Energy Efficiency Implies Cost Efficiency? Revisiting Design and

Operations of Combined Heat and Power Systems

Wenbin Wang†, Owen Q. Wu∗, Gilvan C. Souza∗

† Shanghai University of Finance and Economics, Shanghai, China, wang.wenbin@shufe.edu.cn

∗ Kelley School of Business, Indiana University, Bloomington, IN 47405, owenwu, gsouza@indiana.edu

May 29, 2020

The combined heat and power (CHP) technology produces both heat and electricity from a single fuel input,

achieving an efficiency (total useful energy output divided by fuel input) as high as 90%. Because the high

energy efficiency is achieved by utilizing exhaust heat, it is a common practice to design a CHP system to

match the thermal demand it serves. Energy efficiency, however, does not necessarily imply cost efficiency

when fuel price is variable. In this paper, we study the problem of optimizing the design (including capacity

and power-to-heat ratio) and operations of a CHP system for an industrial firm facing variable fuel price.

We identify two drivers for improving the overall cost efficiency of CHP: 1) the flexibility of operating the

CHP system at various output levels in response to the fuel price, enabled by properly over-sizing the CHP

system relative to the heat demand, and 2) the flexibility of co-operating the CHP system and the legacy

boiler. Numerical examples with realistic settings demonstrate that our recommended CHP system design

and operations can yield substantial long-run cost savings for the firm. The results of this paper call for

revisiting the current practice of designing CHP systems, as recommended by the government and industry

associations.

Key words: Combined heat and power (CHP), energy efficiency, fuel price variability, capacity investment

1. Introduction

Combined Heat and Power (CHP), also referred to as cogeneration, is the concurrent production of

electricity and useful thermal energy, for heating and industrial processes, from a single source of

energy. According to the U.S. Department of Energy (DOE 2016), there are 83 gigawatts of CHP

capacity in the U.S., accounting for 8% of the total electricity generation capacity in the U.S. and

producing 12% of all electricity generated. The estimated CHP technical potential in the U.S. is about

240 gigawatts, which can supply over 30% of the total electricity and meet a significant portion of

the thermal needs. Different types of CHP technologies exist for different applications: reciprocating

engines, gas turbines, steam turbines, microturbines, fuel cells, among others. According to the

Department of Energy CHP Installation Database (https://doe.icfwebservices.com/chpdb/),

72% of all CHP capacity in the U.S. is natural gas-fired, among which 71% is configured in a

combined cycle (explained below). Descriptions for other CHP technologies are provided by the

Environmental Protection Agency (EPA 2017).

Figure 1 schematically shows a typical natural gas-fired CHP in a combined cycle configuration,

which corresponds to 51% of all CHP capacity in the U.S. and is the topic of this paper. Starting from

the top left, natural gas is burned with compressed air, and the resulting high-pressure gas drives

a turbine connected to a generator to produce power. A heat recovery steam generator (HRSG)

uses the hot (800 to 1100F) exhaust gas to heat water into high-pressure steam. A portion of that

steam is used in manufacturing processes or for heating purposes, whereas the other portion is used

at a steam turbine to generate more electricity. The percentage split between electricity and steam

Figure 1: Typical Natural Gas Combined-Cycle CHP System

Combustion

chamber

Generator

Condenser

Heat recovery steam

generator (HRSG)

Natural gas

Fresh air

Hot exhaust gas

Cooling water

Exhaust

AC

power

AC

power

Water pumpWater

Steam

Heat

CompressorGas

turbineGenerator

Steam

turbine

1

output is configured during the CHP design, and is one of the key decisions we study in this paper.

This design decision determines the CHP’s power-to-heat ratio (total power output divided by the

steam output), an important measure in practice. The spent low-pressure steam is condensed and

pumped back to the HRSG. Typical applications for such large capacity CHP include universities,

hospitals, and industries with high power and heat demands, such as chemical plants.

Because of the heat recovery, the energy conversion efficiency of CHP, defined as the produced

energy (useful heat and electricity) divided by the energy input, is typically around 65 to 90%,

which is considerably higher than the typical 50% achieved by separately producing electricity by a

natural gas power plant and producing steam via a separate boiler (EPA 2017). In addition to higher

efficiency, other cited advantages of CHP include lower local and greenhouse gas emissions, reduced

electricity transmission losses, and higher power reliability (Athawale and Felder 2014, Blankinship

2004, Hampson et al. 2013). Thus, from both economic and environmental considerations, CHP

offers significant benefits, and is a technology that should be encouraged.

CHP plants are typically “local”—they deliver electricity and heat (which cannot travel too far)

to facilities in the vicinity. The vast majority of CHP capacity in the U.S., 97%, is privately owned

either by the firm where the CHP is located, or by a third-party that contracts with the firm (DOE

2012). This paper aims to optimize CHP design and operations from an industrial firm’s perspective.

The most critical cost driver in operating CHP for a firm is the fuel price. Figure 2 shows the

monthly average natural gas prices for industrial establishments in Indiana, which display consider-

able variability. Long-term fixed-price natural gas contracts are possible only with a high premium,

as described in DOE (2017, p. 48): “While it may be possible to get a [fuel] supplier to enter a fixed

Figure 2: Monthly Industrial Natural Gas Prices in Indiana 1989-2017 (Source: www.eia.gov)

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Na

tura

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s P

rice

($

/mm

Btu

)

2

price contract for the useful life of the project, this is unlikely and usually very expensive.” Thus,

firms typically partially hedge fuel price risk through short-term contracts and futures contracts (e.g.,

Henry Hub monthly natural gas futures), which remove price uncertainty but leave the firm exposed

to price variability over the lifetime of CHP. This paper’s primary focus is to analyze how an energy

efficient technology, CHP, should be implemented in practice under fuel price variability.

An important design decision for a CHP system is its power-to-heat ratio. Several methods are

used in practice to achieve various power-to-heat ratios (EPA 2017). For example, one can choose

the size of the steam turbine that uses a portion of the steam to generate electricity, delivering the

rest for manufacturing processes. Another method is to use an extraction steam turbine that has one

or more openings in its casing for extracting a portion of the steam at some intermediate pressure

for manufacturing processes.

To guide the design and installation of CHP, government and industry associations often rec-

ommend CHP as a replacement for legacy boilers (EPA 2013) and designing the CHP to match

the thermal demand: “It is generally recommended to size a CHP system based on a site’s thermal

load” (EPA 2017, p. 3-6). This is backed by engineering studies, e.g., Joskow and Jones (1983) and

the references therein. Intuitively, since the thermal output (steam) is used locally, oversizing CHP

relative to the thermal demand seems inefficient. This is also supported by DOE (2016, p. A-3):

“sizing to the thermal load leads to the most efficient CHP system operation.”

The energy conversion efficiency, however, is not equivalent to operating cost efficiency, due to

natural gas price variability. When the natural gas price is high, a CHP system designed to match

the steam demand must continue to run to support the firm’s operations. As a result, the CHP may

generate electricity at a cost higher than the electricity price charged by the utility company. The

CHP system cannot reduce its electrical output because the power-to-heat ratio is fixed at the design

phase. The inability to react to the natural gas price fluctuations prompts us to rethink the CHP

system design and operations.

In this paper, we consider the problem of optimally sizing, configuring, and operating CHP for a

private firm. As seen in the next section, only portions of this problem have been considered in the

literature, while we propose an integrated approach. We find that the optimal CHP system design

may be matching CHP’s maximum power with electrical demand while oversizing CHP’s thermal

capacity relative to the steam demand. Furthermore, the leggacy boiler, if not too inefficient, can

reduce the CHP’s exposure to the fuel price spikes. Thus, our research calls for a revisit to the

current CHP design recommendations provided by, for example, EPA (2013, 2017) and DOE (2016).

3

2. Literature Review

The academic literature on CHP can be categorized as (i) optimal design and operation of CHP

systems, (ii) inclusion of CHP in a utility’s portfolio optimization, and (iii) CHP adoption and related

regulatory incentives. There is also related literature on optimal capacity and operation of other types

of sustainable technologies, such as waste-to-energy, fuel-saving technologies and renewable energy.

Furthermore, CHP operation is also related to the literature on co-production with proportional

production technology. We review each stream, and position our paper accordingly.

In terms of CHP design, Joskow and Jones (1983) propose a simple economic model considering

fixed and operating costs to study a firm’s investment in cogeneration, compared to investing only

in a conventional boiler and purchasing electricity from the grid. Using a sensitivity analysis, they

show that the profitability of investing in cogeneration is highly dependent on electricity, fuel prices,

and steam demand. Wickart and Madlener (2007) also consider the two alternative investments, but

explicitly model electricity and fossil fuel prices as stochastic processes, and use an options-based

framework to numerically decide on the timing and type of investment. Unlike our paper, these

models do not consider the optimal size of the CHP or its configuration. In terms of optimal CHP size,

Gibson et al. (2013) find the optimal size of a steam turbine CHP under different scenarios of carbon

pricing in Australia. On optimal operations, there is some research that proposes mathematical

programming models to minimize the operating cost of a CHP while satisfying the total energy

demand; see Cho et al. (2014) for a review.

There is a significant body of literature on the optimization of portfolio of generation technologies

(e.g., Banal-Estanol and Micola 2009, Chao 1983), with more recent literature including renewable

energy and associated incentives (e.g., feed-in tariffs) or operational issues related to intermittency

(e.g., Kok et al. 2018, 2019, Aflaki and Netessine 2017, Alizamir et al. 2016, Carley et al. 2017).

These papers do not explicitly consider cogeneration. Other research explicitly include CHP in the

portfolio, showing its benefits (e.g., Kwun and Baughman 1991, Weber and Woll 2006, Sundberg and

Henning 2002). Malguarnera and Razban (2015) consider the economics of investing in a CHP for a

utility, assuming that the CHP only produces the baseload heat for an industrial user, thus always

operating at full capacity. There is also research on the design of contracts between a utility and

an independent CHP generator for selling excess power generated by CHP (e.g., Daniel et al. 2013,

Hall and Parsons 1990). Different from these papers, we focus on optimal sizing, configuration, and

operation of a single CHP facility for a private firm.

Regarding CHP adoption, Mueller (2006) examines the relatively low adoption of CHP among

4

firms, and determines that regulatory uncertainty plays a large role in the low adoption. Athawale

and Felder (2014) empirically study the “CHP gap” and show that CHP facilities have considerably

lower average capacity factors, between 47% and 62%, than the 80% or higher used in investment

decisions. They propose an assurance payment to the CHP owners in the situation of low capacity

utilization. Zhang et al. (2016) study the effectiveness of different policy instruments (capital grants,

utility credits) in reducing the payback period for investments in CHP in buildings. In our model, we

explicitly consider the optimal operating policy (hence its expected capacity factor) in determining

the CHP investment.

Our paper is also related to the research on optimal investment decisions in sustainable technolo-

gies. Ata et al. (2012) consider the optimal operation of waste-to-energy firms, with two revenues

sources, waste generators and renewable energy consumers, under different policy mechanisms. A

stream of research considers a firm’s investments in a portfolio of alternative technologies to meet

random demand, where one technology is more sustainable with (typically) lower fuel consumption

but higher investment costs (e.g., Drake et al. 2016, Hu et al. 2015, Wang et al. 2013). In this

paper, the cleaner technology, CHP, is more economical to operate when the fuel price is not too

high. Furthermore, in the industrial settings we consider, there are two demand streams (heat and

electricity) that may be met by co-operating CHP and legacy technologies.

