k located at i only if such a generator was actually built. Constraints (8) require at most one generator to be placed at any location. This model is a two-stage stochastic integer program, and the formulation given in (5)–(10) is called the extensive form formulation. This problem can be solved by a Benders’ decomposition approach, called the L-shaped method (Benders, 1962; Van Slyke and Wets, 1969). This method decomposes the problem into two stages. The first stage determines where the generators are to be located. The second stage is solved for every scenario, and optimally delivers power for that scenario given the location of the generators. Information about the cost of this arrangement flows back to the first-stage problem. This procedure is iterated until an optimal decision is found. We illustrate this decomposition approach using the L-shaped method of Van Slyke and Wets (Van Slyke and Wets, 1969). Let θ represent the second-stage cost. The restricted master problem is as follows: min





cik xik + θ,

(11)

i∈I k∈K

subject to
k∈K

xik
1,

∀i ∈ I,

(12)

LOCATING HYBRID FUEL CELL–TURBINE





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gk xik
max

1

r

k∈K i∈I





h Eik xik




 hj (ξ ) ,

(13)

h = 1, . . . , H,

(14)

j ∈J

+θ e , h

k∈K i∈I

xik ∈ {0, 1},

∀i ∈ I, ∀k ∈ K.

(15)

Constraint (13) forces any solution to this restricted master problem to allow feasible solutions in the second stage. Clearly, any solution satisfying (13) must allow sufficient power to be generated for every scenario. This constraint eliminates the need for the feasibility cuts that are typically required in the L-shaped method. Let x¯ be a solution to this restricted master problem on iteration H . For every scenario 1

r, the following subproblem is solved:

dij (ξ )yij (ξ ), (16) min i∈I j ∈J

subject to




yij (ξ ) = hj (ξ ),

i∈I




yij (ξ )


i∈I




gk x¯ik ,

∀j ∈ J, 1

r,

(17)

∀i ∈ I, 1

r,

(18)

∀i ∈ I, ∀j ∈ J, 1

r.

(19)

k∈K

yij (ξ )  0,

Let πj  be the dual variables on constraints (17) and let σi be the dual variables on constraints (18). Then for all ri ∈I , all k ∈ K, and all scenarios , let Eik = r p (−σ g ) and let e = i k  j ∈J p πj  hj (ξ ). If the constraint in (14) cuts =1  =1 off the current solution, it is added to the restricted master. Otherwise, the solution is optimal and the algorithm terminates. The multi-cut version of Birge and Louveaux (1988) creates a theta variable for every scenario, and then creates a cut similar to (14) for every scenario. These cuts are added for those scenarios for which they are binding. Numerical experiments for stochastic linear programs indicate that the multi-cut version is preferred when the number of realizations r is not significantly larger than the number of first-stage constraints (Birge and Louveaux, 1988, 1997; Gassmann, 1990). However, Smith, Schaefer, and Yen (2002) found that when the first-stage problem is an integer program, the single cut version was preferable even with many fewer scenarios relative to the number of first-stage constraints. 5.

Data collection and generation

Our formulated city is based on a real urban area, and our model considers historical data as well as projections as to hybrid power generation capabilities. The model evaluates different network node placements, random power demands, and customer distributions,

