DOI: 10.3303/CET25120079 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 03 July 2025; Revised: 29 August 2025; Accepted: 21 October 2025 
Please cite this article as: Isafiade A.J., 2025, Application of Game Theory to Interplant Heat Exchanger Networks Involving Multiple Unequal 
Period Durations, Chemical Engineering Transactions, 120, 469-474  DOI:10.3303/CET25120079 
  

 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 120, 2025 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Bing Shen How, Viknesh Andiappan, Denny K.S. Ng, Hon Loong Lam, Petar S. Varbanov 

Copyright © 2025, AIDIC Servizi S.r.l. 

ISBN 979-12-81206-21-2; ISSN 2283-9216 

Application of Game Theory to Interplant Heat Exchanger 

Networks Involving Multiple Unequal Period Durations 

Adeniyi J. Isafiade 

Department of Chemical Engineering, University of Cape Town, South Africa  

niyi.isafiade@uct.ac.za  

The integration of heat exchange networks across multiple plants, known as interplant heat integration, offers 

significant opportunities for energy savings and cost reduction. However, most existing studies have focused 

on plants with single operational periods, with only a few addressing multiperiod scenarios. In cases where 

these periods vary in duration, there is a need for methods to assess the economic benefits accruing to each 

plant participating in the integrated network. This study explores the application of game theory to evaluate the 

economic advantages of coalition formation among plants in a multiperiod interplant network, where integration 

may occur through an energy hub and operational periods are unequal. In the three co-located process plant 

case study considered in this paper, each plant has a 2-period operational profile consisting of daytime and 

nighttime with unequal durations across plants. The three plants can form a coalition involving interplant process 

heat exchange and heat exchange with hot and cold utilities at the energy hub. To determine the benefits that 

accrue to each plant for participating in the coalition, Shapley values are computed to establish the marginal 

contribution of each plant to the total annual cost of the integrated network. The result obtained for the total 

annual cost for the grand coalition comprising the three plants is $ 418,348 with marginal contributions for plants 

1, 2 and 3 being $ 369,997, -$ 30,339, and $ 78,690.   

1. Introduction 

Heat integration in process plants can be used to reduce greenhouse gas emissions. Two key methods for 

process heat integration are pinch technology and mathematical programming. For complex heat integration 

problems involving multiple process plants, each operating over several periods, applying pinch technology can 

become tedious. In such cases, mathematical programming offers a more practical and efficient alternative. 

Although most Heat Exchanger Network (HEN) synthesis problems to which mathematical programming has 

been applied have mostly involved single periods of operation, real-life heat integration problems may involve 

multiple periods of operation. In such cases, stream supply and target temperatures and flow rates may vary 

periodically due to upstream process upsets, changes in feedstock quality or environmental conditions, plant 

start-ups and shutdowns, etc. Problems of this nature have been addressed by Aaltola (2002) using an average 

heat exchanger area approach and by Verheyen and Zhang (2006) using a maximum heat exchanger area 

approach. In some problems, variations in stream parameters across periods may be unequal, necessitating 

the use of a correct weighting approach in the objective function, as done by Isafiade (2023). 

Heat integration problems involving multiple plants, known as Interplant Heat Integration (IPHI), have received 

less attention in the literature despite their potential benefits in providing opportunities for heat sharing among 

co-located process plants. Song et al. (2016) used an interplant shifted composite curve with an intermediate 

fluid to integrate heat among multiple plants, where each plant has a single period of operation for its process 

parameters. Song et al. (2017) used a pinch technology-based screening algorithm combined with the 

theoretical maximum heat recovery potential for heat integration among multiple plants. The energy hub 

approach for IPHI, where only interplant process heat exchange can occur at the energy hub, was adopted by 

Chang et al. (2018). Isafiade et al. (2022) modified the energy hub approach for IPHI problems involving multiple 

periods of operations by restricting intra-plant process heat exchange to the individual plant levels, while the 

energy hub is reserved for both inter-plant process heat exchange and heat exchange with utilities. Isafiade 

469



(2023) extended the work of Isafiade et al. (2022) for IPHI problems involving multiple periods of operations to 

the case where periodic durations across plants are unequal.  

In IPHI problems, where individual plants form coalitions to share energy, game theory helps guide strategic 

decisions about coalition formation. The Shapley value (Shapley, 1953), a concept from game theory, can be 

used to determine each plant’s marginal contribution to the coalition’s total annual cost (TAC). Cheng et al. 