Finally, there is literature on multiproduct firms with proportional production technology, where

a primary input gives rise to multiple products in fixed proportions, such as cocoa beans transformed

into cocoa liquor, butter, and powder. For a review, see Chen et al. (2013). This literature, however,

focuses on random yields, demand uncertainty (Boyabatli 2015), or product substitution (e.g., Hsu

and Bassok 1999). In our case, there is no random yield in electricity and heat production using

CHP, and electricity and heat are not substitutes or stored.

3. The Model

3.1 Combined Heat and Power (CHP) System

We consider a situation where a manufacturing firm that uses both steam and electricity in its

primary process is considering installing a natural gas-fueled CHP system on its premise. The firm

needs to decide (i) the size of the CHP system, k ≥ 0, which is the maximum natural gas input rate,

and (ii) the configuration parameter z ∈ [0, 1], which is the proportion of steam used directly in the

manufacturing process. After the CHP system is up and running, the firm also needs to decide (iii)

the actual fuel input rate a ∈ [0, k] based on the fuel price and demand.

In accordance with Figure 1, the Sankey diagram in Figure 3 shows the energy flows of a typical

5

Figure 3: Energy Flows of a CHP System

Power: 1 −

Energy loss: 1 − 1 −

Power: 1 −

Steam:

Energy loss: 1 − 1 −

Steam:

Input

1

, , ∈ 0,1

∈ 0,1

CHP with combined cycles. For clarity of exposition, from this point onward, we measure both

electrical and thermal energy using the same unit (e.g., MWh). As illustrated in Figure 3, one unit

of natural gas input first produces β units of steam and (1−β)µ1 units of electricity. A proportion z

of the β units of steam is sent to manufacturing processes, while the remaining steam drives a steam

turbine to produce an additional β(1 − z)µ2 units of electricity.

For a given CHP project, it is reasonable to assume that the performance parameters, β, µ1, and

µ2 are fixed. In most practical situations, µ1 > µ2, because the electrical efficiency of the gas turbine,

(1−β)µ1, is on par with µ2, the electrical efficiency of the steam turbine (Masters 2013), and β < 1.

Thus, we assume µ1 > µ2 throughout our analysis.

The proportion of processing steam, z ∈ [0, 1], can be configured at the time when the CHP

facility is built. If z = 1, all steam is sent to manufacturing processes. If z = 0, all steam is used to

generate electricity, and the CHP becomes a combined-cycle power generator. Thus, the two decision

variables at the design and installation phase are the CHP system’s size, k ≥ 0, and the proportion

of steam for direct use, z ∈ [0, 1]. In practice, z can indeed be configured in a continuous range,

especially if an extraction steam turbine is used to extract a desirable portion of processing steam

(see the discussion in §1). Although the size choice is discrete, due to the maturity of the CHP

technology, many sizes are available for a given application. In our analysis, we assume that k and

z are continuous decision variables.

Let xe ∈ (0, 1) and xs ∈ [0, 1) respectively denote the electrical and thermal energy (subscript s

for steam) produced from one unit of energy input (refer to Figure 3):

xedef= (1− β)µ1 + β(1− z)µ2 and xs

def= βz. (1)

As z ∈ [0, 1], we have xe ∈ Xe ≡ [xe, xe], where xe ≡ (1 − β)µ1, xe ≡ (1 − β)µ1 + βµ2, and

6

xs ∈ Xs ≡ [0, xs], where xs ≡ β. Furthermore, xe and xs are linearly related:

xe = xe − µ2xs and xs =xe − xe

µ2. (2)

The definition in (1) also allows us to treat xe (or xs) as a decision variable in lieu of z, which

facilitates the analysis in §4. In sum, the design decisions are k and z (or xe or xs).

The last decision is about CHP operations. Once the CHP system is installed (i.e., k and z

are fixed), the firm needs to decide the actual input rate a ∈ [0, k] in response to the fuel prices

and possible demand changes. Recent CHP systems have outstanding part-load performance: the

efficiency changes very slightly when operating at part load (see, e.g., https://www.gentec.cz/en/).

We assume that operating at part load will not affect the CHP system’s efficiency. Thus, the

electricity and steam output rates, axe and axs, are proportional to the input rate a.

To model part-load efficiency loss and size-dependent efficiency, we can extend the model to

include a fixed energy input required for running CHP. In such an extension, our state-dependent

policy for operating the CHP system (and the legacy boiler) achieves more cost-savings than the

must-run policy of a CHP system that matches the thermal demand (thus must incur the fixed

operating cost all the time), which strengthens our results. For expositional simplicity, we present

the model without fixed operating cost.

3.2 Demand and Cost Structure

The firm currently obtains electricity and steam from separate sources to meet its demands.

Demand : The firm’s demands for electricity and steam are assumed to be stationary processes,

represented by random variables De and Ds (measured in MW) with stationary distributions. To

clearly illustrate the main insights, the main part of our analysis focuses on the case of constant

demands, and we consider fluctuating demand in the numerical analysis in §5.

Separate steam production cost : The firm currently produces steam using a natural gas-fired

boiler, which is a typical way of generating steam for industrial processes. Producing 1 MBtu of steam

requires γ > 1 MBtu of natural gas, i.e., the efficiency of the boiler is 1/γ. The firm purchases natural

gas from a gas supplier on a variable-price contract, i.e., the contract price is adjusted monthly. This

reflects the reality that most large commercial and industrial users are exposed to natural gas price

variability; refer to Figure 2 and the discussion in §1. Similar to the demand processes, we assume

that the natural gas price process is stationary, represented by a random variable Pg (measured in

$/MWh; 1 MWh = 3.412 MBtu) with a stationary distribution. We assume the firm does not have

a natural gas storage facility, which is realistic for most firms.

Electricity cost : The firm purchases electricity from the utility at a fixed price pe (in $/MWh),

7

which is the regulated price set by the regulatory entity (e.g., state utility commission). In contrast

to the monthly-adjusted natural gas price, it is common that the regulated electricity price pe is

much more stable over the years, because any rate adjustment requires regulatory approval.

Energy cost without CHP : In the absence of CHP, given the realized demand rates, de and

ds (MW), and the electricity and natural gas prices, pe and pg ($/MWh), the firm’s (hourly) energy

cost is simply the sum of the costs from the two separate sources: depe + dsγpg.

CHP investment and maintenance cost : Constructing and maintaining a CHP system of

size k > 0 costs vf + vk, where vf is the fixed cost for constructing the facility and v is the variable

cost per unit of capacity (discounted to the time of investment). This two-part cost model reflects

the cost structure in reality; see, e.g., EPA (2017) for the construction cost for CHP of various sizes.

CHP operations and energy cost : With the presence of CHP, the energy cost depends on

how the CHP is operated and whether the boiler is retired. For a given CHP input rate a ∈ [0, k],

the hourly input fuel cost is apg. Unmet electricity demand, (de − axe)+ can be met by the utility

at the regulated price pe. Overproduced electricity, (axe − de)+, may be sold back to the utility at a

predetermined price re ≤ pe. Note that the case of re = pe corresponds to the net-metering policy in

some states, whereas re = 0 means that the utility will not credit the firm for any excess electricity.

Some states allow the energy to be sold back to the grid at re ∈ (0, pe).

Because CHP is typically designed to match thermal demand, retiring the legacy boiler is common

in practice. In this paper, however, we look for the optimal investment by exploring all possible

options, including keeping the legacy boiler operational. Let indicator 1b denote whether the legacy

boiler remains operational (1b = 1) or is retired (1b = 0). When 1b = 1, CHP can output steam

axs < ds and the remaining demand, (ds − axs)+, can be met by the legacy boiler at cost of γpg

per unit of energy. If 1b = 0, we must have axs ≥ ds. In either case, the overproduced steam,

(axs − ds)+, has no value. All notations are summarized in Table 1.

3.3 Formulation of the CHP Operations and Design Problem

Given the cost parameters in §3.2, the firm considers investing in a CHP to supply the energy for

its manufacturing process. The firm needs to decide the capacity of the CHP system, k, and the

proportion of processing steam, z (or equivalently, xe or xs, as discussed in §3.1). Once the CHP is

installed, the firm operates the CHP over a T -period horizon. At the beginning of each period (e.g.,

month), the firm observes the realized natural gas price pg and the demand rates de and dg, and

decides the input rate a. Within each month, pg, de, and ds remain constant. The planning horizon

T is typically at least 180 months (15 years) or longer.

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Table 1: Notations

Decision variables:

k Input capacity of CHP, i.e., the maximum natural gas input rate, one-time decision

a Input rate, a ∈ [0, k], per-period decision

z Proportion of steam used for manufacturing processes, one-time decision

xe(z) Electrical energy generated by one unit of input, an equivalent decision variable for z

xs(z) Steam energy generated by one unit of input, an equivalent decision variable for z

Engineering parameters:

β Proportion of input energy converted into steam in the first stage

(1− β)µ1 Efficiency of the first-stage power generation

µ2 Efficiency of the second-stage steam turbine

γ Amount of natural gas needed for the legacy boiler to produce 1 unit of steam

Demands:

De, de Demand rate for electricity and its realization

Ds, ds Demand rate for steam and its realization

Prices and costs:

Pg, pg Price of natural gas for industrial users and its realization

pe Regulated price of electricity paid by the firm to the utility

re Price of excess electricity that the utility buys back from the firm

vf , v Fixed and variable capacity investment and maintenance cost, discounted to period 1

We first consider how the firm should optimally set the input rate a ∈ [0, k] in response to the

realized demand, de and ds, and natural gas price pg. Under a given CHP design (k, xe), the firm

chooses the input rate a ∈ [0, k] by minimizing the (hourly) operating cost:

C(k, xe;1b, pg, de, ds)def= mina∈[(1−1b)ds/xs,k

](de − axe)

+pe − (axe − de)+re + (ds − axs)

+γpg + apg

. (3)

The firm’s current cost is C(0, xe; 1, pg, de, ds) = depe+ dsγpg, which is the energy cost without CHP

discussed in §3.2. When 1b = 0 and k < ds/xs, we set C(k, xe; 0, pg, de, ds) = ∞ as it involves a

infeasible range for a. This means that when boiler is retired, we must have k ≥ ds/xs, since CHP

is the only source of steam.

The CHP design problem aims to find the optimal capacity k and configuration xe that minimize

the sum of the investment cost and the total expected discounted operating cost:

mink≥0, xe∈Xe

vf1k>0 + vk +

T∑

t=1

δt−1E[nC(k, xe;1b, Pg,De,Ds)

], (4)

where δ ∈ (0, 1) is the discount factor per period and n is the number of operating hours per period.

In optimizing (4), we consider two cases: 1b = 0 and 1b = 1. We do not treat 1b as a decision

variable, because the retirement decision also depends on other factors, such as the salvage value

9

of the retired boiler. Our analysis will demonstrate that the optimal CHP design may be different

depending on whether the boiler is retired or not and that the current industry practice has not

explored all possible options.

In the remainder of this section, we rewrite the problem in (4) for analytical convenience. In

(4), the expected operating cost per period, E[nC(k, xe;1b, Pg,De,Ds)

], is time-invariant. Defining

N ≡∑T

t=1 δt−1n = n(1−δT )

1−δ , we can rewrite (4) as

mink≥0, xe∈Xe

NE[C(k, xe;1b, Pg,De,Ds)

]+ vf1k>0 + vk. (5)

Without investing in CHP, the firm will incur an expected operating cost of NE[Depe + DsγPg

].