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and considers how the system functions for each of these scenarios. Our belief is that power networks that explicitly consider these random factors will perform better than those that do not. 5.1. Fuel cell costs and capabilities Ten types of hybrid generation modules were selected, ranging from 250 kW to 10 MW in size. For each of these sizes, it is assumed that the gas turbine provides less than 25% of the power. It is also assumed that the generation capability of the units does not significantly decrease over the life of the plant. This is a reasonable assumption since individual fuel cells have been found to degrade at a rate of 0.1% for 1000 hours of life, and faulty or cracked cells can be replaced through routine maintenance (Forbes, 2001). Cost and efficiency estimates for hybrid power generation modules vary widely. The Solid State Energy Conversion Alliance (SECA) predicts that fuel cells will cost $400/kW by 2010 (Williams and Surdoval, 2000). After incorporating a gas turbine and balance of plant components, this cost rises to approximately $825/kW. Other sources predict hybrid unit costs of $1491/kW and $2236/kW (Khandkar, Hartvigsen, and Elangovan, 2000; MacKerron, 2000). One source predicted a cost of ¥526,000/kW (approximately $4015/kW), but this value has been treated as an outlying point and ignored for the purpose of this analysis (Tanaka, Wen, and Yamada, 2000). Maintenance costs should also be included in the first cost estimates, and have been calculated to have a present worth of 22% of the first costs (Riensche, Stimmung, and Unverzagt, 1998). The efficiency of hybrid units have been predicted as ranging between 64% and 75%. To simplify our analysis, we have assumed that higher efficiency systems will cost more than lower efficiency units. This is a valid assumption, since high efficiency systems may operate at higher pressures, which increases turbine and balance of plant costs. The correlation between efficiency and cost has also been demonstrated in the literature (Khandkar, Hartvigsen, and Elangovan, 2000). A summary of unit efficiencies and first costs (including maintenance) is provided in table 1. 5.2. Location costs In addition to the equipment and maintenance costs, land must also be purchased for the establishment of distributed generation units. Land costs will vary by location and zoning levels. For example, land zoned for industrial purposes that is far from the city center will have a significantly different cost than land in a central residential area. The general topology of each generated case was assumed to be fairly uniform. Table 1 Hybrid generation unit efficiencies and costs. Efficiency

75%

73%

71%

69%

67%

64%

Cost ($/kW)

2806

2400

1900

1550

1220

1000

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Furthermore, it is unlikely that potential residential, commercial, and industrial sites will overlap geographically. To account for this variation, a three-level staggered grid was developed for the placement of the possible hybrid generation units. The grid length scale was designed so that the maximum potential transmission loss from a plant to a consumer is 4% (Rastler, 2002). Recent real estate sales in the Pittsburgh area were examined, and representative land values were assigned to the grid points. Land sizes were selected based on current technology, in which a Siemens–Westinghouse 250kW hybrid generation unit has a footprint of 37.8 ft × 11.7 ft (Forbes, 2001). Industrial locations were found to range in value from $25,000 to $150,000, commercial sites from $65,000 to $350,000, and residential areas from $40,000 to $300,000. 5.3. Raw fuel costs Currently, hybrid power generation units are designed to be primarily fueled by natural gas. Over the past three years, natural gas prices in the United States have fluctuated widely. Gas prices in Pennsylvania have followed this national trend. In October 1999, an average residential customer in Pennsylvania paid $9.07 for a thousand cubic feet of gas. In October 2001, the same customer paid $12.06 (Kass, 2002). Gas prices also vary during the course of a year. During 2001, Pennsylvania residential consumers paid between $10.09 and $16.83 per thousand cubic feet. Different types of consumers also pay different rates. Residential users pay the highest rates, followed by commercial and then industrial customers. Electric utilities pay the lowest rate. (Electric utility costs occasionally rise above industrial rates.) Because of the large power generation levels that can be provided by hybrid generation, it will be assumed that hybrid units in residential areas can purchase power for commercial rates, units in commercial areas can purchase power for industrial rates, and units in industrial areas can purchase power at utility rates. Furthermore, using historical data, ten scenarios have been developed that will each occur with a given probability that determine the level of each rate (Perritt, 2002). These rates vary depending on the season and on the likelihood of a regular, inexpensive, or expensive natural gas market during each season. The probability of each scenario is directly correlated to these variations in the price of natural gas. These values are given in table 2. Table 2 Natural gas costs (dollars per thousand cubic feet). I = inexpensive, R = regular, E = expensive.