(2014) applied Game Theory and a four-step approach to design an interplant heat-integrated network that 

maximizes the financial benefit for each participating plant. Jin et al. (2017) utilized cooperative Game Theory 

and a risk-based Shapley value to allocate co-construction costs among multiple plants in a coalition. For IPHI 

problems involving plants with multiperiod process parameter profiles and unequal periodic durations, it is 

essential to ensure fair cost allocation, an area where Game Theory offers valuable tools. To the best of the 

author's knowledge, this aspect has not been addressed in existing literature. This paper advances the state of 

the art by applying Game Theory to determine equitable cost distribution in IPHI scenarios where both the 

process parameters and the durations of their variations differ across plants and periods. 

2. Problem statement  

The problem addressed in this paper involves a set of co-located process plants, P, where each plant contains 

a set of hot, H, and cold, C, process streams. Each stream is characterized by a supply temperature (Ts), a 

target temperature (Tt), and a heat capacity flow rate (FCPH for hot streams and FCPC for cold streams). The 

process parameters in each plant follow a multiperiod profile, represented by set T, with unequal durations 

across the different plants. Additional parameters include interplant layout data, such as distances between 

plants and between each plant and f as well as stream-specific heat transfer coefficients and cost-related data, 

including cost functions for heat exchangers and for hot and cold utilities available at the energy hub. The 

objective is to synthesize an interplant heat integration network that minimizes the TAC by optimally integrating 

the heat demands of the multiple co-located plants with each other and with the energy hub. This must be 

achieved while accounting for variations in process parameters and unequal period durations across plants and 

ensuring a fair allocation of both annual operating and annual capital costs among the coalition members. 

3. Methodology  

The methodology adopted and the case study investigated in this paper is based on the work of Isafiade (2023) 

with the superstructure for the multi-plant network shown in Figure 1. In the figure, a stream within any plant can 

exchange process heat with another stream from the same plant (intra-plant heat exchange). At the energy hub, 

it can also exchange heat with a stream from a different plant (inter-plant heat exchange) or with a utility. The 

dashed lines indicate that the streams originate from process units, enter the heat exchange network to transfer 

heat, and then proceed to other process units. Since the case study includes multiperiod profile with unequal 

period duration within each plant and across plants, Isafiade (2023) developed the mapping approach shown in 

Figure 2. In Figure 2, although each plant has daytime and nighttime periods, it’s only in plant 1 that the duration 

of daytime period (0 – 12 h), represented by green arrow, is equal to that of nighttime (12 – 24 h), represented 

by orange arrow. For plants 2 and 3, the daytime period duration is less than nighttime. This necessitates the 

use of the Time Slice periodic mapping approach developed by Isafiade (2023) for IPHI with unequal period 

durations across plants. The objective function based on the maximum area approach used by Isafiade (2023) 

which is also used in this paper is shown in Eq(1). In this equation, AF (0.2 /y) represents annualization factor 

for heat exchangers and pipes, CF ($ 5,500) represents installation cost for heat exchangers, AC (700 $/m2) 

represents cost per heat exchanger area, AE represents cost exponent for heat exchanger areas, binary 

variable yp,i,j,k indicates whether a match exists between streams i and j as intra-plant heat exchanger in plant p 

or as interplant heat exchanger at the energy hub. In Eq(1), the average interplant distance and distance 

between each plant and the energy hub is represented by Disti,j (m), while PCp,i ($/m) and PCp,j ($/m) represent 

the per unit costs of pipes connecting each plant to the energy hub (same parameters used by Chang et al. 

(2018)), DOPts represents the duration of each periodic time slice ts, while the number of periodic time slices is 

represented by NOP. The unit costs of cold and hot utilities are represented by CUC ($/(kW∙y)) and HUC 

($/(kW∙y)) while heat exchanger heat load is represented by qp,I,j,k,ts (kW) and area of the representative heat 

exchanger is represented as Ap,I,j,k (m2). To adequately account for the fractional contributions of quantity of 

utilities consumed per period per plant in Eq(1), Isafiade (2023) applied the weighting terms that are multiplied 

by annual operating costs of cold and hot utilities and summed over the time slices.   

This paper builds on the work of Isafiade (2023) by regarding the 3-plant network as players in a coalition and 

then applying Shapley values to determine the economic benefits gained by each plant in participating in the 

coalition. For a detailed description of the Time Slice periodic mapping approach, the reader is referred to 

Isafiade (2023). For the case considered in this paper, the computation of the Shapley value of plant p is as 

470



shown in Eq(2). In the equation, ∅𝑝 denotes payoff for plant p in the game defined by the value function v, s 

represents the size of coalition S while N represents the set of all players and n the total number of players in 

N. Eq(1) is first applied to all possible coalitions to determine their TAC, after which the Shapley values are 

computed using Eq(2). 