Thus, (5) is equivalent to maximizing the expected net long-term benefit of a CHP system:

maxk≥0, xe∈Xe

NE[B(k, xe;1b, Pg,De,Ds)

]− vf1k>0 − vk, (6)

where B function represents the operating benefit of CHP, defined as

B(k, xe;1b, pg, de, ds)def= depe + dsγpg − C(k, xe;1b, pg, de, ds). (7)

The problem in (6) formalizes the cost-benefit tradeoff involved in constructing and operating a

CHP system. The joint optimization of (k, xe) in (6) can be solved as a sequential decision problem by

either (A) optimize the size k for every given configuration xe and then optimize xe, or (B) optimize

the configuration xe for every given size k and then optimize k. Interestingly, these two approaches

complement each other in the following way. Approach A identifies an initial set of candidates for the

optimal CHP design, but optimizing xe is very difficult in both approaches. However, approach B

can effectively narrow down the candidates identified by approach A and provide conditions under

which certain CHP design is optimal.

Before we analyze the problem in detail, we impose the following condition to ensure that the

optimal solution is finite:

Assumption 1 NE [(xere − Pg)+] < v.

If Assumption 1 does not hold, then the firm can configure the CHP to produce electricity only, sell

back to the grid, and still make profit. This unrealistic situation is precluded by this assumption.

4. Optimal Sizing and Configuration of a CHP System

As the main insights of the paper can be obtained from a model without demand variability, we

assume that the demand is constant, de and ds, throughout this section. Industries with continuous

processing and high steam requirements (e.g., pulp and paper mills and basic chemical processing)

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have steady demand for electrical and thermal energy, and applications to these industries are good

economic targets for CHP deployment (EPA 2017). Assuming constant demand also removes the

dependence of CHP design on demand variations, allowing us to focus on analyzing the effects of

market and technology factors on CHP design decisions. To provide a comprehensive analysis, our

numerical analysis in §5 includes studies with demand variability.

4.1 Overview of the Analysis

Our analysis proceeds as follows. First, for any given CHP design (k, xe), we solve for the optimal

CHP operations problem in (3). To this end, we consider four regions shown in Figure 4, depending

on whether the CHP can overproduce steam and/or electricity. Second, we use approaches A and B

discussed after (7) to identify the candidates for the optimal size and configuration (k∗, x∗e) and

compare them with the design choices often adopted in the current industry practice. Third, we study

the market conditions under which certain candidates for the optimal CHP design dominate or are

dominated. Four candidate designs are discussed in detail. Finally, we consider the situation when

the capacity choice in practice is not continuous and provide additional insight on CHP configuration.

Figure 4: CHP Design Space: k ≥ 0 and xe ∈ [xe, xe] or xs ∈ [0, β]

, input capacity

, electricity

output per

unit of input

Ω

≤

≤

Ω

≥

≤

Ω

≥

≥

Ω

≤

≥ = (1– )"

= (1– )"+ "

=

= 0

, steam

output per

unit of input

0

Before diving into optimization, it is useful to eliminate sub-optimal solutions.

Lemma 1 It is not optimal to design a CHP with configuration xek > de and xsk > ds. That is,

the optimal CHP design (k∗, x∗e) must satisfy x∗ek∗ ≤ de or x∗sk

∗ ≤ ds.

Lemma 1 states that it is never optimal to install a CHP system that can overproduce both steam

and electricity. (All proofs are included in the appendix.) Thus, we only need to focus on regions

Ω1, Ω2, and Ω3 in Figure 4. Furthermore, if the legacy boiler is retired (1b = 0), then the CHP must

satisfy all the steam demand (kxs ≥ ds), implying that we only need to focus on the curve kxs = ds

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and region Ω3 when 1b = 0.

4.2 Optimal Operations of the CHP Generator

In this section, we solve the optimal operations problem in (3). The optimal input rate a∗ stays

constant within each period (month), and so does the hourly benefit function; both are given in the

following proposition.

Proposition 1 (Optimal CHP Operations) For a given CHP design (k, xe) and realized fuel

price pg, the optimal input rate for the CHP and the associated operating benefit are given as follows.

In all expressions, xs is a function of xe, expressed in (2).

Case 1: The boiler remains operational (1b = 1):

(i) If (k, xe) ∈ Ω1, then

a∗ =

k, if (1− xsγ)pg ≤ xepe,

0, otherwise,(8)

and the operating benefit of CHP is

B(k, xe; 1, pg, de, ds) = kxe

(pe +

xsxe

γpg −pgxe

)+

. (9)

(ii) If (k, xe) ∈ Ω2, then

a∗ =

k, if (1− xsγ)pg ≤ xere,

dexe, if xere < (1− xsγ)pg ≤ xepe,

0, if (1− xsγ)pg > xepe,

(10)

and the operating benefit of CHP is

B(k, xe; 1, pg, de, ds) = de

(pe +

xsxe

γpg −pgxe

)+

+ (kxe − de)

(re +

xsxe

γpg −pgxe

)+

. (11)

(iii) If (k, xe) ∈ Ω3, then

a∗ =

k, if pg ≤ xepe,

dsxs, if xepe < pg ≤ xepe + xsγpg,

0, if (1− xsγ)pg > xepe,

(12)

and the operating benefit of CHP is

B(k, xe; 1, pg , de, ds) = ds

(γpg +

xexs

pe −pgxs

)+

+ (kxs − ds)

(xexs

pe −pgxs

)+

. (13)

Case 2: The boiler is retired (1b = 0):

(iv) If kxs = ds and (k, xe) ∈ Ω2, then a∗ = k and the operating benefit of CHP is

B(k, xe; 0, pg, de, ds) = depe + dsγpg + (kxe − de)re − kpg. (14)

12

(v) If (k, xe) ∈ Ω3, then

a∗ =

k, if pg ≤ xepe,

dsxs, if pg > xepe,

(15)

and the operating benefit of CHP is

B(k, xe; 0, pg, de, ds) = ds

(γpg +

xexs

pe −pgxs

)+ (kxs − ds)

(xexs

pe −pgxs

)+

. (16)

The benefit functions in Proposition 1 can be intuitively explained as follows. In region Ω1, to

fulfill one unit of electrical demand, the CHP needs an input of 1/xe and produces xs/xe units of

steam, incurring an input cost of pg/xe and bringing cost savings of pe+xsxe

γpg. Thus, the net benefit

of fulfilling one unit of electrical demand is pe +xsxe

γpg −pgxe

. If this net benefit is positive, the firm

should meet as much electrical demand as possible using CHP. The expression in (9) and the first

term in (11) represents the benefit of fulfilling electrical demand. After satisfying de, the net benefit

of overproducing electricity is re +xsxe

γpg −pgxe

. Thus, in region Ω2, the CHP should operate at full

capacity (thus overproducing electricity) if and only if this benefit is positive. This corresponds to

the second term in (11). Using the same logic, in region Ω3 when overproducing steam is possible,

we can similarly interpret (13) and (16), in which γpg +xexs

pe −pgxs

is the net benefit of fulfilling one

unit of steam demand.

When the boiler remains operational (1b = 1), Proposition 1 states that the CHP should not run

(a∗ = 0) if and only if pg > xsγpg + xepe, seen from (8), (10), and (12). Intuitively, the CHP should

not operate when its input fuel cost pg exceeds the marginal benefit—the reduced input cost for the

boiler, xsγpg, and reduced cost of purchasing electricity, xepe. In the case where the boiler is retired

(1b = 0), the CHP should always run to meet the steam demand.

Because the theoretical analysis on optimizing CHP design with boiler retirement is much simpler

than the case with boiler operations, in the rest of this section, we focus on the latter case and define

B(k, xe;Pg) ≡ B(k, xe;1b = 1, Pg , de, ds).

In §5, we will numerically study and compare both cases.

Proposition 1 prescribes that the optimal operation of the CHP follows a price-control-band

policy: the CHP should be running at full capacity, partial capacity, or not running when the

natural gas price falls in the respective price bands. The following corollary shows how these price

bands vary with the design decision xe.

Corollary 1 Suppose the boiler remains operational. The CHP configuration xe (or xs) affects the

optimal price bands as follows:

13

(a) Suppose xs < γ−1. Then, as xs decreases (or xe increases),

(i) In Ω1, the full-run price band[0, xepe

1−xsγ

]shrinks, and there is no partial-run;

(ii) In Ω2, the full-run price band[0, xere

1−xsγ

]and the partial-run price band

[xere1−xsγ

, xepe1−xsγ

]shrink;

(iii) In Ω3, the full-run price band [0, xepe] expands, while the partial-run price band[xepe,

xepe1−xsγ

]

shrinks.

(b) Suppose xs ≥ γ−1. Then, a CHP configured in Ω1 ∪ Ω2 should always run at full capacity.

For a CHP configured in Ω3, the full-run price band [0, xepe] expands as xe increases, while the

partial-run price band[xepe,∞) shrinks as xe increases.

In practice, CHP usually cannot generate more steam per unit of input than the legacy boiler,

i.e., β < γ−1, which ensures xs < γ−1 in Corollary 1. It is also possible that β > γ−1 when the

legacy boiler is very inefficient. In such case, both xs < γ−1 and xs ≥ γ−1 are possible.

Corollary 1(a) (i) and (ii) imply that, in Ω1 ∪ Ω2, the CHP operation increases as xs increases.

Intuitively, in Ω1 ∪Ω2, the steam output of CHP brings a benefit that co-fluctuates with the natural

gas price, partially offsetting the negative impact of high fuel price. Thus, for larger xs, the CHP

runs for more months and runs at full capacity more frequently. Interestingly, Corollary 1(a) (iii)

implies that, in Ω3, the CHP operates at full capacity more frequently as xe increases. This is because

the overproduced steam by a CHP with configuration in Ω3 has no benefit. The partial-run price

band still shrinks, since the steam demand is exactly met by CHP running at partial capacity. These

insights will be useful for understanding the optimal design in the following analysis.

4.3 Optimizing Capacity k for Given xe (Approach A)

We begin with considering the optimal capacity k for any given configuration xe:

B(xe)def= max

k≥0NE

[B(k, xe;Pg)

]− vf1k>0 − vk. (17)

The following proposition characterizes the objective function in (17).

Proposition 2 (Marginal benefit of capacity) (i) The expected benefit function is piecewise lin-

ear and concave in k with the following slopes within the interior of each region:

∂E[B(k, xe;Pg)]

∂k=

E[(xepe + xsγPg − Pg)

+], (k, xe) ∈ Ω1,

E[(xere + xsγPg − Pg)

+], (k, xe) ∈ Ω2,

E[(xepe − Pg)

+], (k, xe) ∈ Ω3,

E[(xere − Pg)

+], (k, xe) ∈ Ω4.

(18)

(ii) Under given xe (or xs), the optimal CHP size k∗ ∈0, de/xe, ds/xs

.

14

Figure 5: Optimal capacity k for given configuration

Candidates for optimal capacity under given

=

= (1–)

=

= 0

=

= (1–)

=

= 0

Capacity

Capacity

Ω Ω

Ω

Ω

Ω

Ω

0

0

=

=

(a)

(a)

(b)

(c)

(c)

(d)

Ω

Proposition 2 (ii) narrows down the candidates for the optimal CHP design to either matching

electrical demand or matching steam demand. Figure 5 top panel illustrates these candidates when

the firm’s steam demand is relatively high, so that the curves kxe = de and kxs = ds do not intersect

(i.e., Ω3 does not exist); Figure 5 bottom panel shows the case where the two curves intersect.

As discussed in §1, the current industry practice is to size the CHP to match the firm’s steam

demand. However, the main takeaway of Proposition 2 is that matching steam demand is not the

only candidate for the CHP design. Actually, in region Ω3, if the slope E[(xepe − Pg)

+)]> v, i.e.,

the CHP is an economic source of electricity even if it overproduces steam, then matching electricity

demand is preferred over the industry practice of matching the steam demand.

To further analyze when matching electrical or thermal demand is desirable, we need to solve

for the optimal configuration problem by maximizing B(xe) defined in (17). Unfortunately, B(xe)

does not exhibit properties amenable to analysis or interpretation. In fact, if the optimal capacity

matches electrical demand, the expected operating benefit E[B(k = de

xe, xe;Pg)

]is neither convex nor

concave in our numerical experiment. Neither does E[B(k = ds

xs, xe;Pg)

]exhibit useful properties.