Residential rate Commercial rate Industrial rate Probability Scenario number

I

Winter R

E

I

Spring E

6.77 4.07 2.94 0.05 1

8.67 6.04 6.67 0.15 2

10.1 8.99 10 0.1 3

6.98 4.19 3.1 0.2 4

12.14 9.19 7.29 0.1 5

I

Summer R

E

R

Fall E

7.92 3.51 2.36 0.025 6

8.99 4.43 5.09 0.1 7

12.05 6.89 7 0.025 8

7.08 3.71 2.95 0.15 9

11.55 6.14 6 0.1 10

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5.4. Power costs To calculate the cost of generating electricity, a number of factors must be considered. These include the heating value of the natural gas, the efficiency of the hybrid module, the fuel utilization factor, and the transmission losses between the hybrid generation unit and each customer. The heating value of natural gas varies by location and by the nature of the reforming process. An average value of 1019 Btu per thousand cubic feet of natural gas was used for this analysis. Additionally, it is physically impossible for any cycle to perfectly convert one form of energy into another. This is quantified through the cycle efficiency η, which is given in table 1. The fuel utilization factor (FU) can be considered to be a design parameter in the creation of hybrid power modules. For every thousand cubic feet of natural gas that is provided to a hybrid power unit, only a fraction of that gas will actually be utilized (Appleby, 1993). In this analysis, a fuel utilization factor of 85% has been selected (Haynes and Wepfer, 2000). Finally, the distance between the generation unit and the customer also affects the cost of power. The transmission level varies between 95–99% of the generated power as a function of distance from the hybrid plant: TL = 0.99 − 0.01414 dist(i, j ),

(20)

where TL is the transmission level and dist(i, j ) is the normalized distance between plant i and customer j . Bringing all of these factors together, the cost of generating electricity for a given plant, scenario, and customer is: Transmission Cost ((Natural Gas Cost · 3412 Btu/kWh)/(1000 ft3 · 1019 Btu/ft3 )) . = TL · η · FU

(21)

So for a 71% efficient plant located at (0.2, 1.2) in a commercial zone operating under scenario 7, the cost of delivering power to a customer located at (1.9, 1.7) is 2.55  c/kWh. This cost includes fuel, maintenance, and time-averaged capital costs. Additional factors, such as profitability and the ability to remain competitive in an active market will further affect the final price that customers will pay. In 2000, the average U.S. residential customer paid 8.22  c/kWh, the average commercial customer paid 7.22  c/kWh, and the average industrial customer paid 4.46  c/kWh (Schnapp and Quade, 2001). 5.5. Additional costs In addition to economic costs, future work could also incorporate environmental emissions. These costs could be integrated through the assignment of carbon trading permit prices, or through environmental cost accounting. Life-cycle assessment (LCA) is a useful tool for evaluating the total environmental impact of a system. Instead of simply examining the emissions generated during a system’s working life, LCA also looks at the issues associated with the creation and disposal of the structure. Coupling LCA

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Figure 3. U.S. electricity consumption.

and environmental accounting creates a powerful tool for total system decision-making (Shapiro, 2001). The authors have already begun a comprehensive survey of LCA inventory information, and are in the process of cataloguing the data (Dicks and Martin, 1998; Inaba et al., 1997; Ippommatsu, Sasaki, and Otoshi, 1996; Itoh et al., 1994; Singhal, 1998; Ghiocel and Rieger, 1999; Tryon and Cruse, 2000; Manninen and Zhu, 1999; Yokokawa et al., 2001). 5.6. Customer demand National electricity consumption is steadily increasing. Figure 3 shows the national trend over the past twenty years (EIA, 1999). Electricity consumption in Pennsylvania and the Pittsburgh region has also increased over this time period, although greater fluctuations in the consumption levels can be seen due to varying population density. In 2000, Pennsylvania consumed over 98 billion kWh of electricity, and the Pittsburgh area consumed over 10 billion kWh (Dunn, 2001). The level of customer demand is a function of a number of factors, including weather, population growth and density, and socio-economic status. Based on historical data, the demand for each sector and each scenario was specified (Perritt, 2002). These demand levels (in million kWh) are given in table 3. For each customer, a type (industrial, commercial, or residential) was assigned and a random location and level of power consumption was generated. The percentage used by each type of customer was set to sum to one over the entire subset.