   

Figure 1: Interplant heat integration superstructure (Isafiade, 2023)  

4. Case study 

The case investigated is taken from Isafiade (2023). Tables 1 and 2 show the process hot stream data for plants 

1, 2 and 3 for daytime and nighttime periods, while the equivalents for process cold streams are shown in Tables 

3 and 4. For data used for hot and cold utilities and average distances between plants and the energy hub, the 

reader is referred to Isafiade et al. (2022).     

Min 𝑇𝐴𝐶 = 𝐴𝐹 {𝐶𝐹. ∑ ∑ ∑ ∑ 𝑦𝑝,𝑖,𝑗,𝑘 + 𝐴𝐶

𝑘∈𝐾𝑗∈𝐶𝑖∈𝐻𝑝∈𝑃

. ∑ ∑ ∑ ∑[𝐴𝑝,𝑖,𝑗,𝑘]
𝐴𝐸

𝑘∈𝐾𝑗∈𝐶𝑖∈𝐻𝑝∈𝑃

+ ∑ ∑ ∑ ∑ (𝑦𝑝,𝑖,𝑗,𝑘 ∙ (𝐷𝑖𝑠𝑡𝑖,𝑗 ∙ 𝑃𝐶𝑝,𝑖 + 𝐷𝑖𝑠𝑡𝑖,𝑗 ∙ 𝑃𝐶𝑝,𝑗))

𝑘∈𝐾𝑗∈𝐶𝑖∈𝐻𝑝∈𝑃

}

+ ∑ (
𝐷𝑂𝑃𝑡𝑠

∑ 𝐷𝑂𝑃𝑡𝑠
𝑁𝑂𝑃
𝑡𝑠=1

∙ ∑ ∑ ∑ ∑ 𝐶𝑈𝐶 ∙ 𝑞𝑝,𝑖,𝑗,𝑘,𝑡𝑠

𝑘∈𝐾𝑗∈𝐶𝑖∈𝐻𝑝∈𝑃

)

𝑡𝑠∈𝑇𝑆

      

+ ∑ (
𝐷𝑂𝑃𝑡𝑠

∑ 𝐷𝑂𝑃𝑡𝑠
𝑁𝑂𝑃
𝑡𝑠=1

∙ ∑ ∑ ∑ ∑ 𝐻𝑈𝐶 ∙ 𝑞𝑝,𝑖,𝑗,𝑘,𝑡𝑠

𝑘∈𝐾𝑗∈𝐶𝑖∈𝐻𝑝∈𝑃

)

𝑡𝑠∈𝑇𝑆

   

(1) 

 

∅𝑝
∗ (𝑣) = ∑

𝑠! (𝑛 − 𝑠 − 1)!

𝑛!
𝑠⊑𝑁\𝑝

[𝑣(𝑆 ∪ 𝑝) − 𝑣(𝑆)] (2) 

Since the case study comprises 3 plants, there are 7 possible coalitions. Coalition 1 comprises only plant 1 (P1), 

coalition 2 comprises only plant 2 (P2) and coalition 3 comprises only plant 3 (P3). The other four possible 

coalitions, which are P1-P2, P1-P3, P2-P3 and P1-P2-P3 as well as their TACs and heat exchange profiles, are 

shown in Table 5. The Time Slice model of Isafiade (2023) was modified and solved for each of the possible 

coalitions. For the coalition involving only plant 1, the TAC is $414,764. This involves two intra-plant heat 

exchangers and three utility exchangers. The equivalent TACs for P2 only coalition and P3 only coalition are 

$51,039 and $106,027. The P2 only coalition involves two intra-plant heat exchangers and two utility heat 

s

t

s

t

t
s

s

t

t

t

s

s

s

t

t

t

s

s

s

t

t

t

471



exchangers at the hub. The P3 only coalition involves three intra-plant heat exchangers and one utility 

exchanger at the hub. Coalitions P1-P2, P2-P3 and P1-P2-P3 all involve inter-plant heat exchange at the energy 

hub as shown in Table 5. 