The next section carries the analysis forward using approach B and sheds light on the optimal CHP

configuration choices.

15

4.4 Optimal CHP Designs (Approach B)

In this section, we consider choosing the configuration xe under given CHP input capacity k:

maxxe∈Xe

E[B(k, xe;Pg)

]. Approach B furthers the analysis in three ways:

1) We show that E[B(k, xe(k);Pg)

]is convex in k when xe(k) is chosen along the segments (a),

(b), and (c) in Figure 5, which allows us to further narrow down the candidates for the optimal

design. Note that approach A is still essential to our analysis, because approach B alone cannot

identify the initial candidate set shown in Figure 5.

2) Approach B identifies the market conditions under which certain CHP design options are dom-

inating or dominated. We will discuss market conditions for not installing CHP, installing CHP

that matches steam demand, and installing CHP that matches electrical demand.

3) In practice, the choice of capacity k is not as continuous as the choice of xe. Thus, designing a

CHP system requires optimization of xe for given k, which falls under approach B.

The rest of this section addresses the above three areas of analysis in sequence.

4.4.1 Candidates for the Optimal CHP Design

First, we show that the benefit function evaluated along the segments (a), (b), and (c) in Figure 5

is convex in k, as summarized in the following lemma.

Lemma 2 Segment (a): For de/xe ≤ k ≤ minko, de/xe

, E

[B(k, de/k;Pg)

]is convex in k;

Segment (b): For ds/β ≤ k ≤ ko, E[B(k, xe − µ2ds/k;Pg)

]is convex in k;

Segment (c): For k ≥ maxko, ds/β

, E

[B(k, xe − µ2ds/k;Pg)

]is convex in k, with limiting slope

limk→∞

d

dkE[B(k, xe − µ2ds/k;Pg)

]= E

[(xere − Pg)

+].

The convexity in Lemma 2 implies that we only need to consider the endpoints of the segments

(a), (b), and (c) as candidates for the optimal CHP design. Furthermore, the benefit function along

segment (c) has a slope increasing in k but no more than E[(xere−Pg)

+]< v (Assumption 1), which

precludes an infinite capacity to be optimal. The benefit function along segment (d) in Figure 5 is

not necessarily convex or concave. We will discuss this segment in detail later. Proposition 2 and

Lemma 2 lead to the following set of candidates for the optimal CHP design.

Proposition 3 (CHP candidates) The optimal CHP capacity and configuration choice set is

(k∗, x∗e) ∈

(0,NA),

(dexe

, xe

),(dsβ, xe

),(dexe

, xe

), if

dsβ

≥dexe

,

(0,NA),

(dexe

, xe

),(dsβ, xe

)∪(

k,dek

)∣∣∣k ∈[ko,

dexe

], if

dsβ

<dexe

.

(19)

16

Figure 6: Optimal CHP design choice set, compared with traditional choice set

Ω Ω

= (1–)

=

= 0

=

= (1–)

=

= 0

Capacity

Capacity

=

Ω

Ω

Ω

Ω

=

=

Ω

Traditional choice set Optimal choice set

0

0

Figure 6 illustrates the candidates for the optimal CHP design in large dots and a thick curve.

We also compare the optimal CHP design choice set with the choice set traditionally considered in

industry practice (marked as dashed curves). As discussed in §1, in practice, CHP projects typically

aim to match the CHP steam output with the steam demand, while electricity output can either

exceed or fall short of the electrical demand.

Importantly, Figure 6 reveals that the optimal CHP choice set is significantly different from

the traditional choice set. If the CHP output were matched with the steam demand, then the

two candidates would be (k = ds/β, xe = xe) and (k = ko and xe = de/ko). Expanding the

industry choice set, we show that matching CHP output with the electrical demand may be optimal.

In particular, practitioners should explore the design option of matching electrical demand while

oversizing CHP relative to the steam demand (the thick solid curve shown in Figure 6). Indeed, we

show that such design can be optimal under realistic settings in our numerical analysis.

4.4.2 Market Conditions and Optimal CHP Design

Proposition 3 has identified the candidates for the optimal CHP design under any market conditions.

Next, we derive market conditions for some candidates to emerge as the optimal.

17

Consider how the objective E[B(k, xe;Pg)

]varies in configuration xe within region Ω1. In this

region, one unit of energy from the steam can either generate µ2 units of electricity, thus saving µ2pe,

or meet one unit of thermal demand, saving the boiler a fuel cost of γpg. Thus, if the CHP were

running at full capacity regardless of natural gas price, the marginal value of xe would be

∆1def= µ2pe − γE[Pg]. (20)

The actual marginal value of xe is no lower than ∆1, because the firm has the option of not running

the CHP. Thus, if ∆1 ≥ 0, the actual marginal value of xe must be nonnegative, and it is optimal to

set xe as high as possible. This intuition is formalized in the following lemma.

For a given k, define the feasible set for xe in region i as Xe,i(k)def= xe ∈ [xe, xe] : (k, xe) ∈ Ωi.

Lemma 3 (Benefit as function of xe in Ω1) For given k with nonempty Xe,1(k), the objective

E[B(k, xe;Pg)

]has the following properties for all xe ∈ Xe,1(k):

(i) E[B(k, xe;Pg)

]is convex in xe;

(ii) If γE[Pg ] ≤ µ2pe (or ∆1 ≥ 0), then E[B(k, xe;Pg)

]increases in xe;

(iii) If γE[Pg ] > µ2pe (or ∆1 < 0), then

E[B(k, xe;Pg)

]increases in xe if β < γ−1 and

∫ xepe1−βγ

0(γpg − µ2pe)dF (pg) ≤ 0, and

E[B(k, xe;Pg)

]decreases in xe if

∫ xepe

0(γpg − µ2pe)dF (pg) ≥ 0.

Lemma 3 (ii) shows that ∆1 ≥ 0 is sufficient for the benefit function to increase in xe, confirm-

ing the intuition discussed earlier. When ∆1 < 0, directly using steam is more beneficial, which

intuitively drives the benefit function to decrease in xe. Interestingly, under ∆1 < 0, the benefit

function may still increase in xe, as stated in part (iii). This is mainly because a larger xe provides

the firm with more capacity to produce power when the natural gas price is low, while the firm stops

running CHP when the natural gas price is high (see Proposition 1 (i)). In other words, a larger xe

provides a higher option value of profiting from low fuel prices and shutting down during high fuel

prices. In fact, the value of the shutdown option increases when xe increases. This can be seen from

Corollary 1 (i): the firm is more likely to exercise this option under a higher xe. This explains why

the benefit function is convex in xe, as stated in Lemma 3 (i).

The above option value can be enhanced by a more dispersed distribution of the natural gas price,

which is embodied in the condition in Lemma 3 (iii),

∫ pg

0

(pg − pg

)dF (pg) ≤ 0, where pg ≡

xepe1−βγ

and pg ≡ µ2peγ . This condition, together with the case (iii) condition E[Pg ] > pg , implies that the

price is distributed with a significant density in pg : pg < pg or pg > pg. On the other hand, if

the natural gas price is not so dispersed such that

∫ xepe

0(pg − pg )dF (pg) ≥ 0, then the average price

18

condition γE[Pg] > µ2pe implies a higher value in using the steam directly and, therefore, a lower xe

becomes more desirable. If neither of the conditions in part (iii) holds, the benefit function generally

decreases first and then increases in xe.

Consider a CHP configured in region Ω2. Similar to the reasoning for Ω1, if the CHP were running

at full capacity regardless of the natural gas price, the marginal value of xe would be

∆2def= µ2re − γE[Pg]. (21)

If ∆2 ≥ 0, the actual marginal value of xe must be nonnegative (due to the option value). This

is formalized in Lemma 4 below, which is similar to Lemma 3, but the proof is considerably more

difficult due to the complicated benefit function in (11).

Lemma 4 (Benefit as function of xe in Ω2) For given k with nonempty Xe,2(k), the objective

E[B(k, xe;Pg)

]has the following properties for all xe ∈ Xe,2(k):

(i) E[B(k, xe;Pg)

]is convex in xe;

(ii) If ∆2 ≥ 0, then E[B(k, xe;Pg)

]increases in xe;

(iii) If ∆2 < 0, then E[B(k, xe;Pg)

]generally decreases and then increases in xe; it decreases in xe

for xe ∈ Xe,2(k) ∩[xe, xe − µ2/γ

], if this set is nonempty.

Now consider a CHP configured in Ω3. The analysis in this region is complicated by the fact that

E[B(k, xe;Pg)

]generally does not exhibit monotonicity or convexity/concavity in xe for xe ∈ Xe,3(k).

To see this, note that in (13), the net benefit of fulfilling one unit of steam demand is

γpg +xexs

pe −pgxs

= γpg − µ2pe +xepe − pg

xs, (22)

which is concave increasing or convex decreasing in xs depending on the price realization pg. This,

together with the operator ()+, makes the benefit function in (13) complicated to analyze. Never-

theless, the expected benefit function still increases in xe when ∆1 ≥ 0, as stated in the next lemma.

When ∆1 < 0, the optimal xe for given k may be in the interior of Ω3. We explore the properties of

the optimal configuration and the resulting benefit function next.

Lemma 5 (Benefit function and optimal xe in Ω3) For given k with nonempty Xe,3(k), we have

(i) If ∆1 ≥ 0, then E[B(k, xe;Pg)

]increases in xe for xe ∈ Xe,3(k);

(ii) If ∆1 < 0, let x†e(k) denote the minimum of the optimal solutions to the following problem:

G(k)def= max

xe∈Xe,3(k)E[B(k, xe;Pg)

]. (23)

Then, there exists a threshold k such that, for k ∈[dsβ , k

], x†e(k) increases in k, and x†e(k) = de/k

19

for k ∈[k, dexe

].

It is worth noting that, for given k ∈[dsβ , dexe

], the optimal configuration xe (or xs) may be such

that the CHP running at full capacity will overproduce steam (kxs > ds) but does not overproduce

electricity (kxe ≤ de). This is surprising, since overproduced steam has no value while overproduced

power can be sold at re. We discuss the reason below. Consider the net benefit of fulfilling one unit

of steam demand in (22). When the realized fuel price pg > xepe, the net benefit in (22) increases in

xs (or decreases in xe). Intuitively, when the CHP operates at part load to meet the steam demand,

a higher xs allows the CHP to satisfy the steam demand using less fuel input and thus producing

less electricity, which is preferred when purchasing electricity from the utility at pe is relatively

inexpensive. This can render kxs > ds to be a preferred configuration.

Using the results in Lemmas 3, 4, and 5, we now revisit the optimal CHP design choices provided

in Proposition 3 and identify conditions under which certain designs are dominated or optimal.

First, we consider whether or not it is optimal to install CHP at all. To this end, it is useful to

define the net benefit of CHP excluding the fixed investment cost vf :

Bdef= max

k≥0, xe∈Xe

NE[B(k, xe;Pg)

]− vk. (24)

If B ≤ 0, the firm need not consider CHP investment, whereas if B > 0, the firm should compare B

with the fixed cost vf to make an installation decision. The next proposition provides a necessary

and sufficient condition for B > 0. Importantly, this condition does not require the firm to evaluate

the benefit of CHP.

Proposition 4 B > 0 if and only if

v < v0def= max

E[(xepe − Pg)

+],E

[(xepe − (1− βγ)Pg)

+]

. (25)

In (25), the two expected values are the marginal benefit of k in region Ω1 when xe = xe and

xe = xe; refer to Proposition 2. The proof of Proposition 4 shows that v0 is the maximum marginal

benefit of capacity, which is much easier to evaluate than the benefit of CHP. Comparing v0 and the

variable CHP capacity cost v determines whether or not the firm should consider CHP investment.

