PROCEEDINGS OF ECOS 2016 - THE 29TH INTERNATIONAL CONFERENCE ON

EFFICIENCY, COST, OPTIMIZATION, SIMULATION AND ENVIRONMENTAL IMPACT OF ENERGY SYSTEMS

JUNE 19-23, 2016, PORTOROŽ, SLOVENIA

Mixed integer optimization of an LNG supply chain in the Baltic Sea region

Alice Bittantea, Raine Jokinenb, Jan Krooksc , Frank Petterssona and Henrik Saxéna

a Åbo Akademi University, Turku, Finland, alice.bittante@abo.fi (CA), frank.pettersson@abo.fi,

henrik.saxen@abo.fi b Pöyry, Vantaa, Finland, raine.jokinen@poyry.com,

c Wärtsilä-Energy Solutions, Vaasa, Finland, jan.krooks@wartsila.com

Abstract:

A numerical tool for designing optimal small-scale supply chains of liquefied natural gas (LNG) is presented. The main problem is formulated as a supply task, where LNG is delivered from a set of supply terminals to a set of receiving (satellite) terminals by ship transportation and by land-based truck transports from the terminals to customers on or off the coast. The objective is to minimize the overall cost, considering the price of LNG, investment cost of the receiving terminals and LNG trucks, rental costs of the ships and delivering costs. The problem is written as a mixed integer linear programming (MILP) problem. The optimization results give information about the placement of the satellite terminals and their capacity, the optimal fleet (ship size and number), the number of trucks, travelling routes of the ships and trucks, and the amount of LNG to supply to the demand sites. The system developed is illustrated by a set of examples designed to shed light on the future LNG supply in the region around the Gulf of Bothnia. The supply chain is optimized under different price of the alternative fuel and the arising solutions are analysed. It is demonstrated that there is a general consensus on where to build the satellite terminals, even though the delivered quantities of LNG vary depending on the price difference to the alternative fuel. Given the short computational time required to solve the examples of the paper, the model can easily tackle more complicated supply chain problems in the future.

Keywords:

Energy Systems, MILP, Optimization, Small Scale LNG, Supply Chain.

1. Introduction Natural gas (NG) is the fastest growing energy source among the fossil fuels. Its global demand is

growing at an average rate of 1.8 % per year against 0.9 % for oil [1]. According to the future

scenario presented by IEA, natural gas consumption is expected to reach 5.4 trillion cubic meter

(tcm) in 2040, replacing coal as the second largest fuel source after oil [2]. Within the natural gas

trade, liquefied natural gas (LNG) is playing a very important role. By 2035 LNG’s share of the

world energy demand is expected to reach 15 % (from 10 % in 2014), surpassing NG supplies by

pipeline [1].

LNG is produced by cooling NG below ̶ 162 °C (at atmospheric pressure), which reduces the

volume to approximately one six-hundredth of the original one. NG in liquid form can be

economically transported over long distances by specially designed vessels, which are able to

maintain the fuel in the liquid state. The traditional LNG supply chain is designed for large volume

deliveries transported over distances of thousands of kilometers with LNG cargo capacities ranging

between 125,000 m3 and 140,000 m3 [3]. In recent years a new market segment of small-scale LNG

logistic chains has become more important. The increasing energy demand and the constraints

imposed by environmental regulations have in many countries promoted the utilization of LNG for

medium- and small-scale applications, where NG pipelines are absent or unpractical, i.e., for small

and sparsely distributed demands. In a small-scale supply chain LNG is shipped from supply

terminals to customers through a network of satellite terminals with a combination of sea- and land-

based transport. The design of such supply networks is a challenging task and the high investment

cost involved in both infrastructure and operation makes it a relevant problem for mathematical

optimization.

The model presented in this paper is aimed at aiding decision making on tactical and strategical

aspects of designing a small-scale LNG logistic chain. Tactical planning addresses the vehicle

routing problem (VRP), for both the maritime and land transports, while strategic planning deals

with decisions regarding location of satellite terminals and size of the optimal fleet to solve the

overall transportation problem. Vehicle routing problems are a well-established area of research

with a rich literature tackling different aspects of the problem. A recent review on VRP [4]

classifies 277 papers from 2009 to 2015. The resulting classification reveals a broad range of

problems as variants of the classical VRP. Recently, researchers have paid more attention to

introducing real-life characteristics and less restrictive assumptions to their tasks thus creating more

realistic models which can be applied in practice. The problem we study in this paper is also

inspired by real-life cases and is therefore a special variant of a combination of classical problems.

