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1
The Design of By-product Hydrogen Supply Chain
Considering Large-scale Storage and Chemical Plants:
A Game Theory Perspective
Qianni Cao, Student Member, IEEE, Boda Li, Student Member, IEEE, Mengshuo Jia, Member, IEEE, and
Chen Shen, Senior Member, IEEE
Abstract—Hydrogen, an essential resource in the decarbonized
economy, is commonly produced as a by-product of chemical
plants. To promote the use of by-product hydrogen, this paper
proposes a supply chain model among chemical plants, hydrogen-
storage salt caverns, and end users, considering time-of-use
(TOU) hydrogen price, coalition strategies of suppliers, and
road transportation of liquefied and compressed hydrogen. The
transport route planning problem among multiple chemical
plants is modeled through a cooperative game, while the hydrogen
market among the salt cavern and chemical plants is modeled
through a Stackelberg game. The equilibrium of the supply
chain model gives the transportation and trading strategies of
individual stakeholders. Simulation results demonstrate that the
proposed method can provide useful insights on by-product
hydrogen market design and analysis.
Index Terms—Hydrogen market, large-scale storage, Stackel-
berg game, cooperative game, supply chain
NOMENCLATURE
Indices
i, j Index of chemical plants.
tIndex of time periods during the day.
n
Index of hydrogen processing equipment, in-
cluding liquefiers and compressors.
I+ 1 Index of the salt cavern.
Parameters
INumber of chemical plants.
TNumber of time periods.
po
Retail price purchased by customers from the
salt cavern.
pt, pt
Lower and upper bound of the buying price
offered by the salt cavern to chemical plants.
Qtrans Maximal injection rate of the salt cavern.
NC
Number of compressors with different capacity.
NDNumber of liquefiers with different capacity.
Qi,t
By-product hydrogen quantity produced by
chemical plant iin period t.
Qpr
Capacity set of hydrogen processing equip-
ment(kg/h), Qpr ={Qn
pr},∀n.
QC, QD
Capacity of a tube trailer and a tanker truck
(kg/trip).
wtElectricity price in period t.
γc, γd
Electricity consumption for unit compressed
hydrogen and liquefied hydrogen (kwh/kg).
K1
Initial investment set of hydrogen processing
equipment, K1={Kn
1},∀n.
Kc
2, Kd
2
Initial investment cost of a tube trailer and a
tanker truck.
K3
Operation cost of a tube trailer (or a tanker
truck) in each period.
TaTi,j
a
represents duration from chemical plant
i
to j(j=I+ 1 represents the salt cavern).
βL1
1 - hourly evaporation rate during the tanker
truck loading.
βL2
1 - hourly evaporation rate during transit by a
tanker truck.
Decision variables of the salt cavern
pt
Buying price the salt cavern offers to chemical
plants in period t.
qtrans
i,t
Hydrogen transaction amount of chemical plant
i
in period
t
, measured as hydrogen shipped
from chemical plant
i
at the end of the time
period t.
ui,I+1
Binary variables. Equals to 1 when products
from chemical plant
i
is shipped directly to the
salt cavern. Otherwise, ui,I+1 equals to 0.
Decision variables of chemical plants
qpr
i,t
Hydrogen quantity chemical plant
i
com-
pressed/liquified in period t.
xixi={xi},∀n
is a set of binary variables.
xn
i= 1
when the type of hydrogen processing
equipment is selected to purchase. Otherwise,
xn
i= 0.
Ncars
i
Integer variables of number of tube trailers (or
tanker trucks) purchased by chemical plant i.
ui,j
Binary variables. Equals to 1 when products
from chemical plant
i
is shipped to chemical
plant j. Otherwise, ui,j equals to 0.
qstore
i,t
Hydrogen quality in the tube trailer (or tanker
truck) left at chemical plant
i
before filled to
capacity in period t.
qunpr
i,t
Hydrogen quantity temporarily stored in low-
pressure storage tanks before compression or
liquefication in period t.
ncars
i,t
Integer variables of tube trailers (or tanker
trucks) leave chemical plant iin period t.
arXiv:2211.03118v1 [eess.SY] 6 Nov 2022
2
I. INTRODUCTION
A. Motivation
I
N the context of emission peak and carbon neutrality,
hydrogen is not only regarded as a critical alternative to
fossil fuel to achieve carbon neutrality but offers versatility
and flexibility that renewables cannot reach [1]. As one of
the most cost-effective options, hydrogen produced as a by-
product from many chemical plants serves as a cheap and
large-scale source of hydrogen. Moreover, by-product hydrogen
is usually sufficiently clean and well suited for a wide range
of applications, such as fuel cell (FC)-based cogeneration,
FC vehicles, domestic heating, and so on [2]. However, the
potential of by-product hydrogen has yet to be realized, which
is emitted and thus wasted in most cases. Therefore, it presents
opportunities as a new revenue stream for chemical plants
and promisingly delivers on announced pledges of energy
conversion nationwide in the mid-term. However, the lack of
infrastructure development such as large-scale storage, logistical
supply chain establishment and unexplored market have slowed
down its further development.
Salt cavern storage is one of the most promising technologies
to achieve large-scale, fast and secure hydrogen storage
[3],which offers the most promising option owing to their low
investment cost, high sealing potential and low cushion gas
requirement [4]. Notable projects are the salt cavity storages for
hydrogen in Teeside, UK, and Texas, USA [5], demonstrating
the operation feasibility on a full industrial scale. However, the
business of acquiring, storing and selling by-product hydrogen
has not yet been presented as an option by salt cavern operators,
which inspires the work to design by-product hydrogen supply
chain considering large-scale storage and chemical plants in
this paper.