Finally, we synthesize the conditions in Lemmas 3, 4, and 5 to refine the optimal CHP choice set

derived in Proposition 3.

Proposition 5 Suppose v < v0 and a CHP facility is to be built. Then,

(a) If E[Pg] ≤µ2reγ (or ∆2 ≥ 0), the optimal CHP design is (k∗, x∗e) =

(dexe, xe

).

(b) If E[Pg] ∈(µ2re

γ , µ2peγ

](or ∆1 ≥ 0 > ∆2), then

(dsβ , xe

)cannot be optimal when ds

β < dexe, and

20

the optimal CHP design can be chosen from the following set:

(k∗, x∗e) ∈

(dexe, xe

),(dsβ , xe

),(dexe, xe

), if ds

β ≥ dexe,

(dexe, xe

)∪(

k, dek)∣∣∣k ∈

[ko, dexe

], if ds

β < dexe.

(26)

(c) If E[Pg] ≥µ2peγ (or ∆1 < 0), then

(i) If β < γ−1 and

∫ xepe1−βγ

0(γpg − µ2pe)dF (pg) ≤ 0, the optimal choice set coincides with (26).

(ii) If

∫ xepe

0(γpg − µ2pe)dF (pg) ≥ 0, then

(dexe, xe

)cannot be optimal, and the optimal CHP

design can be chosen from the following set:

(k∗, x∗e) ∈

(dsβ , xe

),(dexe, xe

), if ds

β ≥ dexe,

(dsβ , xe

)∪(

k, dek)∣∣∣k ∈

[ko, de

xe

], if ds

β < dexe.

(27)

(iii) If neither condition in (i) nor (ii) holds, the optimal choice set is (27) and(dexe, xe

).

Based on Proposition 5, we provide intuitive explanations for when each of the CHP design

choices may be preferred.

Design I:(dexe, xe

). Matching electrical demand and no thermal output. This choice maintains

separate (rather than combined) heat and power generation. It is a preferred choice when the legacy

boiler is relatively efficient (e.g., γ−1 close to 80%) and the fuel price is relatively low on average, so

that it is most economical to configure CHP as a combined-cycle power generator to meet the firm’s

electrical demand while keeping the legacy boiler.

Design II:(dexe, xe

). CHP with maximum thermal output but matching electrical demand. This

configuration is a preferred choice when the fuel price is medium to high, electricity buyback rate is

low, and the firm’s thermal demand is high (dsβ ≥ dexe) such that a CHP matching thermal demand

(current industry practice) will generate excess electricity that cannot be sold at a high price. Note

that this design is also discussed as part of Design IV below if the steam demand is not too high.

Design III:(dsβ , xe

). CHP with maximum thermal output and matching thermal demand. This

design is commonly adopted in the industry practice. The inefficient legacy boiler can retire after

installing the CHP. This choice is preferred if the electrical demand is low (dsβ ≥ dexe) but excess

electricity can be sold back at a reasonable price, or if the electrical demand is high (dsβ < dexe) but

the fuel price is relatively high so that purchasing power from the utility is economical.

Design IV: k ∈[ko, dexe

], xe = de

k . CHP matching electrical demand and overproducing steam.

CHP of this configuration can run at full capacity to satisfy electrical demand but overproducing

steam, or run at part load to satisfy thermal demand while meeting the residual electricity from the

utility. The value of this operational flexibility will be demonstrated in the next section. This design

21

is preferred when the firm has a relatively high electrical demand (dsβ < dexe) and the natural gas price

is medium to high.

Competing design choices often exist, in which case we recommend evaluating multiple design

choices listed above.

4.4.3 Optimal Configuration when Capacity Choice Set is Discrete

As discussed at the beginning of §4.4, in practice, the choice of capacity k is not as continuous as

the choice of xe. Thus, the firm may have to choose a design that is close to the theoretical optimal

design. For a chosen capacity k, the optimal xe can be found under the guidance of Lemmas 3, 4, 5,

and numerical evaluation.

When the available CHP capacity choices do not exactly match the optimal design, one might

expect that the CHP should be configured to match either electrical or thermal demand, as suggested

by Proposition 2 when the design choices are continuous. However, given capacity k, the optimal xe

may not match either demand, for two reasons explained below. First, although Lemmas 3 and 4

show that the benefit function is convex in xe within regions Ω1 and Ω2, Lemma 5 reveals that the

benefit function lacks convexity in Ω3. Second, although Lemma 1 eliminates the interior of Ω4 from

the optimal CHP design when k is a continuous variable, Ω4 may contain the optimal configuration

for certain choices of k, as shown in the numerical examples in the next section.

5. Numerical Analysis

In this section, we provide a numerical study that has three objectives. First, we show that under

realistic parameter values, it can be optimal to design a CHP in configurations that are not common

in practice, and we estimate the long-run net benefit of the optimal design. Second, we analyze the

value of co-operating the CHP and the legacy boiler, in contrast to the practice of retiring the boiler.

Third, we consider the impact of random electricity and steam demands and show that our insights

based on deterministic demand continue to hold.

5.1 Parameter Values: Baseline Case

We consider a basic petrochemical manufacturing process—ethylene production. The U.S. produc-

tion of ethylene was about 1.6 million barrels per day in 2018. In a typical ethylene production

process, the required power-to-heat ratio is between 2.5 and 3; see McMillan et al. (2016) for the

details of the process description and energy requirements. Matching this power-to-heat ratio, in

our baseline case, the firm’s demands are at constant levels: de = 10 MW and ds = 3.5 MW. We

consider a 20-year operating horizon and assume that the firm’s energy consumption is stable over

22

the entire horizon. Each period is a month, and thus there are a total of T = 240 periods with

n = 720 operating hours per month. We use a monthly discount factor δ = 0.999. The discount

factor takes into account both energy price inflation and the firm’s discounting of future cash flows.

Thus, the scaling factor in (5) is N = n(1−δT )1−δ = 153, 696.

We use an empirical distribution of the natural gas price Pg based on monthly commercial natural

gas prices in Indiana, as shown in Figure 2, where each monthly observation is taken with equal

probability. As a baseline, we consider the period from March 2008 to February 2010, including a

peak near $17 per MMBtu. The average and standard deviation of the price in this period is $10.64

and $3.33 per MMBtu, respectively, or alternatively $36.30 and $11.36 per MWh.

The electrical efficiency of of the first-stage gas turbine in typical CHP configurations, (1− β)µ1

ranges between 24% and 36% (EPA 2017, pp. 3-6). The electrical efficiency of the second-stage

steam turbine µ2 ranges between 0.3 and 0.4, with an average value of 0.33 in the U.S. (Masters

2013). The parameter β is the fraction of energy the CHP is able to recover relative to the total

gas input, and it ranges between 0.4 and 0.6 (Masters 2013). Consistently with these benchmarks,

we set µ1 = 0.6, µ2 = 0.3, and β = 0.567, implying that xe = (1 − β)µ1 = 0.26, xe = 0.43. The

overall efficiency when the CHP is configured to produce only electricity (i.e., a combined cycle

power generator) is xe = 43%, while when configured for maximum steam output the CHP’s overall

efficiency is xe + β = 83%. We consider an electricity price of pe = $90 per MWh, and a rate for

selling back to the grid of re = pe/3, as in Hu et al. (2015).

Using the total investment cost estimated in EPA (2017) for various sizes of CHP, we find that the

investment cost can be well approximated by a linear function vf + vk, where the fixed cost is vf =

$5.04 million and the variable cost is v = $0.41 million per MW of natural gas input. To encourage

energy efficiency investments, government subsidies are often provided to CHP construction. These

subsidies are typically proportional to the CHP capacity and vary across regions. In our base case,

we assume a subsidy that offsets 50% of the variable investment. As a result, investing a CHP with

input capacity k costs a total of vf + vk, where v = 0.5v = $0.205 million per MW.

5.2 Optimal CHP Design with Boiler Retirement

In practice, a CHP is typically built to match steam demand and replace the legacy boiler. Thus,

we first find the optimal CHP design under the constraint that the boiler must retire.

Figure 7 (a) shows the operating benefit function in regions Ω3 and Ω4. (As the boiler is retired,

Ω1 and Ω2 are infeasible design regions.) The contours are shown on the surface and in the contour

plot Figure 7(b). The benefit function in Ω4 stays constant as k increases, because in this example

23

Figure 7: (Color online) CHP designs for the baseline case: boiler with γ = 1.67 is retired

(a) Hourly operating benefit E[B(k, xe;Pg)] andoptimal configuration, given k

(b) Optimal vs. industry configurations, given k

Input capacity

(MW)Configuration

Optimal configuration, given capacity k

Input capacity (MW)

Co

nfi

gu

rati

on

Configuration matching steam demand, given capacity k

ΩΩΩ

Ω

($)

(c) Net long-term benefit of CHP under industry solutions and the optimal design

Input capacity (MW)∗0 10 20 30 40 50

5

10

15

(Million $)

CHP matching steam demand

with boiler retirement

CHP with optimal configuration

and boiler retirement

re is relatively small and there is no benefit of increasing capacity only to overproduce electricity.

The industry practice of matching CHP’s thermal output with the steam demand is shown as the

blue curve in Figure 7(b). We evaluate all possible industry solutions and show their corresponding

net long-term benefit (defined in (6)) in the blue curve in Figure 7(c). The industry optimal solution is

to build a CHP with input capacity ko = 25.7 MW (costing vf + vko = 5.04 + 0.205× 25.7 = $10.31

million), configured to exactly match both thermal and electrical demand: xoe = de/ko = 0.389

24

and xos = ds/ko = 0.136. The expected long-run discounted operating benefit of such a CHP is

N (γdsE[Pg] + depe − koE[Pg]), which is the avoided legacy boiler’s fuel cost, plus the avoided cost

of purchasing electricity from the utility, minus the CHP operating cost. This gives CHP’s expected

long-run operating benefit as 19.52γ − 5.056 million dollars, which increases linearly in γ because

replacing a less efficient boiler yields a higher benefit. The net long-run benefit with γ = 1/0.6 is

thus 19.52(1/0.6) − 5.056 − (vf + vko) = $17.17 million, corresponding to the highest point of the

blue curve in Figure 7(c). When k > ko = 25.7 MW, the blue curve rapidly declines, because the

cost of overproducing electricity erodes the benefit. This is also the reason why the industry practice

tends to design CHP to match steam demand while underproducing electricity.

We now turn to the optimal CHP design, assuming that the boiler must retire. Figure 7(b)

shows the optimal configuration as a function of k. For a wide range of k, it is optimal to configure

the CHP to enable it to overproduce steam when running at full capacity: the red curve, when not

overlapping the blue curve, corresponds to configurations that can overproduce steam.

The optimal solution (peak of the red curve in Figure 7(c)) outperforms the industry optimal

solution by $1.47 million in net long-run benefit. The optimal CHP size is k∗ = 27.23 MW > ko,

which is configured to match electricity demand but overproduce steam at full activation: x∗e =

0.367 = de/k∗ and x∗s = 0.209 > ds/k

∗ = 0.129, as shown in Figure 7(b).