The single tactical aspect of the problem is very similar to the so-called fleet size and mix vehicle

routing problem (FSMVRP) while the addition of the strategic part makes it similar to the location

routing problem (LRP). A review paper on both maritime and land transport for the FSMVRP was

presented by Hoff at al. [5]. Most of the literature in this field is based on heuristic methods, as in

[6], which differs from our deterministic approach. The exact formulation has also been presented;

Jokinen et al. [7] proposed a mixed integer linear programming (MILP) model for an LNG

transportation problem along a coastline, while Baldacci et al. [8] designed an MILP model to solve

a problem similar to ours, but where each customer is associated with a single route. The model

proposed in the present paper, on the other hand, allows for multiple visits to the customers by

multiple vehicles, as it has been developed on the basis of the MILP formulation presented by

Bittante et al. [9]. The strategic feature regarding the location of the satellite terminals is addressed

in problems known as two-echelon location routing problem (2E-LRP). An extensive review on

recent papers in this specific field is included in a general survey of the classical LRP by Drexl and

Schneider [10]. Gonzalez-Feliu [11] proposed a mixed integer programming (MIP) formulation for

the generic NE-LRP based on set-partitioning problems and three sets of variables indicating the

activation of the satellite terminal (binary), the activation of the route (binary) and the delivery

associated to the route (floating point). The same sets of variables are also used in the formulation

of problem tacked in the present paper, but we substitute the route activation binary variable with an

integer variable, thus allowing for multiple visits to the same satellite terminal or customer.

The paper presents an MILP model where both the tactical and strategic aspects of designing a

small-scale LNG supply chain are optimized simultaneously. The model is illustrated on the

problem of delivering LNG to an emerging gas market in the northern part of the Baltic Sea region.

2. Problem description In this paper we propose a mathematical model to solve a regional supply of LNG from a set of

potential supply ports to inland end customers, through a set of potential satellite terminals. The

LNG is transported from supply terminals to satellite terminals by ship and from the supply or

satellite terminals to the inland customers by truck. Potential satellite terminals and inland

customers have given demands for the time horizon considered. If the satellite terminal is activated

(built), the total amount of LNG to be delivered from the supply ports is the sum of the satellite

terminal’s demand and the demands of the customers associated to it in the optimized solution.

Alternatively, demands can be fully or partially fulfilled by a distributed alternative fuel, for which

transportation cost is not considered. This fuel is merely used to allow the model to partly or fully

exclude customers from the LNG supply chain. The maritime transportation is performed by a

heterogeneous fleet of vessels, each of which has a given cruising speed, capacity, fuel consumption

and loading/unloading rate. Some types of vessels can perform split delivery. None of the ships are

associated with a specific port and therefore they are not forced to return to the same supply port

they departed from. On the other hand, some vessels can be restricted from visiting certain ports due

to incompatibility with the port specifications (i.e., port depth). Restrictions regarding the maximum

amount of LNG available at the supply ports are included in the formulation and can parametrically

be activated. Land transportation is carried out by a homogeneous fleet of LNG tank trucks of given

capacity and fuel consumption. Trucks are associated to a single supply or receiving terminal, are

restricted to a maximum distance they can cover and are not allowed to perform split delivery.

Customers are assumed to have large enough storage capacity to stock the full demand for the time

period and no investment costs are considered at their sites. By contrast, when a satellite terminal is

activated, a storage tank must also be constructed and the associated size-dependent investment

costs are imposed. This storage size is optimized. Maritime distances between all ports are given

and road distances between terminals and customers are also given. The aim of the model is to solve

the overall LNG distribution problem, selecting the most suitable ports, if any, where satellite

terminals are built, and the size of the LNG storages, the optimal fleet and routing for the maritime

transportation, the number of LNG tank trucks for each port and the port-customer connections to

satisfy the inland demand.

3. Mathematical model In this section we introduce the mathematical model designed to solve the fuel procurement

problem described in the previous section. First we declare the sets and variable in subsection 3.1,

followed by the description of the objective function in subsection 3.2 and ultimately we present all

the constraints in subsection 3.3.