B. Literature Review
As demand and production capacity for hydrogen grows
robustly in recent years, the outlines of hydrogen markets
are starting to emerge worldwide. Initial trade and market
price discoveries come first on a regional and local basis [6].
Infrastructure development, transparent pricing benchmark and
logistical supply chain establishment are key growth challenges
faced by this new traded commodity just becoming established
in energy commodity markets [1].
Presently, the hydrogen market is far from mature but is
showing great potential. Many researchers focus on the planning
of the hydrogen supply chain, considering various market
scales, hydrogen sources and transportation modes. Life cycle
analysis to estimate the economic and environmental benefits
was conducted on global [7], regional [8] or national [9] scales.
For different hydrogen sources, steam methane reforming
(SMR) [10] , coal gasification (CG) [11], biomass gasification
(BG) [12], [13] and electrolysis (ELE) [14] are common
production technologies in recent researches. Considering
hydrogen production based on different feedstocks and energy
sources, an optimal structure of the hydrogen, biomass and
CO2
networks were determined in [15]. To make comparisons
of different transportation modes, Ref. [16] considered four
common options with various criteria and scenarios. Ref.
[17] introduced a method for comparing different transport
possibilities of tube or liquid trailer vs. pipeline delivery.
The results showed that each transportation technology had
a maximally cost-efficient niche and there was no single
perfect solution for the entire system. Recently, large-scale
storage for liquid hydrogen is of great attention. Ref. [18]
considered integrated bulk storage of hydrogen and concluded
that a centralized storage structure and liquefication in central
production plants can reduce the overall cost. Similarly, the
status and key gaps for the commercialization of hydrogen
liquefication technology with large-scale storage were discussed
in [19]. A combination of the hydrogen supply chain with
other energy sources has also attracted the attention of many
researchers. Ref. [20] established a local energy market for
electricity and hydrogen. Ref. [21] proposed a methodological
design framework for hydrogen and methane supply chains
based on Power-to-Gas systems.
In particular, by-product hydrogen has seen growing attention
these years. Ref. [22] for the first time assessed the economic
advantages, the techno-economic feasibility and the central role
of reusing by-product hydrogen in the early phase of hydrogen
infrastructure in the northern Spain region. A multi-period
programming was designed in [23] to make use of existing
infrastructure for by-product hydrogen and natural gas (NG)
pipelines, which demonstrated the economic benefits of by-
product hydrogen. Even though, the potential of by-product
hydrogen remains to be discovered.
Meanwhile, most of the literature focuses on maximizing
the total benefit of the whole hydrogen supply chain. Ref.
[24] aimed to maximize social welfare in Korea by planning
both capacity and technology of production, storage as well
as transportation in an envisioned nationwide hydrogen supply
chain. Ref. [25] assessed the effects that hydrogen grades
play in the development of a cost-effective hydrogen supply
chain. Ref. [26] incorporated the concept of biophysical limits
of the planet to address the optimal design of the hydrogen
supply chain. An optimization method was proposed in [27]
for an integrated value chain of carbon dioxide and hydrogen.
Individual rationality was introduced in [28], where the peer-
to-peer transaction, endogenous market-clearing price, and
uncertainties in hydrogen production were considered in detail.
However, most works failed to consider the strategic behaviors
and the profit of individual participants, which differed from the
usual practice that suppliers and retailers are private companies
and operate with a profit-driven mode.
The research gaps for the existing works are:
1)
The potential of by-product hydrogen is yet to be realized
and its corresponding market is waiting for further
exploration.
2)
The dynamic process of chemical plants and salt caverns
considering hydrogen generation, compression (or lique-
faction), and the transaction is waiting to be modeled.
3)
The interactions and dynamic strategic behaviors of each
stakeholder desire a more dedicated modeling frame-
work that captures profits and rationality of individual
participants.
3
C. Contribution
In this work, we study the by-product hydrogen supply chain
considering large-scale storage and multiple chemical plants.
The main contributions are threefold:
1)
We establish a business model for salt caverns to acquire
and store by-product hydrogen from chemical plants and
sell them to end-users. The by-product hydrogen supply
chain composed of each stakeholder in the business
model is investigated.
2)
The hour-by-hour decision-making process of each
stakeholder, i.e., chemical plants and the salt cavern,
is investigated and mathematically modeled under the
proposed business model, providing a foundation for the
TOU hydrogen pricing strategy.
3)
The by-product hydrogen market is formulated as a game,
considering the individual rationality of each stakeholder.
The planning problem among multiple chemical plants
is modeled through a cooperative game. The hydrogen
market among the salt cavern and chemical plants is
modeled through a Stackelberg game, in which the salt
cavern is the leader and chemical plants are the followers.
II. A BUSINESS MODEL OF SALT CAVERNS AND CHEMICAL
PL AN TS
In this section, we develop a business model for salt caverns
to acquire by-product hydrogen from chemical plants and
sell them to end-users. Generation, large-scale storage, and
consumptive way of by-product hydrogen in the business
model is introduced first. Then, the comparison between the
by-product hydrogen supply chain and the present hydrogen
supply chain is made. Followed by this, the structure of the by-
product hydrogen market under the proposed business model
is introduced in the following section.
A. Generation, Large-scale Storage and Consumptive Way of
By-product Hydrogen
By-product hydrogen is a cost-competitive and widely
distributed source of hydrogen. The process of generation of
by-product hydrogen and its consumptive ways are illustrated
in Fig.1.