Note that k∗x∗e = de = 10 MW but k∗x∗s = 5.70 MW, which exceeds ds by 63%. To understand

how such a design outperforms the industry optimal solution, we decompose the net benefit as

follows. According to Proposition 1, when the natural gas price is high, pg > xepe = 0.367(90) =

$33.03/MWh, the CHP operates at part load to produce 3.5 MW of steam output (satisfying ds) and

10(3.5/5.70) = 6.14 MW of electricity; the remaining electricity will be purchased from the utility at

pe. Using the empirical distribution of natural gas prices, we calculate that this part-load operating

strategy saves $4.571 million compared to the industry solution of satisfying both demands using

CHP. On the other hand, when the natural gas price is low, pg ≤ $33.03/MWh, it is optimal to

operate the CHP at full capacity, generating 10 MW of electricity (satisfying de) but overproducing

steam, which results in an extra fuel cost of $2.787 million compared to the industry solution. Thus,

the operational flexibility brings a benefit of $4.571 − $2.787 = $1.784 million. Importantly, this

benefit is achieved by investing in an additional k∗ − ko = 1.53 MW of capacity, which costs only

1.53 × 0.205 = $0.314 million. In other words, the net long-run benefit is 1.784 − 0.314 = $1.47

million, and the return on (additional) investment is substantial: $1.47/$0.314 = 469%.

25

Figure 8: (Color online) CHP designs for the baseline case: boiler with γ = 1.67 remains operational

(a) Hourly operating benefit E[B(k, xe;Pg)] andoptimal configuration, given k

(b) Optimal vs. industry configurations, given k

Input capacity

(MW)Configuration

Optimal configuration, given capacity k

Input capacity (MW)

Co

nfi

gu

rati

on

Configuration matching steam demand, given capacity k

Ω

Ω

Ω

Ω

($)

Ω

Ω

Ω

Ω

(c) Net long-term benefit of CHP under boiler co-operation and boiler retirement

0 10 20 30 40 50

5

10

15

20

Input capacity (MW)

= ∗

(Million $)

CHP matching steam demand

with boiler retirement

CHP with optimal configuration

and boiler co-operation

5.3 Optimal CHP Design with Boiler Co-operation

We now examine the case where the legacy boiler will remain functional after the CHP is installed.

For the setting described in §5.1, we find that if γ > 2.96, i.e., the boiler efficiency is below 34%,

it will never operate even if it remains functional. In such cases, the optimal CHP design coincides

with the optimal design in §5.2 and the boiler is effectively retired.

When γ < 2.96, the legacy boiler can still have operational value. To illustrate the joint operations

26

of a CHP and the legacy boiler, we consider the case of γ = 1.67, i.e., the boiler efficiency is 60%,

and the industry optimal design discussed in §5.2: ko = 25.7 MW and xoe = 0.389 (xos = 0.136).

According to Proposition 1, when the price of natural gas exceeds xoepe/(1 − xosγ) = $45.30/MWh,

it is cheaper to supply the steam using the legacy boiler and buy electricity from the utility instead

of running the CHP. Using our natural gas price empirical distribution, we estimate the long-run

savings from this operational flexibility to be $5.92 million, shown in Figure 8(c) as the gap between

the peaks of the red and blue curves, where the blue curve is the same as in Figure 7(c).

Figure 8(a) and (b) show the operating benefit function and its contours. For this parameter

setting, the industry optimal design of ko = 25.7 MW and xoe = 0.389 coincides with the optimal

CHP design, as illustrated in Figure 8(c). Co-operating the CHP and the boiler rather than retiring

it brings a long-run savings of $5.92 million, which improves the net long-run benefit from $17.17

million (see §5.2 for calculation of this value) to $23.09 million, an improvement of 34%.

The value of the legacy boiler certainly depends on its efficiency. A boiler of 80% efficiency

can bring $8.89 million operating benefit over the retirement option. As a result, the net long-run

benefit of CHP increases from $9.05 million to $17.94 million, which is almost doubled. In another

example, a boiler of 50% efficiency is quite inefficient and will almost certainly be replaced by the

CHP under the current industry practice. However, even this inefficient boiler can still bring $3.54

million operating benefit over the retirement option. Co-operating the CHP with this boiler will

enhance the net long-run benefit of CHP from $23.70 million to $24.24 million, or 15% improvement.

The key message is that a seemingly inefficient boiler can still bring significant value by reducing the

CHP’s exposure to the high natural gas prices.

In sum, we have analyzed two operational flexibilities: 1) flexibility of running the CHP at part

load to satisfy steam demand while meeting the residual electricity demand by purchasing from the

utility, and 2) flexibility of temporarily shutting down CHP and supplying energy using the legacy

boiler and buying electricity from the utility. Our analysis in §5.3 shows that when the legacy boiler

remains operational, the second flexibility is more prominent and, in our parameter setting, there is

no need to pursue the first flexibility. We have conducted extensive numerical analysis and found

some cases where both flexibilities exist and contribute significant values. Because such settings are

relatively rare, we do not present them as representative cases.

5.4 Impact of Demand Variability

We now relax the assumption that the firm’s energy demands are constant. We consider a set-

ting where demands fluctuate over time, taking three possible levels: (de, ds) = (8 MW, 3.2 MW),

27

Figure 9: (Color online) CHP designs for the baseline case with uncertain demand

0.25

0.30

0.35

0.40

0 10 20 30 40 50

Configuration matching maximum steam demand, given capacity k

Optimal configuration, given capacity k

∗

Input capacity (MW)

Co

nfi

gu

rati

on

(10 MW, 4 MW), and (12 MW, 4.8 MW) with stationary probabilitiess 0.25, 0.5, and 0.25, respec-

tively. The firm is able to operate the CHP (and the boiler) in response to the demand realization

in each period. We assume the demand is independent of the natural gas price, which follows a log-

normal distribution with mean and standard deviation of $31.73 and $16.72 per MWh, respectively.

The dashed curves in Figure 9 represent the curves kxe = de and kxs = ds for the three demand

levels. We evaluate all industry solutions that match the CHP’s thermal output with the maximum

steam demand 4.8 MW. The best industry solution is to size the CHP at ko = 26.7 MW and configure

it at xoe = 0.374 or xos = 0.185. Note that this industry optimal solution does not satisfy the maximum

electricity demand.

The optimal CHP design takes into account demand uncertainty and maximizes the expected

long-term net benefit. The optimal configuration for various capacity k is shown in the red curve

in Figure 9. The optimal CHP capacity is larger: k∗ = 33.6 MW, with x∗e = 0.357 (x∗s = 0.243).

The optimal CHP would operate at full capacity when the natural gas price is low and the realized

demand is high. When running at full capacity, it has a power output that matches the maximum

electricity demand (12 MW) and a thermal output of k∗x∗s = 8.16 MW, which exceeds the maximum

steam demand 4.8 MW by 70%. The optimal CHP design improves the net benefit from $59.41

million (best industry solution) to $61.09 million. The investment cost difference is $1.415 million.

28

Hence, the return on (additional) investment is 119%. In sum, the fundamental insights about the

optimal CHP design strategy identified under the deterministic demand case continue to hold under

stochastic demand.

6. Conclusion

CHP technology provides a high energy efficient solution to industrial firms, but its cost efficiency

depends on many factors, including fuel price variability, utility industry regulations (buyback rate),

the firm’s load characteristics, among others. We study the optimal capacity, configuration (choosing

the mix between electrical and thermal output), and operations of CHP systems for industrial firms.

Common industry practice designs CHP with thermal capacity matching the steam demand, which

appears to be the most energy efficient solution, and the inefficient legacy boiler can be retired. We

consider a range of alternative CHP designs, including those that match the electrical demand and

may overproduce steam. We also allow for the possibility of jointly operating the legacy boiler with

the CHP system. We demonstrate substantial cost savings from these improvement options.

For a given a CHP design, the operational flexibilities are realized through an operating policy

characterized by control bands. Depending on the natural gas prices and demand levels, the CHP

runs at full capacity, or partial capacity to exactly meet electricity or steam demand, or shut down.

In terms of CHP system design, we find that the optimal design options can be significantly

different from the industry practice of matching CHP output with the steam demand. The candidates

for the optimal CHP designs can be narrowed down to four: (i) produce electricity only and match

electrical demand; (ii) match electrical demand and maximize steam output (supplying remaining

steam from legacy boiler); (iii) maximize steam output and match thermal demand; and (iv) match

electrical demand and overproduce steam. We numerically demonstrate that the cost savings of the

optimal design option relative to the industry standard design can be significant, especially under

variable fuel prices.

Based on our findings, we provide two recommendations for CHP design in practice. First,

whenever the legacy boiler must retire due to space and technical constraints, the CHP design

should consider matching CHP’s power output with electricity demand while oversizing the thermal

output relative to the steam demand. The common design of matching CHP’s thermal output to the

steam demand will force the CHP to run at full capacity all the time, while our recommendation will

allow the CHP to run at part load when the fuel price is high. Second, if the boiler does not have

to retire, keep it operational and co-operate it with the CHP. This allows the CHP to temporarily

shut down and the plant goes back to the separate heat and electricity supply mode, which is more

29

economical when the fuel price for CHP surges. Through meticulous design and operations, energy

efficiency can meet cost efficiency.

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31

Appendix: Proofs

Proof of Lemma 1: Consider a CHP design (k, xe) with kxe > de and kxs > ds. We prove its

suboptimality by comparing it with an alternative design (k, xe) with k < k, kxe > de, and kxs > ds,

i.e., the alternative design is also in Ω4, has the same xe but a smaller capacity k.

Denote the objective function in (3) as

c(a)def= (de − axe)

+pe − (axe − de)+re + (ds − axs)

+γpg + apg. (28)

Note that c(a) is convex in a because (de−axe)+pe− (axe−de)

+re and (ds−axs)+γpg are piecewise

linear in a with their slopes increasing at the respective kinks a = dexe

and a = dsxs.

Let a∗(pg) ∈ [0, k] and a∗(pg) ∈ [0, k] denote the optimal decision rules for operating the CHP

systems (k, xe) and (k, xe), respectively. The decision rules a∗(pg) and a∗(pg) are related as follows:

If a∗(pg) < k, then a∗(pg) = a∗(pg). To see this, note that the convexity of c(a) implies that

if a∗(pg) < k minimizes c(a) over [0, k], it must also minimize c(a) over [0, k]. Thus, the two CHP

systems have the same operating cost when a∗(pg) < k.

If a∗(pg) = k, then a∗(pg) = k. To see this, note that the optimality of k implies that c′(k) =

−rexe + pg ≤ 0. When a ≥ k, we have c(a) = −(axe − de)re + apg, which is linear in a with slope

c′(a) = −rexe+ pg ≤ 0. Thus, the operating cost of the CHP system (k, xe) is lower than the system

(k, xe) by (rexe − pg)(k − k) ≥ 0.

Combining the above two cases, we see that the expected operating cost of the CHP system

(k, xe) is lower than the system (k, xe) by E[(rexe − Pg)

+](k − k). That is, every additional unit

of capacity reduces the expected operating cost by E[(rexe − Pg)

+]. However, this cost-reduction

benefit is outweighed by the investment cost:

E[(rexe − Pg)

+]≤ E

[(rexe − Pg)

+]< v, (29)

where the first inequality follows from xe ≤ xe, and the second inequality is due to Assumption 1.

Therefore, the system (k, xe) cannot be an optimal CHP design.

Proof of Proposition 1: The objective function of the problem in (3) is defined as c(a) in (28).

(i) In region Ω1, (28) becomes

c(a) = depe + dsγpg − a(xepe + xsγpg − pg).

To minimize the above linear function in a, the optimal operating policy is

a∗ =

k, if xepe + xsγpg − pg ≥ 0,

0, otherwise,

which proves (8) in the lemma. The corresponding minimum operating cost is

C(k, xe; 1, pg, de, ds) = depe + dsγpg − k(xepe + xsγpg − pg)+.

1

Using the definition in (7), we have the benefit function in (9).