3.1. Sets and variables

In the mathematical formulation, let P be the index set of ports p and L denote the set of customers l

of given demand 𝐷𝑙. Then, let 𝐽 ⊂ 𝑃 be the subset of satellite terminals i which receive LNG from

supply ports 𝑠𝜖𝑆 through a fleet of ship types 𝑘𝜖𝐾. As 𝐽 is also subset of L, the energy demand in

the satellite terminal i can also be satisfied by trucked LNG and/or by an amount of alternative fuel

𝑞𝑙𝑎, similarly to inland customers 𝑑𝜖𝐷. Ship routing is modelled with three sets of variables. Let

integer variables 𝑦𝑝,𝑚,𝑘 and 𝑧𝑘 denote the number of times a ship of type k travels between ports p

and m, and the number of ships of type k needed, respectively. Let variables 𝑥𝑝,𝑖,𝑘 indicate the

number of LNG loads of ship of type k that is transported between ports p and i. The land transport

is also expressed by the use of three sets of variables. Let variables 𝑞𝑝,𝑙 indicate the total amount of

LNG transported by truck from port p to customer l. Let integer variables 𝑧𝑝 denote the number of

trucks allocated to port p, and integer variables 𝑧𝑝,𝑙 give the total number of trips undertaken

between p and l. Two sets of variables are introduced to handle the activation of satellite terminals.

Binary variables 𝑤𝑖 specify the activation of the satellite terminal i while variables 𝑠𝑖 give the size

of the storage at terminal i.

3.2. Objective function

The goal is to minimize the total combined cost associated with the fuel procurement

min 𝐶𝑡𝑜𝑡 = 𝐶𝐹 + 𝐶𝑇 + 𝐶𝐼 , (1)

where the three cost terms are expressed as

𝐶𝐹 = ∑ ∑ ∑ 𝐶𝑠𝐿𝑄𝑘

𝑥𝑠,𝑖,𝑘𝑘∈𝐾𝑖∈𝐽 + ∑ ∑ 𝐶𝑠𝐿𝑞𝑠,𝑙𝑙∈𝐿𝑠∈𝑆 + ∑ 𝐶𝑎𝑞𝑙

𝑎𝑙∈𝐿𝑠∈𝑆 , (2)

𝐶𝑇 = ∑ ∑ 𝐶𝑝𝑦𝑝,𝑚,𝑘𝑘∈𝐾 +(𝑝,𝑚)∈𝑃 ∑ 𝐶𝑘𝑟𝑧𝑘 + ∑ ∑ 𝐶𝑘

𝑓𝑑𝑝,𝑚𝑦𝑝,𝑚,𝑘𝑘∈𝐾 +(𝑝,𝑚)∈𝑃𝑘∈𝐾

2 ∑ ∑ 𝐶𝑓𝑑𝑝,𝑙𝑙 𝑧𝑝,𝑙𝑙∈𝐿𝑝∈𝑃 , (3)

𝐶𝐼 = γ(𝐼 ∑ 𝑧𝑝𝑝𝜖𝑃 + ∑ (𝐼𝑤𝑤𝑖 + 𝐼𝑠𝑠𝑖)𝑖∈𝐽 ). (4)

The first term represents the fuel cost as the amount of LNG and/or alternative fuel used multiplied

by the specific fuel price. The second term accounts for the transportation cost as the sum of port

call costs, chartering of the ships, ship propulsion cost and truck fuel consumption cost. Finally, the

investment cost term includes the trucks purchase and the investment of the construction of the

satellite terminals and associated storages. This last investment cost is expressed by a fixed cost and

a capacity dependent factor. The parameter 𝛾 rescales the total investment cost to the contribution

for the time horizon H considered in the optimization, including discounted interest.

3.3. Constraints

Constraints (5) and (6) ensure that the demand is fulfilled at the satellite terminals and at the inland

customers, respectively. At the satellite terminal, the net amount of LNG shipped to the port

reduced by the LNG transported by truck from the port must at least equal to the demand at the port.

Alternatively, the demand can be supplied by alternative fuel or by LNG by truck in case the

satellite terminal is not activated.