Fig. 1. The process of by-product hydrogen generation and its consumptive
ways
Electrochemical processes, such as the industrial production
of steel, caustic soda and chlorine, produce hydrogen as a
by-product, burnt or emitted as the current practice. However,
they can be made available for applications outside chemical
plants as a future consumptive way. To transport products
from the production facilities to storage sites, by-product
hydrogen should be compressed or liquified in advance, which
collectively are referred to as “hydrogen secondary processing”.
Two common transportation modes are compressed gaseous
hydrogen via tube trailers (CH2) and liquid hydrogen via tanker
trucks (LH2). To alleviate the imbalance between supply and
demand of hydrogen, underground cavities like salt caverns are
potential to offer natural infrastructure to realize cost-effective
and reliable hydrogen storage. At the last link in the supply
chain, by-product hydrogen is sold and distributed to various
end-users. The proposed generation, storage and consumptive
way of hydrogen give rise to a promising by-product hydrogen
business model consisting of chemical plants as suppliers, a
salt cavern as a retailer and end-users as consumers.
B. Characteristics of By-product Hydrogen Supply Chain
Differences between the by-product hydrogen supply chain
under the proposed business model and most hydrogen supply
chains found in literature can be mainly concluded as twofold:
1) composition of major costs; 2) flexibility to coordinate
between planning and scheduling. These differences will lead
to a distinct focus and a smaller timescale for the formulation
of the by-product hydrogen supply chain, which is analyzed
as follows:
TABLE I
MAJOR COSTS OF HYDROGEN SUPPLY CHAIN
Major costs Hydrogen Supply Chain
Traditional By-product
Production Investment X
Operation X
Storage Investment X
Operation X
Transportation Investment X X
Operation X X
Secondary
processing
Investment X
Operation X
TABLE II
MAJ OR CO ST S AND T HE IN FLU ENC E FACT ORS
Major costs Influence factors
Production 1) Production technology
2) Scale of production
Storage 1) Storage technology
2) Storage capacity
Transportation
1) Transportation mode
2) Hydrogen volume
3) Transport distance
Secondary
processing
1) Type of processing equipment
2) TOU electricity price
3) Hydrogen volume
1) Different composition of major costs: Major costs of
hydrogen supply chain and their influence factors are demon-
strated in Table I and II, respectively. Unlike the present
hydrogen supply chain, producers in the by-product hydrogen
supply chain benefit from very low-cost generation. Thus, the
major cost comes from secondary processing and transportation.
4
Power is the major cost for secondary processing. If the
liquefier or compressor operates at low-price periods, it may
potentially reduce operating costs. Since electricity price
fluctuates by hours, the strategic behaviors of each stakeholder
should also be modeled by hour.
Transport cost is determined by transportation mode, hydro-
gen volume and the transport distance. For two transportation
modes considered in this paper, LH2 features large transport
capacity (often 10-20 times as CH2), high initial investment
cost (several times as CH2) and hourly volatile losses. On
the contrary, CH2 features low transport capacity, low initial
investment cost and zero loss. Usually, for long-distance
transportation of a large amount of hydrogen, CH2 is less
economical since it requires long rides of much more vehicles
than LH2. However, for mid- or short-distance of a small
amount of hydrogen, CH2 is more economical since there is no
volatile loss. Obviously, a reasonable decision of transportation
mode would largely reduce the cost of each chemical plant.
2) Less flexibility to coordinate between planning and
scheduling : For suppliers in the by-product hydrogen supply
chain, the generation scale of hydrogen is limited by the
production plan of their main products. Moreover, their location
is less likely to be optimized for the transportation of by-product
hydrogen.
Therefore, there may be a mismatch between each supplier’s
location and generation scale. Specifically, for distant (to the salt
cavern) and medium-yield chemical plants, if CH2 is adopted,
long-distance transport of more tube trailers may result in high
transportation costs. Nevertheless, if LH2 is adopted, substantial
volatile losses would happen due to hours of filling time. This
situation results in a dilemma since both transportation mode
leads to a revenue decline in some way. Therefore, we envision
a scenario where several chemical plants in proximity to each
other form a coalition and select a transit hub between them
to lower transportation costs, instead of shipping individually
to the salt cavern. Two examples of envisioned transportation
routes are highlighted in color in Fig.2. Moreover, to lower
transportation costs, chemical plants destined for the transit
hub adopt the CH2 transportation mode, while the transit hub
destined for the salt cavern adopt the LH2 transportation mode.
In this way, the dilemma between high transportation costs of
CH2 and large volatile loss of LH2 is mitigated.
Fig. 2. Possible routes for the salt cavern to acquire hydrogen from the
chemical plants
To sum up, cost structure differences and the lack of
flexibility to coordinate between production scale and location
lead to a gap between the by-product hydrogen supply chain
and the present ones. Therefore, it is essential to model the by-
product hydrogen supply chain according to its characteristics
rather than simply applying the model of the traditional
hydrogen supply chain.
C. The Structure of By-product Hydrogen Market
The structure of the proposed by-product hydrogen market is
provided in this subsection, followed by the basic assumptions.
Fig. 3. The structure of the by-hydrogen market under investigation
The by-hydrogen market under the proposed business
model has the structure illustrated in Fig.3. Suppliers, namely
chemical plants, process by-product hydrogen by liquefiers or
compressors (depends on the decision results of each supplier)
and deliver it to the retailers. The retailers, namely salt caverns,
sell hydrogen to the customers. To simplify the problem, salt
caverns are regarded as an entity owned by a single company.
This paper focuses on the transaction between suppliers and
retailers. The following assumptions are made without loss of
generality:
1)
The end-users buy all the hydrogen from the retailer
at a fixed price. This may happen when the injection-
production rate of the salt cavern is higher than the
market demand in a region. In order to alleviate the
supplier’s market power to drive up prices, we assume
that the salt cavern and suppliers have reached such an
agreement to bring a fixed price into effect.