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Table 3 Electricity demand for the Pittsburgh region (106 kWh). Sector Industrial Commercial Residential

1

2

3

4

265.8 332.9 196.9

267.4 348.6 219.6

269.7 357.7 226.7

266.9 332.3 159.1

Scenario 5 6 278.8 357.6 169.9

282.2 394.6 211.0

7

8

9

10

286.6 410.4 241.5

296.0 428.1 250.6

281.9 366.4 176.6

285.7 396.1 219.7

5.7. Generation of data sets Six arrays were generated for each data set. The size of the evaluated data sets will be discussed in section 6. The first array specifies the number of demand points, supply points, generator types, and scenarios. The second array assigns a probability to each scenario. The third array is two-dimensional and gives the fixed costs for each type of generator at each location. The fixed cost is a function of the first cost, which is equal to the cost of the land plus the size of the generator multiplied by the capital cost per kW. The first cost is then multiplied by the a/p factor in order to evaluate the costs on a yearly basis. An interest rate of 8% and a plant lifetime of 15 years are assumed. The fourth array is a three-dimensional expression of transmission costs. For each potential supply location, the cost of delivering power to each customer under each scenario is tabulated. The methodology for calculating these transmission costs is given above. The fifth array gives the demand for each customer and scenario in kWh. Finally, the last array contains the transmission capacity of each type of generator. It is assumed that each hybrid power module can roughly operate 24 hours per day and 365.25 days per year. Routine maintenance may slightly decrease this availability. 6.

Results

6.1. Comparison of solution techniques Data sets of various sizes were generated and then solved using the approach outlined in section 4. Each instance was solved using three methods. The first was the single-cut L-shaped method of Van Slyke and Wets (1969). The second was the multicut L-shaped method of Birge and Louveaux (1988). The third method solved the extensive-form formulation directly. All of the mixed-integer and linear programs were solved using CPLEX version 7.0 (ILOG, 2001). We provide the results for the single cut, multicut and extensive-form (EF) algorithms for five instances in table 4. Two of the instances were medium-sized instances, and three were large instances. Recall that |I | is the number of potential supply points, |J | is the number of demand points, |K| is the number of types of generators, and r is

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Table 4 Comparison of solutions for non-penalized model.

M1 M2 L1 L2 L3

|I |

|J |

|K|

r

# EF rows

# EF cols

Single cut

Multicut

EF

VSS

EVPI

48 48 162 162 162

48 48 108 108 108

10 10 10 10 10

10 10 10 10 10

1008 1008 2862 2862 2862

23520 23520 176580 176580 176580

140 211 15615 15504 15275

136 235 16260 15424 21068

2056 647 58207 52235 71246

4.6% 6.0% 5.1% 5.1% 5.1%

6.5% 5.8% 7.4% 7.4% 7.4%

Table 5 Comparison of solutions for penalized model.