∑ ∑ 𝑄𝑘𝑥𝑝,𝑖,𝑘𝑘∈𝐾 − 𝑝∈𝑃 ∑ ∑ 𝑄𝑘𝑥𝑖,𝑗,𝑘𝑘∈𝐾 𝑗∈𝐽 + ∑ 𝑞𝑝,𝑖 − ∑ 𝑞𝑖,𝑙𝑙∈𝐿𝑝∈𝑃 + 𝑞𝑖𝑎 ≥ 𝐷𝑖 ∀ 𝑖 ∈ 𝐽. (5)

At the inland customer, the demand can be satisfied either by LNG from land transports or by

alternative fuel

∑ 𝑞𝑝,𝑑 + 𝑞𝑑𝑎

𝑝∈𝑃 ≥ 𝐷𝑑 ∀ 𝑑 ∈ 𝐷. (6)

The activation of the satellite terminal is defined by

∑ ∑ (𝑦𝑝,𝑖,𝑘 + 𝑦𝑖,𝑝,𝑘)𝑘∈𝐾 ≤ 𝑀 𝑤𝑖 ∀ 𝑖 ∈ 𝐽𝑝∈𝑃 , (7)

where M is a big-M parameter. When the satellite terminal is activated (𝑤𝑖 = 1), the variable 𝑠𝑖

indicating the size of the tank storage is allowed to take values greater than zero, thus implying the

existence of a storage. This is controlled by the constraint

𝑠𝑖 ≤ 𝑀 𝑤𝑖 ∀ 𝑖 ∈ 𝐽. (8)

The size of the storage is determined by constraints (9), by imposing 𝑠𝑖 to be greater than the net

LNG shipped to the port plus a 10 % storage heel

(1 − 𝑓𝑠)𝑠𝑖 ≥ ∑ ∑ 𝑄𝑘𝑥𝑝,𝑖,𝑘𝑘∈𝐾𝑝∈𝑃 − ∑ ∑ 𝑄𝑘𝑥𝑖,𝑗,𝑘𝑘∈𝐾 𝑗∈𝐽 ∀ 𝑖 ∈ 𝐽. (9)

Land transportation is regulated by four sets of constraints. Constraints (10) ban land transportation

of LNG from a satellite terminal which has not been activated.

∑ 𝑞𝑖,𝑙 ≤𝑙∈𝐿 𝑀 𝑤𝑖 1[GWh] ∀ 𝑖 ∈ 𝐽. (10)

The set of constraints (11) is used to determine the integer variable 𝑧𝑝,𝑙 expressing the number of

trips required to deliver the LNG amount 𝑞𝑝,𝑙 by truck. This constraint is required because trucks

are allowed to have partial loads and therefore 𝑞𝑝,𝑙 is not defined as a multiple of the truck capacity.

𝑧𝑝,𝑙 ≥𝑞𝑝,𝑙

𝑄 ∀ 𝑝 ∈ 𝑃, 𝑙 ∈ 𝐿. (11)

The number of trucks at each port must be sufficient to carry out the delivery of LNG within the

available time horizon. An availability factor 𝑎 is applied to adjust the time horizon according to

specific restrictions on the vehicles. An average velocity is used to calculate the time needed to

travel the port-to-customer distance. The resulting constraint is

𝑎𝐻𝑧𝑝 ≥2

𝑣∑ 𝑑𝑝,𝑙

𝑙 𝑧𝑝,𝑙𝑙∈𝐿 ∀ 𝑝 ∈ 𝑃, (12)

where v is the speed of the trucks. A set of constraints was introduced to limit the number of truck

voyages from port p, based on the number of loading stations and loading time restrictions at the

port, so

∑ 𝑧𝑝,𝑙𝑙∈𝐿 ≤ 𝑍𝑝𝑈 ∀ 𝑝 ∈ 𝑃. (13)

Ship routing is controlled by six sets of constraints. The integer variables 𝑦𝑝,𝑖,𝑘 indicating the

number of voyages undertaken by a ship of type k between ports p and i, are defined based on the

variables 𝑥𝑝,𝑖,𝑘 , connected to the amount of LNG transported on the same arc

𝑦𝑝,𝑖,𝑘 ≥ 𝑥𝑝,𝑖,𝑘 ∀ 𝑝 ∈ 𝑃, 𝑖 ∈ 𝐽, 𝑘 ∈ 𝐾. (14)

Route continuity is guaranteed by

∑ 𝑦𝑚,𝑝,𝑘𝑚∈𝑃 = ∑ 𝑦𝑝,𝑚,𝑘𝑚∈𝑃 ∀ 𝑝 ∈ 𝑃, 𝑘 ∈ 𝐾, (15)

while intermediate loading at the satellite terminals during multistep voyages is banned by