2)
The secondary processing cost and transport cost is
undertaken by suppliers.
3)
The production cost is neglected since hydrogen is a
by-product of the industrial process of chemical plants.
4)
Chemical plants would not adjust their production sched-
ule of their main product for the revenue generated by
by-product hydrogen.
Based on the above assumptions, the retailer’s and suppliers’
problems can be described as follows. To maximize profits, the
salt cavern intends to purchase as much hydrogen as possible
from chemical plants at the lowest cost. If the price is too
low, chemical plants are less likely to be attracted by this
new revenue stream and may waste them as before, which
reduces profits of the salt cavern. On the contrary, if the price
5
is too high, the purchasing cost would increase. Therefore, it
is important for the salt cavern to strike a balance between
the attraction of chemical plants and the purchasing cost. To
maximize profits, chemical plants upstream would like to sell
more hydrogen when the selling price is high on the one hand,
and to reduce processing costs and transport costs on the other
hand.
Taking into account the analysis in the last subsection,
the challenges of modeling the by-product hydrogen supply
chain under the proposed structure are mainly twofold: 1) to
explicitly consider possible coalition structures and transport
route strategies in the timescale of transport duration, electricity
price fluctuation and volatile losses; 2) and to allocate the payoff
among the producers in some fairway.
III. STR ATEG IE S AN D DE CI SI ON -MA KI NG P ROC ES S OF
STAK EH OL DE RS
In this section, the decision-making process of each stake-
holder is investigated and mathematically modeled under the
proposed business model.
A. The Retailer’s Problem
In the price-setting problem of the salt cavern, the retailer
decides its buying price
pt
(offered to the suppliers), while
considering the reactions
qtrans
i,t
from suppliers. The problem
can be formulated as
max
pt
po
I
X
i=1
T
X
t=1
qtrans
i,t ui,I+1 −
I
X
i=1
T
X
t=1
ptqtrans
i,t ui,I+1 (1)
s.t. pt≤pt≤pt,∀t(2)
I
X
i=1
qtrans
i,t−Ti,I+1
a≤Qtrans,∀t(3)
Objective
(1)
is the retailer’s profit in which the first term is
the selling income, and the second term is the purchasing
cost. Inequality
(2)
restricts the price offered to suppliers
to be within the interval
[pt, pt]
in each period. Here we
assume that the retailer and suppliers have already reached
an agreement to bring this constraint into effect. Inequality
(3)
prescribes maximal transaction quantity in each period by
maximal injection rate of the salt cavern.
qtrans
i,t
and
ui,I+1
are the optimal solution to the suppliers’ problem.
B. The Suppliers’ Problem
For the suppliers, the planning of the type of processing
equipment, transportation mode, and the transport route as
well as scheduling of transaction quantity, is formulated in
this subsection. To capture the dynamic process of hydrogen
transactions between each stakeholder in detail, as well as
investigating dynamic strategic behaviors of each stakeholder,
the loading process is elaborately taken into consideration.
Specifically, hydrogen is produced as a by-product along with
main products and has three possible disposal ways:
1)
Hydrogen can be loaded to a tube trailer (or a tanker
truck) after compression (or liquefaction). At the end of
period
t
, tube trailers (or tanker trucks) filled to maximum
capacity should depart from chemical plants. Otherwise,
they stay until filled up in the following periods. There-
fore, the transaction quantity sequence
qtrans
i,t
depends on
hydrogen processing quantity sequence
qpr
i,t
and capacity
of the vehicle (
QC
for a tube trailer and
QD
for a tanker
truck).
2)
Hydrogen can also be temporarily stored in low-pressure
storage tanks before liquefication or achieving an ade-
quate compression rate. It will further be loaded into
tube trailers (or tanker trucks) after being compressed
(or liquified) in the following periods.
3)
Hydrogen may also be discarded by being emitted or
burnt as the current practice, which may happen when
buying price offered by the salt cavern is too low or
low-pressure storage tanks are filled up.
The above three disposal ways offer multiple options for
chemical plants during planning and scheduling. For example,
a chemical plant with a generation volume of 100kg per hour,
may purchase processing equipment of 100kg per hour. Thus,
hydrogen can be processed hour-by-hour. An alternative is to
purchase processing equipment of 1000kg per hour. In this
case, by-product hydrogen can be temporarily stored in low-
pressure storage tanks and will be processed every 10 hours.
The suppliers’ problem is to find optimal solutions for planning
and scheduling while considering possible coalitions with each
other.
In the suppliers’ problem, if the destinations of all suppliers
for hydrogen shipment are the salt cavern, decision variables
should be the type of processing equipment
xi
and hydrogen
processing amount
qpr
i,t
; if the scenario of coalitions of suppliers
is taken into account, transport route
ui,j
of chemical plants
i
and j, which form a coalition.
The decision-making problem, including constraints and
objectives of supplier i, is given as follows.
1) Constraints on transit shipment pattern:
I+1
X
j=1
ui,j = 1,∀i(4)
ui,j +uj,i ≤1,∀i, j (5)
Constraint
(4)
denotes that the destination of each chemical
plant is unique. Constraint
(5)
defines that any pairs of the
chemical plant
(i, j)
wouldn’t select each other as the transit
destination simultaneously.