(ii) In region Ω2, (28) becomes

c(a) = (de − axe)+pe − (axe − de)

+re + (ds − axs)γpg + apg

=

depe + dsγpg − a(xepe + xsγpg − pg), if a ∈[0, de

xe

],

dere + dsγpg − a(xere + xsγpg − pg), if a ∈(dexe, k].

Thus, c(a) is piecewise linear in a with a kink at a = dexe. The slope increases at the kink since

re < pe. Therefore, the optimal operating policy that minimizes c(a) is

a∗ =

k, if xere ≥ pg − xsγpg,

dexe, if xere < pg − xsγpg ≤ xepe,

0, if xepe < pg − xsγpg,

which proves (10) in the lemma. The associated minimum operating cost is

C(k, xe; 1, pg, de, ds) =

dere + dsγpg − kxe

(re +

xsxe

γpg −pgxe

), if xere ≥ pg − xsγpg,

de

(−xsxe

γpg +pgxe

)+ dsγpg, if xere < pg − xsγpg ≤ xepe,

depe + dsγpg, if xepe < pg − xsγpg,

which can be rewritten as

C(k, xe; 1, pg, de, ds) = depe + dsγpg − de

(pe +

xsxe

γpg −pgxe

)+

− (kxe − de)

(re +

xsxe

γpg −pgxe

)+

.

Using the definition in (7), we have the benefit function in (11).

(iii) The proof for region Ω3 is parallel to that for region Ω2.

(iv) In this case, the legacy boiler is retired, and the CHP with kxs = ds must run at full capacity

a∗ = k at all time to meet the steam demand ds. Since (k, xe) ∈ Ω2 in this case, the firm sells excess

electricity at re. The operating cost of CHP is C(k, xe; 0, pg , de, ds) = −(kxe − de)re + kpg.

(v) The proof for this case is parallel to that for cases (ii) and (iii), except that a∗ = 0 is no longer

feasible.

Proof of Corollary 1: When xsγ < 1, it is straightforward to derive all of the price bands from

Proposition 1. To show whether these intervals shrink or expand, note that

xe1− xsγ

=xe − µ2xs1− xsγ

=xe − µ2/γ

1− xsγ+

µ2

γ. (30)

Because xe > µ2 and γ > 1, the above fraction increases in xs or decreases in xe.

If xsγ ≥ 1, the stopping criterion (1− xsγ)pg > xepe never holds. Thus, CHP will always run.

Proof of Proposition 2: (i) Using (9), (11), and (13), we immediately obtain the slopes in (18)

for regions Ω1, Ω2, and Ω3. The proof of Lemma 1 shows the slope for regions Ω4.

To see that E[B(k, xe;Pg)] is concave in k, note that pe ≥ re implies that the slope in Ω1 is greater

than the slopes in Ω2 and Ω3, which, in turn, are greater than the slope in Ω4.

2

(ii) Because the objective function in (17) is piecewise linear and concave in k for k > 0, we can

restrict the search for the optimal size k∗ within the set of kinks:0, de/xe, ds/xs

.

Proof of Lemma 2: (a) For dexe

≤ k ≤ minko, dexe

and kxe = de, we have (k, xe) ∈ ∂Ω1, where

∂ denotes boundary. Using (2) and (9), we have E[B(k, dek ;Pg)

]= E

[kxe

(pe +

( xeγµ2

− 1)Pg

xe−

γµ2Pg

)+]∣∣∣xe=

dek

= deE[(

pe +( xeγ

µ2− 1

)kPg

de− γ

µ2Pg

)+], which is convex in k.

(b) This segment exists if dsβ < de

xe. For ds

β ≤ k ≤ ko and kxs = ds, we have (k, xe) ∈ ∂Ω1.

Using (2) and (9), we have E[B(k, xe − µ2

dsk ;Pg)

]= E

[(k(xe − µ2xs)pe + kxsγPg − kPg

)+]∣∣∣xs=

dsk

=

E[(k(xepe − Pg)− µ2dspe + dsγPg

)+], which is convex in k.

(c) For k ≥ maxko, dsβ

and xe = xe − µ2

dsk (or equivalently, xs = ds

k ), (k, xe) is on the right-side

boundary of Ω2. Then, using (45) in the proof of Lemma 4 and simplifying the terms, the expected

benefit along the right-side boundary of Ω2 is

E[B(k, xe − µ2

dsk ;Pg)

]= de(pe − re) + E

[(kxe − µ2ds)re + dsγPg − kPg

]−

1

µ2E[h(k, Pg)

], (31)

where h(k, pg)def= h(k, xe−µ2

dsk , pg). For E

[B(k, xe−µ2

dsk ;Pg)

]to be convex in k, it suffices to show

that h(k, pg) is concave in k for any realized price pg. Using (46)-(49), we have

h(k, pg)def= f(k, pg) + g(k, pg). (32)

f(k, pg)def= f

(xe − µ2

dsk , pg) = de min

0, µ2pe − γpg +

xeγ − µ2

xe − µ2dsk

pg

, (33)

g(k, pg)def= g

(k, xe − µ2

dsk , pg) = (k(xe − µ2

dsk )− de)min

0, µ2re − γpg +

xeγ − µ2

xe − µ2dsk

pg

, (34)

= min0, kµ2(xere − pg) + (µ2ds + de)(γpg − µ2re)−

xeγ − µ2

xe − µ2dsk

pgde

(35)

Note that ℓ(k) =(xe − µ2

dsk

)−1> 0 is convex and decreasing in k, because ℓ′(k) = − µ2ds

(xek−µ2ds)2< 0

and ℓ′′(k) = 2xeµ2ds(xek−µ2ds)3

> 0. The convexity of ℓ(k) and xeγ > µ2 imply that g(k, pg) is concave in k.

The function inside of the minimum operator in (33) decreases in k with a lower bound:

µ2pe − γpg +xeγ − µ2

xe − µ2dsk

pg > µ2pe − γpg +xeγ − µ2

xepg = µ2

(pe −

pgxe

).

If pg ≤ xepe, then f(k, pg) = 0 and h(k, pg) = g(k, pg) is concave in k.

If pg > xepe, there exists k such that

f(k, pg) =

0, if k ≤ k,

de

(µ2pe − γpg +

xeγ−µ2

xe−µ2dsk

pg

)< 0, if k > k.

(36)

Note that f(k, pg) is convex and decreasing in k for k > k. We need to show that the convex portion

of f(k, pg) does not convexify h(k, pg). To this end, note that the function inside of the minimum

operator in (34) is less than that in (33), i.e., µ2re − γpg +xeγ−µ2

xe−µ2dsk

pg < µ2pe − γpg +xeγ−µ2

xe−µ2dsk

pg < 0

3

when k > k. Therefore, for k > k, we use (51) to obtain

h(k, pg) = (kxe − µ2ds)(µ2re − γpg) + k(xeγ − µ2)pg + deµ2(pe − re), (37)

which is linear in k. Therefore, h(k, pg) is concave in k for any realized price pg.

We next show that

limk→∞

d

dkE[f(k, Pg)

]= 0 and lim

k→∞

d

dkE[g(k, Pg)

]= µ2Emin0, xere − Pg, (38)

which, together with (31), implies that

limk→∞

d

dkE[B(k, xe − µ2

dsk ;Pg)

]= E

[xere − Pg

]− E

[min0, xere − Pg

]= E

[(xere − Pg)

+]

Using (35), we have

g(k, pg) = min0, kµ2xere − (µ2ds + de)µ2re −

(kµ2 − (µ2ds + de)γ +

xeγ − µ2

xe − µ2dsk

de

)pg

For large enough k, g(k, pg) < 0 if and only if pg > pg(k).

pg(k) =kµ2xere − (µ2ds + de)µ2re

kµ2 − (µ2ds + de)γ + xeγ−µ2

xe−µ2dsk

de→ xere, as k → ∞. (39)

Therefore,

limk→∞

d

dkE[g(k, Pg)

]= lim

k→∞

d

dk

∫ ∞

pg(k)kµ2(xere − p) + (µ2ds + de)(γp − µ2re)−

xeγ − µ2

xe − µ2dsk

p dedF (p)

= limk→∞

∫ ∞

pg(k)µ2(xere − p) +

µ2ds(xeγ − µ2)

(xek − µ2ds)2p dedF (p)

=

∫ ∞

xere

µ2(xere − p)dF (p) = µ2Emin0, xere − Pg.

Proof of Lemma 3: (i) In region Ω1, taking expectation on the benefit function in (9), we have

E[B(k, xe;Pg)

]= kE

[(xepe +

xe − xeµ2

γPg − Pg

)+ ](40)

=k

µ2E

[(xe(µ2pe − γPg) + (xeγ − µ2)Pg

)+], (41)

where we used the relation xs =xe−xe

µ2in (2). The term inside of the expectation is convex in xe for

any realized price pg. Thus, E[B(k, xe;Pg)

]is convex in xe.

(ii) Directly proving that (41) increases in xe is difficult. Consider the following auxiliary function:

f(x)def= E

[min

0, x(µ2pe − γPg) + (xeγ − µ2)Pg

]. (42)

By definition, f(x) ≤ 0 for all x. Because µ1 > µ2 (see §3.1), we have xeγ−µ2 > 0. Hence, f(0) = 0.

Furthermore, min0, x(µ2pe − γpg) + (xeγ − µ2)pg

is concave in x for any realized price pg, and

thus f(x) is concave in x. These facts together imply that f(x) must be decreasing in x.

Using f(x) and the fact that y+ ≡ y −min0, y, we can write (41) as

E[B(k, xe;Pg)

]=

k

µ2E[xe(µ2pe − γPg) + (xeγ − µ2)Pg

]−

k

µ2f(xe).

4

The first term linearly increases in xe because of the case condition µ2pe ≥ γE[Pg], while f(xe)

decreases in xe. Therefore, E[B(k, xe;Pg)

]increases in xe.

(iii) For this part, it is easier to work with xs, recognizing that xe and xs are linearly related according

to (2). If β ≥ γ−1, then for xs ∈ [γ−1, β], the objective in (40) can be written as

E[B(k, xs;Pg)

]= kE

[xepe + (xsγ − 1)Pg

]= kE

[xs(−µ2pe + γPg) + xepe − Pg

],

which increases in xs for xs ∈ [γ−1, β] since γE[Pg] ≥ µ2pe. Thus, for the benefit function to decrease

in xs, we must have β < γ−1.

Under β < γ−1, the benefit xepe+(xsγ−1)pg remains positive as long as pg < pgdef= xepe

1−xsγ(recall

xs = β). Thus, we write the benefit function as

E[B(k, xs;Pg)

]= k

∫ pg

0

((xe − µ2xs)pe + (xsγ − 1)pg

)dF (pg).

Then,

∂

∂xsE[B(k, xs;Pg)

]= k

∫ pg

0(−µ2pe + γpg)dF (pg). (43)

Because E[B(k, xs;Pg)

]is convex in xs, a necessary and sufficient condition for E

[B(k, xs;Pg)

]to

decrease in xs is that the derivative in (43) must not be positive when xs = xs = β, i.e.,

∫ xepe1−βγ

0(γpg−

µ2pe)dF (pg) ≤ 0. Similarly, a necessary and sufficient condition for E[B(k, xs;Pg)

]to increase in xs

is that the derivative in (43) must be nonnegative when xs = 0, i.e.,

∫ xepe

0(γpg − µ2pe)dF (pg) ≥ 0,

regardless of β < γ−1 or β ≥ γ−1.

Proof of Lemma 4: (i) In region Ω2, taking expectations on the benefit function in (11), we have

E[B(k, xe;Pg)

]= deE

[(pe +

xsxe

γPg −Pg

xe

)+]+ (kxe − de)E

[(re +

xsxe

γPg −Pg

xe

)+]

=deµ2

E

[(µ2pe − γPg +

xeγ − µ2

xePg

)+]+

kxe − deµ2

E

[(µ2re − γPg +

xeγ − µ2

xePg

)+], (44)

where we used the relation xs =xe−xe

µ2in (2). Note that xe ≥ de/k in region Ω2.