∑ 𝑥𝑝,𝑖,𝑘𝑝∈𝑃 ≥ ∑ 𝑥𝑖,𝑝,𝑘𝑝∈𝑃 ∀ 𝑖 ∈ 𝐽, 𝑘 ∈ 𝐾. (16)

For a specific set of ship types, Ks, split delivery is technically infeasible and a minimum load is

required for safety in operation. The set of constraints (17) ban multistep voyages while constraints

(18) impose the minimum load, expressed as fraction (𝑓) of the total capacity of the ship

∑ 𝑥𝑖,𝑗,𝑘(𝑖,𝑗)∈𝐽 = 0 ∀ 𝑘 ∈ 𝐾𝑠, (17)

𝑥𝑠,𝑖,𝑘 ≥ 𝑓𝑦𝑠,𝑖,𝑘 ∀ 𝑠 ∈ 𝑆, 𝑖 ∈ 𝐽. (18)

Similarly to the constraints (12) for the land transportation, the number of ships of each type is

determined based on the time usage expressed as the time spent on travelling the routes, summed

with the time at the harbour for berthing operation and the loading and unloading processes, so

𝑎𝑘𝐻𝑧𝑘 ≥

1

𝑣𝑘∙ ∑ 𝑑𝑝,𝑚𝑦𝑝,𝑚,𝑘(𝑝,𝑚)∈𝑃 + ∑ (𝑡𝑝 ∑ 𝑦𝑝,𝑚,𝑘𝑚∈𝑃 )𝑝∈𝑃 +

2

𝑟𝑘∑ ∑ 𝑄𝑘𝑥𝑠,𝑝,𝑘𝑝∈𝑃𝑠∈𝑆 ∀ 𝑘 ∈ 𝐾, (19)

where vk is the speed of ship type k. Two extra sets of constraints are available to model possible

terminal restrictions. Constraints (20) limit the amount of LNG available at supply port s, while

constraints (21) impose a limit (𝑄𝑖𝑈) on the maximum ship size allowed to visit port i.

∑ ∑ 𝑄𝑘𝑥𝑠,𝑖,𝑘𝑘∈𝐾𝑖∈𝐽 + ∑ 𝑞𝑠,𝑙𝑙∈𝐿 ≤ 𝑄𝑠𝑈 ∀ 𝑠 ∈ 𝑆, (20)

𝑄𝑘𝑦𝑝,𝑖,𝑘 ≤ 𝑄𝑖𝑈𝑦𝑝,𝑖,𝑘 ∀ 𝑝 ∈ 𝑃, 𝑖 ∈ 𝐽, 𝑘 ∈ 𝐾. (21)

4. Case study LNG has recently attracted large attention in the region around the Baltic Sea as a more

environment-friendly fuel for both ship propulsion and heat and power generation. Several projects

for construction of LNG receiving terminals are under discussion and their location and associated

routing is a relevant question, which makes this area an interesting case for computational studies.

Therefore, the model presented in Section 3 was applied in a case study of LNG delivery in the Gulf

of Bothnia, i.e., the northern part of the Baltic Sea.

The study considers three possible supply terminals (Inkoo, Tornio, Stockholm) and seven potential

satellite terminals (Turku, Pori, Vaasa, Raahe, Luleå, Umeå, Sundsvall) on the coasts of Finland and

Sweden. As inland customers we identified a total of twenty-three clusters distributed in of Finland

and Sweden. A representation of the region and the locations included in the case study are shown

in Fig. 1. Demands were assigned on the basis of the population, extent of industrial activity and

time horizon considered. It should be stressed that these are gross estimates used for the mere

purpose of illustration, and they are not claimed to represent the true demands in these locations.

Remote locations far away from the closest potential satellite terminal were not considered, as a

maximum feasible distance for land transport of 350 km was imposed. Maritime distances were

obtained from an online tool for calculation of distances between sea ports [12]. Road distances

were collected from a web mapping service [13]. Vessels parameters were inspired by small-scale

LNG carrier designs by Wärtsilä [14]. All the ship types can carry out split deliveries. The

optimization was performed for a time horizon of 30 days. The availability of vessels and trucks is a

portion of the total time horizon in order to allow for some extra time in the shipping and to rescale

the total time available to eight-hour work days for the land transport. Tables with the numerical

values of all the parameters used in the model are presented in Appendix A.