2) Constraints on hydrogen processing and transport
scheduling: Chemical plants adopting CH2 satisfy:
ncars
i,t ≤(qpr
i,t +qstore
i,t−1)/Qcar
c≤ncars
i,t + 1,∀t(6)
qtrans
i,t =ncars
i,t QC,∀t(7)
qstore
i,t =qstore
i,t−1+qpr
i,t −qtrans
i,t ,∀t∈ {2, ...T }(8)
Constraints
(6)
and
(7)
indicate that hydrogen transaction
amount in each period is an integer multiple of the capacity
of a tube trailer since only tube trailers filled to maximum
capacity will depart from chemical plants. Constraint
(8)
denotes variations of hydrogen quantity stored in low-pressure
storage tanks.
6
With the remaining proportion of hydrogen after being
shipped from chemical plant
i
to
j
(
j=I+ 1
represents
the salt cavern) written as
βi,j
L2=βL2Ti,j
a
, chemical plants
adopting LH2 satisfy
ncars
i,t ≤(qpr
i,t +βL1qstore
i,t−1)/Qcar
d≤ncars
i,t + 1,∀t(9)
qtrans
i,t =ncars
i,t QD
I+1
X
j=1
ui,j βi,j
L2,∀t(10)
qstore
i,t =βL1qstore
i,t−1+qpr
i,t −qtrans
i,t /
I+1
X
j=1
ui,j βi,j
L2,
∀t∈ {2, ...T }(11)
Constraints
(9)
and
(10)
indicate that the hydrogen trans-
action amount in each period is an integer multiple of the
capacity of a tanker truck. Constraint
(11)
denotes variations
of hydrogen quantity stored in low-pressure storage tanks.
Constraints irrelevant to transportation modes are given in
(12)
-
(15)
, in which the transport duration for chemical plant
i
is written as ti
ar =PI+1
j=1 ui,j Ti,j
a.
t
X
t−2×ti
ar
ncars
i,t ≤Ncars
i,∀t∈2×ti
ar, ...T (12)
qpr
i,t ≤
NC+ND
X
n=1
xn
iQn
type,∀t(13)
qunpr
i,t ≤
NC+ND
X
n=1
xn
iQn
type,∀t(14)
qunpr
i,t ≤qunprocess
i,t−1+Qi,t −qpr
i,t +
I
X
j=1
uj,iqt−Ti,j
a
j,trans,
∀t∈max (1, T i,j
a), . . . , T }(15)
Constraint
(12)
imposes the total number of tube trailers
(or tanker trucks) purchased by chemical plant
i
as the upper
bound of tube trailers (or tanker trucks) in the round trip during
the time period
t−2×ti
ar
. Constraint
(13)
prescribes the
processing capability of each chemical plant. Constraint
(14)
restricts the upper bound of hydrogen stored locally, and the
bound parameter is chosen as
PNC+ND
n=1 xn
iQn
type
. Constraint
(15)
represents variations of hydrogen stored locally, in which
‘
≤
’ indicates that hydrogen as a by-product can be stored
temporarily or directly discarded.
If destinations of all suppliers for hydrogen shipment are the
salt cavern, the objective of each chemical plant is to maximize
its daily profit and is given in
(16)
, in which the income by
selling hydrogen to the consumers, initial investment cost and
operation cost are considered.
max πF i =
T
X
t=1
(ptqtrans
i,t −Ci
O−Ci
T)−Ci
IN V 1−Ci
IN V 2
(16)
where
Ci
O=qpr
i,t
NC+ND
X
n=1
xn
iwtγcxc
i+γdxd
i(17)
Fig. 4. Game models involved in by-product hydrogen supply chain including
salt cavern and chemical plants
Ci
T=ncars
i,t
I+1
X
j=1
ui,j K3Ti,j
a(18)
Ci
IN V 1=
NC+ND
X
n=1
xn
iKn
1(19)
Ci
IN V 2=Ncars
i(xc
iKc
2+xd
iKd
2)(20)
where transportation mode is written as
xc
i=PNC
n=1 xn
i
and
xd
i=PNC+ND
n=NCxn
i
;
Ci
O, Ci
T
represent hourly processing
and transport cost respectively;
Ci
IN V 1
,
Ci
IN V 2
represent
investment cost of processing equipment and tube trailers (or
tanker trucks) after converted into daily cost with a discount
rate, respectively. If the scenario where coalitions of suppliers
are considered, we denote chemical plants in a coalition as
Γ
.
For the chemical plant
i
,
∀i∈Γ
, the objective is to maximize
the daily profit of the coalition and is given as (21).
max πF τ =X
i∈Γ
T
X
t=1
ptqtrans
i,t ui,I+1 −Ci
O−Ci
T
−Ci
IN V 1−Ci
IN V 2(21)
where
ui,I+1 = 1
when chemical plant
i
is chosen as a transit
hub. Otherwise ui,I+1 = 0.
IV. GAME FORMULATION AND SOLUTION
A. Game Formulation for By-product Hydrogen Supply Chain
In this section, the by-product hydrogen market is formulated
as a game, considering the individual rationality of each
stakeholder.
The decision-making process of each individual can be
concluded as follows. The suppliers plan their initial equipment
investment, coalition structure and transport routes in the plan-
ning stage. Then, the hydrogen transaction problem, including
the retailer’s pricing problem and suppliers’ scheduling problem,
is optimized in the scheduling stage.
The overall framework of the game models is illustrated in
Fig.4. Specifically, the planning problem of multiple chemical
plants is formulated as a cooperative game, in which a binding
coalition could be formed to reduce transport costs. The
hydrogen transaction problem between the salt cavern and
chemical plants is formulated as a Stackelberg game, in which
the salt cavern is the leader and chemical plants are the
followers.
1) Coorperative game in the planning stage: As previously
analyzed in subsection
II-B
, coalitions between chemical plants
would potentially lower transportation costs, thus bringing
7
collective payoffs. Moreover, to fairly allocate the payoff
πF τ
among the players, the Shapley value is adopted.