M1 M2 L1 L2 L3

Single cut

Multicut

EF

VSS

EVPI

3.1 5.4 42.1 37.8 38.5

15.0 24.9 117.1 75.5 55.1

2019.6 1059.0 94729.0 67679.7 56794.8

9.1% 5.5% 9.4% 9.4% 9.4%

3.8% 5.1% 5.6% 5.6% 5.6%

the number of scenarios. The times are in CPU seconds on a personal computer with an 866 MHz Intel processor. For each instance we calculated the “expected value of perfect information” (EVPI); that is, the amount the decision maker would pay to avoid uncertain demand. We also calculated the “value of the stochastic solution” (VSS); that is, the difference between the solution obtained by replacing the uncertain parameter with its expectation and the solution found by the stochastic program. The EVPI and VSS are presented in table 4 as percentages of the optimal solution to the stochastic program. Further information about EVPI and VSS is available in Birge and Louveaux (1997). The number of first-stage constraints is |I |+1, which is significantly greater than r, the number of scenarios, for all instances. However, the single cut algorithm has the fastest running time for three of the instances, and was within 3% of the multicut algorithm for the other two. This provides further evidence for the hypothesis put forth by Smith, Schaefer, and Yen (2002) that the single cut method is preferable for stochastic integer programs with continuous second stage even when the number of scenarios is small. We hypothesize that the single-cut procedure would perform even better relative to the other two methods as the number of scenarios increases. The extensive form would clearly become even less competitive as the number of scenarios increased, and the number of additional constraints required in the master would further burden the multicut approach. While it would have been interesting to investigate what happens with more scenarios from an algorithmic standpoint, our available data did not justify the additional scenarios without sacrificing realism.

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6.2. Modeling electricity demand with soft constraints We performed another set of computational experiments in which the energy requirements at the various demand points were relaxed, and any unmet demand was penalized. The penalties were set so that any additional electricity requirements could be purchased at the seasonal average price for each sector. These penalty costs are likely to be even higher during actual operation. The computational results are summarized in table 5. Several interesting observations about the penalty models can be made. The existence of penalties makes the decomposition approaches easier to solve, but not the extensive form. Furthermore, the single-cut approach always outperformed the multicut approach, providing further evidence that in stochastic integer programs, the single cut approach is generally preferable. For the three largest instances, in both the penalized and non-penalized cases, the VSS and EVPI did not differ much across instances. This is due to the fact that the instances only differed in network topology, and that the other underlying data were similar. 6.3. Hybrid generator placement For all of the data sets, the output level of the optimal solution closely matched the maximum total power requirement across scenarios. Furthermore, the majority of the hybrid generation units were chosen to be 10 MW in size, with the remaining sizes selected to equal the total power need. Upon examination, it was found that the first costs (including land and equipment costs) dominated the total cost, so the plant placement suggested by the algorithm is logical. Additionally, the least efficient plants were generally selected over more efficient units. Again, this is because the $/kW cost overshadowed the $/kWh cost of transmitting power. Interestingly, in some of the cases, industrial and commercial sites were chosen for the plant locations even when residential sites were available for less money. However, land costs were one to two orders of magnitude less than equipment costs, so this result could simply come from rounding within the optimization program. To further illustrate these results, two representative cases will be examined. In case 1, 48 supply points, 48 demand points, 10 scenarios, and 10 generator sizes were specified. This is considered to be a medium-sized case. The supply and demand points were evenly split between the industrial, commercial, and residential sectors. For the customer distribution shown in figure 4, where the most expensive land is assigned to point (0, 0), the optimal number and types of plants to build is shown in table 6. The maximum total demand across sectors is 9.7469 · 108 kWh, and occurs during scenario 8. As can be seen from the table, the total amount of power provided by the solution is 111.25 MW, or 9.7522 · 108 kWh per year, which closely matches the simulated city’s requirements. In case 2, a much larger number of supply and demand points were selected (the number of scenarios and generator types remained the same): 162 supply points and 108 customer sets were created. (This is considered to be a large instance.) Again, the

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Figure 4. Customer placement in case 1.

Table 6 Hybrid units proposed for case 1. Size

Efficiency (%)

Land cost ($)

Sector

250 kW 1 MW 4 MW 6 MW 10 MW 10 MW 10 MW 10 MW 10 MW 10 MW 10 MW 10 MW 10 MW 10 MW

64 67 64 67 64 64 67 64 67 64 64 64 64 64

80,000 150,000 40,000 120,000 150,000 120,000 58,500 58,500 25,000 25,000 300,000 200,000 98,340 65,000

Residential Industrial Residential Industrial Industrial Industrial Industrial Industrial Industrial Industrial Commercial Commercial Commercial Commercial

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Figure 5. Customer placement in case 2.