The MILP model was implemented in AIMMS 4.8 using the IBM ILOG CPLEX Optimizer [15].

The problem of the case study results in 772 integer variables and 472 continuous variables. The

solution time of one case was usually less than two minutes on a computer with a 3.5 GHz Intel

Core i7 processor and 16 GB of RAM.

Fig. 1. Location of the demands (•) in the case study, potential satellite terminals (●) and supply

ports (■) on a map of Finland and Sweden.

4.1. Base case

In this section we present the results from a computational experiment termed Base Case, where the

model parameters are assigned the numerical values reported in Appendix A. The price of LNG at

the supply ports is 𝐶𝐿 = 30 €/MWh and the price of alternative fuel at the consumers is 𝐶𝑎 = 38

€/MWh. Figure 2 illustrates the optimal maritime routing (indicated by curved arrowed arcs), port

locations (indicated by name) and port-to-customer truck connections (indicated by straight arrows).

Detailed numerical results from the optimization are reported in Tables 1-2.

The results show that almost all the customers are served partially or entirely with LNG. Only four

demands (40.0 GWh in Karlstad, 20.5 GWh in Kuopio, 12.5 GWh in Kiruna and 8.0 GWh in Mora)

are completely satisfied by alternative fuel. Ten other customers are partially supplied with

alternative fuel, while a total of sixteen customers are entirely supplied by LNG by truck. The total

amount of LNG transported by truck is 579.3 GWh, in a total of 1807 trips. Four of the seven

possible satellite terminals are activated. Their storages vary from 14,000 m3 in Sundsvall to 36,000

m3 in Raahe. Pori and Umeå are assigned similar capacities of about 21,000 m3. The number of

trucks per port is indicated in Table 1. Supply ports have the highest number of trucks as they serve

the majority of the land customers reached with LNG. Among the satellite terminals, Raahe has the

larger number of connections for truck transport, and therefore a high number of allocated trucks.

SWEDEN FINLAND

Table 1. Number of trucks per port and storage size for activated satellite terminals

Port 𝒛𝒑, - 𝒔𝒊, GWh

Inkoo 15 -

Tornio 20 -

Stockholm 18 -

Pori 2 126.1

Raahe 10 209.0

Sundsvall 3 81.6

Umeå 1 122.2

Fig. 2. Optimal satellite terminal locations and LNG distribution from ports. Straight arrows

indicate land transport by truck while arrowed arcs indicate maritime routing. Activated satellite

terminals are indicated by name.

Table 2. Routing results for Base Case. Integers Y indicate the number of voyages undertaken and

X the number of LNG loads for the given distances.

Route 𝒚𝒑,𝒎,𝒌 𝒙𝒔,𝒊,𝒌

Inkoo – Pori 3 2.99

Stockholm – Sundsvall 2 1.94

Tornio – Raahe 5 4.96

Tornio – Umeå 3 2.90

Pori – Inkoo 3

Sundsvall – Stockholm 2

Raahe – Tornio 5

Umeå – Tornio 3

Maritime distribution of LNG is carried out with a single ship of type 1 (6,500 m3 of capacity). The

ship does not perform any split deliveries. The optimal routes are indicated by arcs in Fig. 2, while

the number of voyages on the different routes is reported by variable Y in Table 2. In the specific

case, Pori is supplied three times from Inkoo, Sundsvall is served two times from Stockholm, while

Tornio is the supply port for both Raahe and Umeå, with five and three deliveries, respectively.

The total amount of LNG delivered to customers from the three supply terminals corresponds to

about 240 GWh for both Inkoo and Stockholm and about 460 GWh for Tornio. Assuming a storage

tank of 50,000 m3 (corresponding to a capacity of approximately 290 GWh) at the supply ports, the

Base Case solution would suggest a minimum of two refills per month in Tornio and one refill in

Inkoo and Stockholm. Examining the objective function, the cost of fuel purchase was found to

account for 87.7 % of the total costs, while the cost of transportation and the investment cost,

contribute by 2.8 % and 9.5 %, respectively.

4.2. Effect of alternative fuel price

As a brief investigation of the sensitivity of the solution to changes in fuel prices, a set of runs was

performed varying the alternative fuel price keeping the LNG price at the supply terminals constant.