The following assumptions are made without loss of gener-
ality when considering possible coalition structures:
i) Chemical plants in each coalition select one of them as
a transit hub to which other chemical plants in the coalition
transport hydrogen. Since reducing transport costs is considered
as the key factor behind the coalition, we assume that two
chemical plants destined for the salt cavern lack the motivation
to form a coalition.
ii) The influence of hydrogen price variations on the coalition
structure is neglected since the salt cavern’s buying price is
unknown at the planning stage. Moreover, the driving force in
forming a coalition is to reduce costs rather than to increase
the selling income.
Generally, the planning problem of chemical plants is based
on the cooperative game, where players are the chemical
plants. For chemical plant
i
, decision variables are the type
of processing equipment,
xi={xn
i},∀n
, hydrogen processing
amount
qpr
i,t
and transport route
ui,j
of chemical plants in
the coalition. Payoffs are described as
(16)
and
(21)
for self-
sufficient chemical plants and coalitions respectively.
Note that in the planning stage, the optimal solution
qpr
i,t
is to
roughly estimate operation cost under different transportation
mode and processing equipment type decisions, thus helping
the decision of transport route
ui,j
. Therefore, the solution of
qpr
i,t
here neglects the influence of hydrogen price variations.
Actual hydrogen processing quantity sequence
qpr
i,t
will be
obtained by equilibrium analysis in the scheduling stage.
2) Stackelberg game in the scheduling stage: The problem
in the scheduling is the hydrogen transaction problem between
the retailer and the suppliers. After formulating transport route
decisions of suppliers as a cooperative game, the interaction
between the salt cavern and multiple chemical plants is
formulated as a Stackelberg game, where the salt cavern is the
leader, whose strategy is the TOU hydrogen price, and chemical
plants are followers, whose strategies are hourly transaction.
At this stage, the retailer’s and suppliers’ problem can be
formulated as a bilevel optimization. The retailer determines
the hydrogen price sequence
vt
in the upper level, and the
suppliers decide their optimal transaction pattern
qtrans
i,t
in the
lower level, with respect to the hydrogen price sequence
vt
.
The optimal transaction pattern
qtrans
i,t
would in turn influences
hydrogen price sequence
vt
determined by the retailer in the
upper level. Assume that the information of each chemical plant,
such as transit transport routes, processing equipment type and
by-product hydrogen generation quantities, are accessible to
the salt cavern. Therefore, the optimal solution of
qtrans
i,t
can
be predicted by the salt cavern under any given hydrogen price
sequence vt. The suppliers’ dispatching problem (3)-(21) can
be regarded as constraints of the retailer’s pricing problem.
According to the analysis above, the interactions between
the salt cavern and the chemical plants constitute a Stackelberg
competition. In this competition, the salt cavern is the leader,
whose strategy is the TOU hydrogen price sequence. Chemical
plants are the followers, whose strategy is the hourly hydrogen
transaction quantity. The leader’s pricing problem maximizes
its profit, subject to the bounds of hydrogen price (Eq.
(2)
) and
maximal injection rate (Eq.
(3)
). The followers’ scheduling
problem maximizes individual profits or coalition profits,
subject to constraints given in (9)-(18).
B. Solution of the Problem
In this section, we introduce the solution of the game
formulation of the by-product hydrogen supply chain.
Tractable reformulations of the suppliers’ problem are made
to efficiently calculate the equilibrium in the lower level for
both the planning and scheduling problems. Specifically, for
the suppliers’ problem in both stages, the objective of each
individual player (or coalition) is irrelevant to the strategies
of other individual players (or coalitions), while the strategy
set is influenced by the strategies of other individual players
(or coalitions). According to the potential game theory, the
suppliers’ problem can be regarded as a potential game. The
sum of the objectives of each individual player (or coalition)
is the potential function. Besides, the pure-strategy equilibrium
exists in the transport route planning problem of the suppliers
since there exists at least one pure-strategy equilibrium in
an infinite potential game. Thus, the suppliers’ problem is
formulated as a potential game that can be solved as an
optimization problem.
After the reformulation of the suppliers’ problem, the
planning stage problem is reformulated to a mixed integer non-
linear program (MINLP) with
xi, ui,j , qpr
i,t, N cars
i,∀i∈1,...I
as decision variables,
(22)
as the objective and
(4)
-
(15)
as
constraints. Commercial solvers such as Baron can be used
to solve the problem. The solved optimal strategy
xi
and
ui,j
will be adopted at the scheduling stage.
max πF=
I
X
i=1
T
X
t=1
ptqtrans
i,t ui,I+1 −Ci
O−Ci
T
−Ci
IN V 1−Ci
IN V 2(22)
To solve the bi-level problem at the scheduling stage, Genetic
Algorithm (GA) is adopted. First, for the salt cavern in the
upper level, pieces of hydrogen price sequences are generated
and regarded as individuals. Second, to acquire the fitness of
each individual, the suppliers’ scheduling problems in the lower
level are solved. Since the transit transport routes
xi
and the
processing equipment type
ui,j
are known at the scheduling
stage, the suppliers’ problem becomes a mixed-integer linear
program (MILP), which can be solved efficiently by off-the-
shelf commercial solvers. Thus, daily profits of the salt cavern,
considering the best response of the suppliers, can thus be
calculated and regarded as finesses for given price sequences.