Table 7 Hybrid units proposed for case 2. Size

Efficiency (%)

Land cost ($)

Sector

250 kW 1 MW 10 MW 10 MW 10 MW 10 MW 10 MW 10 MW 10 MW 10 MW 10 MW 10 MW 10 MW

67 64 64 64 64 64 64 64 64 64 64 64 64

25,000 65,000 150,000 140,000 130,000 120,000 100,000 75,000 58,500 41,500 25,000 160,000 98,340

Industrial Commercial Industrial Industrial Industrial Industrial Industrial Industrial Industrial Industrial Industrial Commercial Commercial

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supply and demand points were evenly split between the industrial, commercial, and residential sectors. The locations of the customers are shown in figure 5. Thirteen plant locations were selected, ten of which were placed in industrial areas, and three of which were placed in commercial locations, as shown in table 7. Twelve of the thirteen plants operated at the lowest specified efficiency, while the remaining plant operated at the second to lowest efficiency. As in case 1, the total amount of power provided by the solution is 111.25 MW, which slightly exceeds the maximum amount of power required across the scenarios. 7.

Potential impact

As stated previously, America’s power consumption has grown rapidly over the past few decades (EIA, 1999). The majority of America’s electricity is produced by the combustion of fossil fuels (EIA, 1997). Because most conventional, coal-driven electric plants produce power with an efficiency of only 35%, annual carbon emissions from electric generation have also increased from 418 million metric tons of carbon in 1980 to nearly 550 million metric tons in 1999, as shown in figure 6 (EIA, 2000). With the Kyoto Protocol, global warming became an issue of interest to the public. By using top–down economic models, which include numerous debatable assumptions, economists have calculated the cost for reducing annual carbon emissions to be $100– $200 per ton of carbon. This is approximately equal to the current cost of fossil fuels, meaning the effective cost of energy would double, which is politically unacceptable (Passel, 1997).

Figure 6. Carbon generation from electricity production.

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Conversely, a bottom-up analysis that looks at high efficiency technology shows that the use of such technology can give a neutral or positive economic payoff (Jacard, Bailie, and Nyboer, 1996). Hybrid distributed power generation is one example of this type of technology. If the cost targets for SOFCs are met, hybrid systems will be able to produce electricity at a price equal to or less than current conventional power generation electric rates. Even a cost-neutral result would provide environmental benefits, since hybrid systems can greatly reduce greenhouse gas emissions. Furthermore, if carbon trading is enacted, a hybrid system’s lower emissions would also produce economic credits. 8.

Conclusions

This paper discusses a method for optimizing the placement of distributed hybrid fuel cell–turbine power generation. In order to address the challenges to distributed generation that may arise, it is important to understand hybrid generation’s true economic and environmental costs and benefits. It is also important to develop a flexible model for the optimal placement of distributed generation units so that they can be quickly implemented once the technology reaches a feasible cost and efficiency level. Distributed power systems have vast potential since they may have a smaller adverse environmental impact than current power systems. The potential reduction in greenhouse gas emissions is enormous, and it is likely that this reduction can be accomplished while decreasing consumer costs. The economic and environmental savings of distributed generation are not its only advantages, however. The research that this paper outlines also shows that such systems have further benefits. By placing power generation closer to customers transmission losses are reduced. Additionally, a distributed power network may be designed so that it can absorb the temporary loss of one generator without losing its ability to serve its customers. The research described in this paper can also serve as the basis for future work. One potential direction for future research is the exploration of more sophisticated fuel cell– turbine hybrid systems. These systems include pairing a SOFC with a low-temperature polymer electrolyte fuel cell (PEFC) such that either the SOFC acts as a reformer for the PEFC or the PEFC acts as a chemical bottomer for the SOFC (NETL, 2001). Additionally, using the fuel cell to produce hydrogen during minimal load times could be investigated. Finally, analyses could be performed to compare hybrid generation with technology that is currently considered marginal or cutting-edge, such as wind farms and photovoltaics. Acknowledgments This work was supported in part by NSF grant DMI-0217190 and by a grant from the University of Pittsburgh Central Development Fund. The authors wish to thank two anonymous referees, who have improved the clarity and content of this paper.

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