Thus, the alternative fuel price can be expressed as 𝐶𝑎 = 𝐶𝐿 + ∆𝐶, where 𝐶𝐿 is the average LNG

price at the supply terminals. The results illustrated in Fig. 3 were chosen to depict the main

changes in the solution. Keeping the Base Case as a reference (Fig. 3a), for which ∆𝐶 = +8

€/MWh, the alternative fuel price was increased by ∆𝐶 = +9 €/MWh in Case 1 (Fig. 3b), ∆𝐶 =+11 €/MWh in Case 2 (Fig. 3c) and ∆𝐶 = +21 €/MWh in Case 3 (Fig. 3d). Overall, the solution

evolves as expected, i.e., showing an increasing amount of LNG transported by truck and ship and,

consequentially, a decreasing amount of alternative fuel purchased (Fig. 4). In Case 2 a new satellite

terminal is built (Vaasa), two more customers are supplied by LNG (Kuopio and Mora) and a ship

of type 2 (10,000 m3) substitutes the smaller ship (type 1), which was used in the Base Case and in

Case 1. In Cases 1-3 the selected ship performs one or more split deliveries. When alternative fuel

is 70 % more expensive than LNG (Case 3, ∆𝐶 = +21 €/MWh), also a satellite terminal in Luleå is

built and all the customers are supplied, partially or entirely, by LNG. Figure 4 illustrates how the

share of the different sources of energy supply evolves while increasing the price of the alternative

fuel. The total share of energy from LNG grows from about 80 % to almost 98 %. Within the bars,

three contributions are depicted in Fig. 4. The share of LNG directly trucked from the supply ports

represents 38 % in the Base Case and it is almost constant ( 41 %) in the three other cases. The

total LNG delivered by ship increases considerably along with the change of ship type, and is

higher for Cases 2-3. The share of LNG delivered by ship at the satellite terminals and not further

trucked increases from about 30 % to 39 %, while the share of LNG trucked from the satellite

terminals to land customers reaches a maximum of almost 18 % in Case 2 and then decreases to

16.5 % in Case 3. This can be explained by a general redistribution of the truck capacities in Case 3

and the activation of the satellite terminal in Luleå, which was previously supplied entirely by

trucks from Tornio in Case 2.

Fig. 3. Optimal satellite terminals location and LNG distribution from supply ports for Base Case

(a), Case 1 (b), Case 2 (c) and Case 3 (d). The price of alternative fuel exceeds the price of LNG by

9 €/MWh, 11 €/MWh and 21 €/MWh in Cases 1-3.

Fig. 4. Share of total energy supply with respect to fuel and type of delivery.

0

10

20

30

40

50

60

70

80

90

100

8 9 11 21

Sh

are

of

Tota

ql

En

ergy

ΔC (€/MWh)

Alternative fuel

LNG from supply

ports by truck

LNG from satellite

terminals by truck

LNG by ship and not

further trucked

5. Conclusions and future work This paper has presented an MILP model for the optimal design of a small-scale LNG supply chain.

The objective function that is minimized expresses the total combined cost associated with fuel

procurement. The resulting optimal solution provides information about the locations of satellite

terminals, fleet configuration, number of tank trucks and the associated distribution network. A case

study has illustrated the features and the coherent performance of the model upon parameter

perturbations. The proposed model has proven to be a flexible framework which can be easily

applied to other similar supply chain optimization problems. In the future work the present model

will be extended to a multi-period formulation with the aim to address variation of the demands and

to be able to estimate the optimal storage inventory at the satellite terminals. This will require new

sets of constraints to control the tank storage mass balance and sizing. Also additional constraints

may be included to make the problem formulation more realistic.

Acknowledgments This work was carried out in the Efficient Energy Use (EFEU) research program coordinated by

CLIC Innovation Ltd. with funding from the Finnish Funding Agency for Technology and

Innovation, Tekes and participating companies. The financial support is gratefully acknowledged.

Appendix A

In this appendix we report all the numerical values of the parameters included in the mathematical

model. Table A.1 reports the sea distances expressed in kilometers. Parameters regarding port

specifications are given in Table A.2. The parameter 𝑍𝑝𝑈 limiting the truck trips has been estimated

considering the number of loading stations available at the ports (5 for supply ports, 3 for satellite

terminals), an average two-hour time for loading operations and a ten-hour service at the port for

twenty working days a month. Table A.3 reports parameters of the different ship types. In the ship

rental cost, the parameter 𝐶𝑘𝑟 is expressed as

𝐶𝑘𝑟 = 𝐶𝑟𝑒𝑓

𝑟 (𝑄𝑘

𝑄𝑟𝑒𝑓)

0.7

, (A.1)

where 𝐶𝑟𝑒𝑓𝑟 = 800,000 €/month and 𝑄𝑟𝑒𝑓 = 12,000 m3.