Table 1: Process hot stream data for plants 1, 2 and 3 in daytime period (Isafiade et al., 2022)    

Plants Streams Ts (K) Tt (K) FCPH  

(kg/s) 

h  

(kW/(m2∙K)) 

Cp  

(kJ/(kg∙K)) 

ρ  

(kg/m3) 

DOPts 

(h) 

Plant 1 H1 660 370 2.0 1.0 3.2 780 12 

 H2 590 320 1.6 1.0 3.5 820  

Plant 2 H1 553 333 2.0 1.6 3.7 790 10 

 H2 493 310 2.5 1.6 3.3 810  

Plant 3 H1 520 390 2.0 2.1 3.2 782 8 

 H2 473 370 1.0 4.1 3.0 760  

Table 2: Process hot stream data for plants 1, 2 and 3 in nighttime period (Isafiade et al., 2022)    

Plants Streams Ts (K) Tt (K) FCPH  

(kg/s) 

h  

(kW/(m2∙K)) 

Cp  

(kJ/(kg∙K)) 

ρ  

(kg/m3) 

DOPts  

(h) 

Plant 1 H1 640 350 2.1 1.0 3.2 780 12 

 H2 600 310 1.8 1.0 3.5 820  

Plant 2 H1 543 333 1.9 1.6 3.7 790 14 

 H2 483 300 2.4 1.6 3.3 810  

Plant 3 H1 522 395 2.1 2.1 3.2 782 16 

 H2 479 380 1.8 4.1 3.0 760  

Table 3: Process cold stream data for plants 1, 2 and 3 in daytime period (Isafiade et al., 2022)    

Plants Streams Ts (K) Tt (K) FCPC 

(kg/s) 

h  

(kW/(m2∙K)) 

Cp  

(kJ/(kg∙K)) 

ρ (kg/m3) DOPts  

(h) 

Plant 1 C1 320 650 3.5 2.0 4.0 790 12 

 C2 360 500 3.3 2.0 3.3 770  

Plant 2 C1 303 430 2.0 2.6 4.1 810 10 

 C2 363 413 2.5 2.6 3.4 790  

Plant 3 C1 303 398 3.5 2.1 3.9 790 8 

 C2 308 373 3.9 2.1 3.3 780  

Table 4: Process cold stream data for plants 1, 2 and 3 in night-time period (Isafiade et al., 2022)   

Plants Streams Ts (K) Tt (K) FCPC  

(kg/s) 

h  

(kW/(m2∙K)) 

Cp  

(kJ/(kg∙K)) 

ρ  

(kg/m3) 

DOPts  

(h) 

Plant 1 C1 310 640 3.4 2.0 4.0 790 12 

 C2 350 500 3.5 2.0 3.3 770  

Plant 2 C1 293 420 2.1 2.6 4.1 810 14 

 C2 353 415 2.2 2.6 3.4 790  

Plant 3 C1 299 390 3.4 2.1 3.9 790 16 

 C2 296 373 3.9 2.1 3.3 780  

 

For P1-P2 coalition, exchanger H1,P2-C2,P1 transfers process heat at the hub between hot stream 1 in plant 2 

and cold stream 2 in plant 1. This heat exchange takes place in time slice TS3. In this time slice, plant 1 is in 

the daytime period while plant 2 is in the nighttime period. For the P2-P3 coalition, exchanger H1,P2-C1,P3 is 

involved in interplant heat exchange at the energy hub. For the grand coalition involving P1-P2-P3, there are 

three exchangers involved in interplant heat exchange at the energy hub. The marginal contributions of each 

plant to the TAC of the grand coalition are: P1 = $ 369,997, P2 = -$ 30,339, P3 = $ 78,690. This implies that by 

entering the grand coalition, plant 1 saves 44,767 $/y while plant 3 saves 27,337 $/y. On the other hand, plant 

2 will receive 30,339 $/y to join the coalition, with 25,018 $/y contributed by plant 1 and 5,321 $/y by plant 3. 

From Table 5, the 2 two-plant coalitions, i.e., P1-P2 and P2-P3, having interplant heat exchange at the hub 

involves plant 2 while for the grand coalition, two of the three interplant heat exchange at the hub include plant 

2.  

472



Table 5: TAC, heat exchanger area and heat load distribution among the seven possible coalitions  

Coalitions  Exchanger  Area (m2) Heat load (kW) 

TAC ($/y)    TS1 

(0 - 8 h) 

TS2 

(8 – 10 h) 

TS3 

(10 – 12 h) 

TS4 

(12 – 24 h) 