V. CASE STUDY
To validate the effectiveness of the proposed model and
algorithm, numeric experiments on a by-product hydrogen
supply chain composed of three chemical plants and a salt
cavern are carried out. All of the following tests are conducted
on PCs with Intel Xeon W-2255 processor, 3.70 GHz primary
frequency, and 128GB memory. CPLEX 2.16 is used to solve
related MILP problems.
8
A. System Configuration
Scenario parameters of the envisioned by-product hydrogen
supply chain are given in Table III.
Qi,t
are hydrogen
generation sequences of a typical day produced by a Gaussian
distribution with a mean value of 1000 for the 1st chemical
plant (1500 for the 2nd and 3000 for the 3rd) and a variance of
100. Moreover, in the envisioned by-product hydrogen supply
chain,
Qpr
are a vector consisting of 1200, 2000, 4000 and
8000, the first two and the last two of which are the compressor
capacity and liquefier capacity to choose from, respectively.
Parameters of different processing equipment and transportation
modes refer to [24] and [29] and are given in Table IV.
K1
are
a vector consisting of 774.29, 126612, 18977.17 and 34757.99,
corresponding to each element in
Qpr
. Note that the time scale
involved in the problem is one day. Initial investment costs of
the liquefier, the compressor, and the transportation vehicles are
converted into daily investment costs with a discount rate. The
operation cost of a tube trailer (or a tanker truck) in each period
includes fuel price, driver wage, and maintenance expenses.
TABLE III
SCE NAR IO PAR AME TER S OF T HE BY-PRODUCT HYDROGEN SUPPLY CHAIN
Parameters
I3ND2
T12 Ta[0,0,0,4;0,0,0,4;0,0,0,4]
po15 pt, pt5/13
NC2Qtrans 9000
1) Equilibrium of possible coalition structures of the sup-
pliers: With three chemical plants, there are five possible
coalition structures: no cooperation, cooperation between two
players with the third being self-sufficient (there are three
ways this could occur) and complete cooperation among all the
three chemical plants. The benefits of individual participants
or coalitions are shown in Table V, in which
M
represents the
benefit, and the benefit of each chemical plant and the sum
of them are denoted by
M1, M2, M3
and
Mtotal
respectively.
‘
{}
’ indicates a cooperation, and the chemical plant serving as
the transit hub is marked by a ‘*’.
It can be analyzed from Table V that:
i) In the 1st coalition structure with no cooperation at all, the
total benefit of the three chemical plants is the lowest among
all coalition structures, indicating a potential collective payoff
gained by forming coalitions between chemical plants.
ii) In the 3rd coalition structure, the benefit of the coalition
{1,3∗}
denoted as
M{1,3∗}
equals to 286531 and is lower
than the sum of benefits that they could get on their own,
which is calculated as
M1+M3= 290866
, violating collective
rationality.
iii) In the 5th coalition structure, although collective benefit
is higher than the sum of benefits each coalition member
TABLE IV
PARA MET ER S OF HY DRO GE N TRA NS PORTATI ON
Parameters
QC200 K3($/h)[0,0,0,4;0,0,0,4;0,0,0,4]
QD4000 βL15/13
γc/γd(kwh/kg)1/8.18 βL29000
Kc
2/Kd
2($) 82.20/219.18
TABLE V
PARTICIPANTS/ALLIANCE OPTIMAL INCOME UNDER NON-COOPERATIVE
AND COOPERATIVE GAME MODELS
Number Coalition
structure
Profits($/day)
Individual or
a coalition Mtotal
1{1},{2},{3}
M1= 54052
M2= 81060
M3= 236814
371926
2{1,2*},{3}M{1,2}= 170589
M3= 236814 407403
3{1,3*},{2}M{1,3}= 286531
M2= 107868 394399
4{1},{2,3*}M1= 53562
M{2,3}= 323154 376716
5{1,2,3*}M{1,2,3}= 383925 383925
Fig. 5. Hydrogen price of salt cavern and transaction quantity of chemical
plant
could get on their own, the total benefit of the 5th coalition
structure
Mtotal{1,2,3∗}
is lower than that of the 2nd coalition
structure
Mtotal({1,2∗},{3})
. Therefore, the grand coalition
is not stable since there is a preferred alternative. The analysis
of the 4th coalition structure is analogous.
iv) In the 2nd coalition structure,
M{1,2∗}
, the benefit of the
coalition
{1,2∗}
, equals to 170589 and is higher than the sum
of benefits they could get on their own, which satisfies
M1+
M2= 135112
. Moreover, the total benefit of the 2nd coalition
structure is the highest among the five possible structures, so
there exists no preferred alternatives. Therefore, the coalition
of the chemical plants {1,2∗}is stable.
The insights provided by different coalition structures above
is that for several chemical plants in proximity to each other,
those chemical plants with low or medium generation scale
(chemical plant 1 and 2 in our case) tends to form a coalition,
and to compete with those with larger generation scale.
In order to realize a fair imputation of the collective payoff
of chemical plants
{1,2∗}
, the Shapley value is adopted. The
allocation result is 71790.5,98798.5
$
, which is higher than
the benefit they could get on their own, which are
{$
54052,
$
81060
}
. The coalition between chemical plant 1 and 2 increase
their profits by 24.7% and 18.0% respectively.
B. Equilibrium of Hydrogen Pricing and Scheduling
In this case, the fixed price at which consumers purchase is
set as 15
$
/kg. The equilibrium of the buying price offered by
the salt cavern
pt
and the hydrogen transaction quantity
qpr
i,t
are illustrated in Fig.5. The minimal price takes value at its
lower bound 5
$
/kg, and the maximal value is 11.9
$
/kg . It can
9
Fig. 6. Time of use electricity price and processing mass of chemical plant
be observed from Fig.5 that the variation trend of the hydrogen
transaction quantity goes with the buying price. The higher
the buying price, the higher the transaction quantity. This can
be attributed to the storage capacity of chemical plants, which
can temporarily store by-product hydrogen in low-pressure
storage tanks or tube trailers (or tanker trucks) before filled to
maximal capacity. Therefore, the chemical plants can choose
to sell hydrogen at a higher price.