To prove the convexity of E[B(k, xe;Pg)

]in xe, we use y+ ≡ y −min0, y to write (44) as

E[B(k, xe;Pg)

]=

deµ2

E

[µ2pe − γPg +

xeγ − µ2

xePg

]+

kxe − deµ2

E

[µ2re − γPg +

xeγ − µ2

xePg

]

−1

µ2E[h(k, xe, Pg)

]

= de(pe − re) +k

µ2E[xe(µ2re − γPg) + (xeγ − µ2)Pg

]−

1

µ2E[h(k, xe, Pg)

], (45)

where

h(k, x, pg)def= f(x, pg) + g(k, x, pg). (46)

f(x, pg)def= de min

0, µ2pe − γpg +

xeγ − µ2

xpg

, (47)

g(k, x, pg)def= (kx− de)min

0, µ2re − γpg +

xeγ − µ2

xpg

, (48)

5

To show that E[B(k, xe;Pg)

]in (45) is convex xe for xe ≥ de/k, it suffices to show that h(k, x, pg)

is concave in x for any pg > 0 and x ≥ de/k, which is the focus of the rest of the proof.

Since kx ≥ de, we can write g(k, x, pg) as

g(k, x, pg) = min0, (kx− de)(µ2re − γpg) + k(xeγ − µ2)pg −

xeγ − µ2

xpgde

. (49)

Because xeγ−µ2 > 0 (since µ1 > µ2 and γ > 1), the function inside of the above minimum operator is

concave in x, and thus g(k, x, pg) is concave in x. Furthermore, by definition in (48), when x = de/k,

g(k, x, pg) = 0, and when x ≥ de/k, g(k, x, pg) ≤ 0. Hence, g(k, x, pg) is concave and decreasing in x

for x ≥ de/k.

If γpg ≤ µ2pe, then f(x, pg) = 0 and h(k, x, pg) = g(k, x, pg) is concave and decreasing in x.

If γpg > µ2pe, then define x = xeγ−µ2

γpg−µ2pepg, and we have

f(x, pg) =

0, if x ≤ x,

de

(µ2pe − γpg +

xeγ−µ2

x pg

)< 0, if x > x.

(50)

Note that f(x, pg) is convex and decreasing in x for x > x. We need to show that the convex portion

of f(x, pg) does not convexify h(k, x, pg). To this end, note that the function inside of the minimum

operator in (48) is less than that in (47), i.e., µ2re − γpg +xeγ−µ2

x pg < µ2pe − γpg +xeγ−µ2

x pg < 0

when x > x. Therefore, for x > x, we have

h(k, x, pg) = de

(µ2pe − γpg +

xeγ − µ2

xpg

)+ (kx− de)

(µ2re − γpg +

xeγ − µ2

xpg

)

= kx(µ2re − γpg) + k(xeγ − µ2)pg + deµ2(pe − re), (51)

which linearly decreases in x. This linearity, combined with the fact that limx→x−

∂f(x,pg)∂x = 0 >

limx→x+∂f(x,pg)

∂x , implies that h(k, x, pg) is concave and decreasing in x for x ≥ de/k.

(ii) If ∆2 ≥ 0 or µ2re ≥ γE[Pg], then E[xe(µ2re − γPg)

]increases in xe. Since h(k, xe, pg) decreases

in xe, E[B(k, xe;Pg)

]increases in xe for xe ∈ Xe,2(k).

(iii) The result of decreasing and increasing in xe follows directly from the convexity of E[B(k, xe;Pg)

].

For xe ≤ xe − µ2/γ, the terms inside of ( )+ in (44) are positive for all values of Pg. Then, we

can write the objective as

E[B(k, xe;Pg)

]= de(pe − re) +

k

µ2E[xe(µ2re − γPg) + (xeγ − µ2)Pg

],

which linearly decreases in xe under the condition of ∆2 < 0 or µ2re < γE[Pg].

Proof of Lemma 5: (i) For this proof, it is analytically more tractable to prove the properties

of the benefit function with respect to xs, which is linearly related to xe. Using (2), we define the

expected benefit as a function of xs:

B(k, xs)def= E

[B(k, xe − µ2xs;Pg)

]. (52)

6

Taking expectation of (13), we have

B(k, xs) = dsE

[(γPg +

xexs

pe −Pg

xs

)+ ]+ (kxs − ds)E

[(−bs +

xexs

pe −Pg

xs

)+ ]

= dsE[V1(xs, Pg)

+]+ (kxs − ds)E

[V2(xs, Pg)

+], (53)

where

V1(xs, pg)def= −µ2pe +

(γ −

1

xs

)pg +

xepexs

, V2(xs, pg)def= −µ2pe − bs +

xepexs

−pgxs

. (54)

Note that V1(xs, 0) =xe

xspe > 0, and V1(xs, pg) is linear in pg; V2(xs, 0) =

xe

xspe − bs, which is positive

if bs <xe

xspe, and V2(xs, pg) linearly decreases in pg. Thus, we can write (53) as

B(k, xs) = ds

∫ p1(xs)

0V1(xs, pg)dF (pg)− ds

∫ p2(xs)

0V2(xs, pg)dF (pg)

+ k

∫ p2(xs)

0

(xs(−µ2pe − bs) + xepe − pg

)dF (pg),

(55)

where F (pg) is the cumulative distribution function of Pg, and the upper limits of the integrals are

p1(xs)def=

(xe − µ2xs)pe1− γxs

, if xs < 1/γ

∞, if xs ≥ 1/γ

(56)

p2(xs)def=

(xepe − (µ2pe + bs)xs

)+. (57)

By definition, the upper limits satisfy

V1(xs, p1(xs)) = 0, if p1(xs) < ∞,

V2(xs, p2(xs)) = 0, if p2(xs) > 0,

p1(xs) > xepe > p2(xs), if p1(xs) < ∞. (58)

Assume that B(k, xs) in (55) is differentiable with respect to xs. Then,

∂B(k, xs)

∂xs= ds

∫ p1(xs)

p2(xs)

−xepe + pgx2s

dF (pg) + k

∫ p2(xs)

0(−µ2pe − bs) dF (pg). (59)

For the first integral in (59), we have p1(xs) > p2(xs) from (58), and for pg ∈ [p2(xs), p1(xs)), we

have V1(xs, pg) > 0, which implies that

V1(xs, pg) = −µ2pe + γpg +xepe − pg

xs> 0, for pg ∈ [p2(xs), p1(xs)). (60)

Using (60) and the lemma condition k ≥ ds/xs, we have

∂B(k, xs)

∂xs<

dsxs

∫ p1(xs)

p2(xs)(−µ2pe + γpg)dF (pg) +

dsxs

∫ p2(xs)

0(−µ2pe − bs) dF (pg)

≤dsxs

∫ p1(xs)

0(−µ2pe + γpg) dF (pg)

≤dsxs

∫ ∞

0(−µ2pe + γpg) dF (pg) =

dsxs

(−µ2pe + γE[Pg]) ,

7

where the last inequality follows from

∫ ∞

p1(xs)(−µ2pe + γpg) dF (pg) ≥ 0. This is because if p1(xs) < ∞

and pg ≥ p1(xs), we have γpg ≥ γp1(xs) > γxepe ≥ µ2pe, where we used (58), γ > 1, and xe > µ2.

Therefore, if µ2pe ≥ γE[Pg], the expected benefit function decreases in xs or increases in xe.

(ii) Let x†e(k) denote the minimum of the optimal solutions to G(k) = maxxe∈Xe,3(k) E[B(k, xe;Pg)

].

The benefit function B(k, xs) defined in (52) is submodular in (k, xs) because (59) leads to

∂2B(k, xs)

∂xs∂k=

∫ p2(xs)

0(−µ2pe − bs) dF (pg) ≤ 0. (61)

Therefore, the benefit function E[B(k, xe;Pg)

]is supermodular in (k, xe) for (k, xe) ∈ Ω3.

We next prove the following statement:

For any given ka ∈[dsβ,dexe

], for all kb ∈

[ka,

de

x†e(ka)

], we have x†e(k

b) ≥ x†e(ka). (∗)

Graphically, this means that if we draw the line xe = x†e(ka) indicated as AB in Figure 10, then x†e(k)

must stay at or above line AB for k within the range indicated by AB. The proof uses the technique

in Topkis (1978), but since the region Ω3 is not a lattice, we need to rewrite for our application.

We prove by contradiction. Suppose for some kb ∈[ka, de/x

†e(ka)

], x†e(kb) < x†e(ka). For notation

simplicity, let g(k, xe) ≡ E[B(k, xe;Pg)

], which is supermodular in Ω3. Then, we have

0 ≤ g(ka, x†e(ka))− g(ka, x†e(k

b)) ≤ g(kb, x†e(ka))− g(kb, x†e(k

b)) ≤ 0, (62)

where the first and last inequalities follow from that x†e(k) maximizes g(k, xe), and the second inequal-

ity is due to supermodularity. Then all inequalities in (62) must hold with equality, implying that

x†e(kb) also maximizes g(ka, xe), which contradicts to the fact that x†e(ka) is the minimum maximizer.

Figure 10: Properties of x†e(k) in Region Ω3

= (1–)

=

= 0

()

Ω

=

=

()

It follows that the minimum optimal solution x†e(k) (weakly) increases in k until it reaches the

boundary of Ω3 (we label the capacity at this boundary point as k), and x†e(k) stays on the boundary

for k ≥ k, illustrated in Figure 10. To prove this, notice two facts: First, if x†e(k) ever decreases

8

before reaching the boundary, then the result in (∗) is violated. Second, after x†e(k) reaches the

boundary, if it ever drops below the boundary so that x†e(kc) < de/kc for some kc > k, then we can

redraw the line AB so that it passes in between x†e(kc) and de/kc, and the result in (∗) is violated.

Proof of Proposition 4: If v < v0, then either v < E[(xepe−Pg)

+]or v < E

[(xepe−(1−βγ)Pg)

+].

In the former (latter) case, installing a CHP with maximum power (steam) output yields a positive

net benefit, excluding the fixed cost. This proves the sufficiency of v < v0.

Now suppose v ≥ v0. For a CHP configured in Ω1, Proposition 2 implies that the marginal

value of k is E[(xepe + xsγPg − Pg)

+], which is convex in xe according to Lemma 3. Therefore, the

maximum marginal benefit of CHP in region Ω1 is achieved at either xe = xe or xe = xe. Thus,

v ≥ v0 implies that the unit cost of the CHP capacity exceeds the maximum possible marginal benefit

of CHP and, therefore, CHP would not bring a positive net benefit.

Proof of Proposition 5: (a) ∆2 ≥ 0 implies ∆1 ≥ 0. Then, Lemma 3, 4, and 5 suggest that the

benefit increases in xe for any k. Therefore, the optimal design is (k∗, x∗e) =(dexe, xe

).

(b) When ∆1 ≥ 0, Lemma 3 suggests that the benefit increases in xe for regions Ω1 and Ω3. Therefore,

we can eliminate the design(dsβ , xe

)if ds

β < dexe, leading to the choice set in (26).

(c) When ∆1 < 0, Lemma 3 (iii) provides sufficient conditions for the benefit to increase or decrease

in xe in region Ω1. When the benefit increases in xe,(dsβ , xe

)cannot be optimal if ds

β < dexe, leading

to the candidate set in (i). When the benefit decreases in xe,(dexe, xe

)cannot be optimal, leading to

the candidate set in (ii).

9