P1-P2 Intra-plant heat exchange      

335,634  H1,P1-C1,P1 17 1,856 1,856 1,856 1,856 

  H2,P1-C1,P1 72.4 1,486.4 1,486.4 1,486.4 1,486.4 

  H1,P2-C1,P2 23.8 252 252 347.9 347.9 

  H2,P2-C1,P2 29 535.7 535.7 353 353 

  H2,P2-C1,P2 45.6 253.8 253.8 392.5 392.5 

  H2,P2-C2,P2 11.4 425 425 463.8 463.8 

 Interplant heat exchange at the hub     

  H1,P2-C2,P1 54.6 1,376 1,376 1,376 1,128.4 

 Heat exchange with utilities at the energy hub     

  HU1-C1,P1 28.1 1,277.7 1,277.7 1,277.7 1,277.7 

  H2,P1-CU1 1.2 25.7 25.7 25.7 25.7 

  HU2-C2,P1 3.8 148.6 148.6 148.6 396.2 

  H2,P2-CU2 13 295.3 295.3 240 240 

P1-P3 Intra-plant heat exchange      

498,703  H1,P1-C1,P1 27 1,856 1,856 1,856 1,948.8 

  H2,P1-C1,P1 87.8 1,496 1,496 1,496 1,762.5 

  H1,P3-C1.P3 4.1 497.5 397.1 397.1 397.1 

  H1,P3-C2,P3 3.9 334.5 456.4 456.4 456.4 

  H2,P3-C2,P3 7.9 309 534.6 534.6 534.6 

 Heat exchange with utilities at the energy hub     

  HU1-C1,P1 27.95 1,268 1,268 1,268 776.7 

  H2,P1-CU3 1.1 16.04 16.04 16.04 64.5 

  HU2-C2,P1 16 1,524.6 1,524.6 1,524.6 1,732.5 

  HU2-C1,P3 5.2 799.3 809.6 809.6 809.6 

P2-P3 Intra-plant heat exchange      

61,756  H1,P2-C1,P2 39 828.8 819 666.7 666.7 

  H1,P2-C1,P3 5.9 799.3 809.6 809.6 809.6 

  H2,P2-C1,P2 6 212.7 223 426.8 426.8 

  H2,P2-C2,P2 12.4 425 425 463.8 463.8 

  H1,P3-C1,P3 4.1 497.5 497.5 397.1 397.1 

  H1,P3-C2,P3 3.4 334.5 334.5 456.4 456.4 

  H2,P3-C2,P3 10.1 309 309 534.6 534.6 

 Heat exchange with utilities at the energy hub     

  H2,P2-CU2 14.6 872.1 861.7 558.8 558.8 

P1-P2-P3 Intra-plant heat exchange      

418,348  H1,P1-C1,P1 27 1,856 1,856 1,856 1,948.8 

  H2,P1-C1,P1 87.8 1,496 1,496 1,496 1,762.5 

  H1,P2-C1,P2 34.7 540.9 385 348.8 140.9 

  H2,P2-C1,P2 43.6 213 303.1 242.7 242.7 

  H1,P3-C2,P3 5.1 334.5 456.4 456.4 4564 

  H2,P3-C2,P3 7.9 309 534.6 534.6 534.6 

 Interplant heat exchange at the hub   

  H1,P2-C2,P1 82.3 1,027.1 1,127.6 1,127.6 1,335.5 

  H2,P2-C1,P3 20.2 1,296.8 1,206.7 1,206.7 1,206.7 

  H1,P3-C2,P1 11.6 497.5 397.1 397.1 397.1 

 Heat exchange with utilities at the energy hub     

  HU1-C1,P1 27.95 1,268 1,268 1,268 776.7 

  H2,P1-CU3 1.1 16 16 16 64.5 

  HU2-C1,P2 5.5 287.5 353.3 502 710 

  HU2-C2,P2 3.7 425 425 463.8 463.8 

  H1,P2-CU1 2.3 60 115.5 - - 

473



 

Figure 2: Duration of time slices for case study investigated (Isafiade, 2023). 

5. Conclusions 

This paper has demonstrated the integration of Game theory, through Shapley values computation, with IPHI 

problems involving multiple periods of operations with unequal durations. The method adopted builds on the 

energy hub approach where both heat exchange with utilities and interplant process heat exchange can take 

place at the energy hub. The results obtained show that heat sharing among multiple plants through interplant 

heat exchange at the energy hub has the potential to reduce the TAC that a plant would incur if it were not 

involved in a coalition with other plants. Future work will include detailed calculations of costs of pumping 

between plants and the energy hub and detailed heat losses along pipes transporting fluids. Also, the marginal 

contributions of each plant to environmental impacts of the grand coalition will be investigated simultaneously 

with the marginal contributions to the TAC through multi-objective optimisation.    

Acknowledgments 

The financial support of the University of Cape Town is acknowledged. 

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	0231.pdf
	Application of Game Theory to Interplant Heat Exchanger Networks Involving Multiple Unequal Period Durations