Moreover, due to the influence of the TOU electricity price,
the operating cost of the processing equipment fluctuates. The
TOU electricity price and the equilibrium of the total processing
quantity are plotted in Fig.6.
It can be observed from Fig.6 that the variation trend of the
hydrogen processing quantity and the TOU electricity price
go oppositely. This is because chemical plants tend to process
hydrogen when the electricity price is low, thus reducing the
processing cost of hydrogen.
According to the above results, it can be noted that the
equilibrium of salt cave pricing encourages chemical plants to
process and trade hydrogen when the electricity price is lower.
As a result, the salt cavern can purchase hydrogen with lower
processing cost, thus reducing the purchase cost of hydrogen
per unit. For chemical plants, the hydrogen price is higher
during 1-2 periods after periods with lower electricity prices
than in other periods, thus reducing the hydrogen processing
cost.
The result of profits and total transaction quantities are
plotted in Fig.7 considering different fixed prices. The optimal
price offered by the salt cavern is about 9
$
/kg, and its profit
is
$
287884.8 for a day. However, the profit of the salt cavern
reaches to
$
343947.16 at the optimal TOU hydrogen price.
Hence, a TOU hydrogen price strategy for the salt cavern
increases its profit by 19.5%.
Generally, the equilibrium of the Stackelberg game between
the salt cavern and the chemical plants benefits all the players.
It also indicates the positive response of the salt cavern and
chemical plants to TOU electricity price, and reflects the role
of chemical plants in peak shaving and valley filling, which
benefits the safe and stable operation of power grid.
C. Sensitivity Analysis
1) Impact of per period transportation operation cost: The
reduction in operation cost of a tube trailer (or a tanker truck)
per period
K3
reduces the transport cost, thus bringing down
the collective payoff brought by coalitions of chemical plants.
Fig. 7. The result of profits and total transaction quantities with time-invariant
hydrogen price
Fig. 8. The result of profits and total transaction quantities with time-invariant
hydrogen price
Based on the first assumption in section
IV-A
1, each coalition
must take one of them as a transit hub, and two chemical
plants destined for the salt cavern lack the motivation to form
a coalition. Consequently, the collective payoff declines as the
transport cost reduces, until collective rationality no longer
holds when the benefits of the coalition are less than the sum
of benefits each individual could get on their own. As shown
in Fig.8, when
K3
decreases from
$
390 to
$
382, the sum of
benefits of chemical plant 1 and 2 under the equilibrium of the
1st and 2nd coalition structure, denoted by
M{1,2}
total , M {1},{2}
total
respectively, gradually increases.
As illustrated in Fig.8, the coalition benefit is more sensitive
to
K3
than individual benefits. When
K3
decreases to about
$
386, the coalition
{1,2∗}
no longer bring additional benefits
to individuals, resulting in a breakdown of the coalition.
2) Impact of maximal injection rate of the salt cavern: The
maximal injection rate
Qtrans
of the salt cavern directly limits
the total transaction quantity per period between the salt cavern
and the chemical plants. Table VI demonstrates the impact of
Qtrans to the equilibrium of the second-stage problem.
TABLE VI
INDIVIDUAL INCOME OF THE EQUILIBRIUM UNDER DIFFERENT MAXIMUM
TR ANS PO RTATION Q UALI TY O F SALT C AVERN G AS P IPE LIN E IN S ING LE
PE RIO D
Qtrans M{1,2}
total /kg M{3}
total/kg
Mtotal
(chemical
plants)/$
Mtotal
(the salt
cavern)/$
12000 23103.96 21387.07 44491.03 342814.83
9000 26203.97 22762.82 48966.80 325379.79
6000 10528.29 1139.11 11667.40 278396.59
10
It can be analyzed from Table VI that
Qtrans
has different
impacts on the participants: the daily income of chemical
plants does not necessarily increase with the increase of
Qtrans
,
whereas the daily income of the salt cavern increases with the
increase of
Qtrans
. Therefore, the salt cavern will be motivated
to determine an appropriate
Qtrans
according to the generation
scale of by-product hydrogen of the chemical plants so as to
increase individual benefits.
VI. CONCLUSION
This paper proposes an equilibrium model of a by-product
hydrogen market with the salt cavern as the retailer and
chemical plants as the suppliers. A business model for large-
scale storage to acquire by-product hydrogen from chemical
plants and sell them to end-users is established for the
first time. The decision-making process of each stakeholder,
i.e., chemical plants and the salt cavern, is investigated and
mathematically modeled considering different transportation
modes, locations of chemical plants and TOU electricity price.
To consider the individual rationality of each stakeholder, the
by-product hydrogen market is formulated as games. The
transport route planning problem between multiple chemical
plants is formulated as a cooperative game. The hydrogen
transaction problem between the salt cavern and chemical plants
is formulated as a Stackelberg game. Numeric experiments on a
by-product hydrogen supply chain composed of three chemical
plants and a salt cavern are carried out. The results show that
a coalition between chemical plants potentially increases their
profits. Moreover, the adoption of TOU hydrogen price in a
Stackelberg formulation also increases the profit of the salt
cavern. The proposed business model and the optimization of
the by-product hydrogen supply chain management not only
presents a new revenue stream for both chemical plants and
salt caverns but increases resource efficiency and accelerates
energy conversion.
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