Other miscellaneous model parameters are listed in Table A.4. Finally, Table A.5 reports the road

distances between ports and customers, as well as the customers’ demands.

Table A. 1. Sea distances between ports [12]

𝒅𝒑,𝒎, km Inkoo Tornio Stockholm Turku Pori Vaasa Raahe Luleå Umeå Sundsvall

Inkoo 0 893 426 193 406 675 800 851 622 579

Tornio 893 0 809 885 559 373 145 130 338 556

Stockholm 426 809 0 324 422 580 802 846 632 495

Turku 193 885 324 0 315 485 824 663 452 407

Pori 406 559 422 315 0 253 466 517 288 383

Vaasa 675 373 580 485 253 0 281 331 115 250

Raahe 800 145 802 824 466 281 0 186 271 463

Luleå 851 130 846 663 517 331 186 0 292 514

Umeå 622 338 632 452 288 115 271 292 0 285

Sundsvall 579 556 495 407 383 250 463 514 285 0

Table A. 2. Port specific parameters

Ports 𝑪𝒑, € 𝑪𝒔𝑳, €/MWh 𝑸𝒔

𝑼, GWh 𝑸𝒊𝑼, MWh 𝒕𝒑, h 𝒁𝒑

𝑼, -

Inkoo 5,000 30 3,000 5 500

Tornio 5,000 30 3,000 5 500

Stockholm 5,000 30 3,000 5 500

Turku 100,000 5 300

Pori 100,000 5 300

Vaasa 100,000 5 300

Raahe 100,000 5 300

Luleå 100,000 5 300

Umeå 100,000 5 300

Sundsvall 100,000 5 300

Table A. 3. Ship-related parameters

Ship Type 𝒂𝒌, - 𝑪𝒌𝒇, €/km 𝑪𝒌

𝒓 , €/month 𝑸𝒌, MWh (m3) 𝒓𝒌, MW (m3/h) 𝒗𝒌, km/h

Type 1 0.95 5 520,838 37916.67 (6,500) 4666.7 (800) 24

Type 2 0.95 6 704,147 58333.33 (10,000) 4666.7 (800) 26

Type 3 0.95 7 800,000 70000.00 (12,000) 4666.7 (800) 27

Table A. 4. Other model parameters

Parameter

𝒂, - 0.225

𝒂𝒊,- 0.012

𝒇𝐬 0.1

𝑪𝒂, €/MWh 38

𝑪𝒇, €/km 1

𝑪𝒕, MWh (m3) 320.83 (55)

𝑰, € 2,000,000

𝑰𝒔, €/MWh 200

𝑰𝒘,€ 20,000,000

𝒗, km/h 50

𝛄, - 0.012

Table A. 5. Road distances between ports and customers [13] and customer’s demands for the 30-day time horizon used for evaluation of the model

𝒅𝒑,𝒍𝒍 , km

Tu

rku

Po

ri

Vaa

sa

Raa

he

Lu

leå

Um

eå

Su

nd

sval

l

Kem

i

Kir

un

a

Ko

kk

ola

Mal

mb

erg

et

Ou

lu

Pie

tars

aari

Pit

eå

Tai

val

ko

ski

Ku

op

io

Jyv

äsk

ylä

Häm

een

lin

na

Kaj

aan

i

Tam

per

e

Up

psa

la

Öre

bro

Lin

kö

pin

g

No

rrk

öp

ing

Kar

lsta

d

Bo

rlän

ge

Mo

ra

Ly

ckse

le

So

llef

teå

Öst

ersu

nd

Inkoo 130 245 438 654 925 1,180 1,443 770 1,159 543 1,062 665 520 987 816 449 328 147 617 224 446 619 641 604 729 634 719 715 785 944

Tornio 778 639 450 206 130 386 649 28 357 330 260 131 368 175 237 418 470 658 313 618 957 1,088 1,219 1,182 1,198 962 945 388 589 696

Stockholm 1,000 455 747 1,228 906 638 375 1050 1,235 868 1,138 1,153 845 850 1,259 767 622 456 1,335 476 70 195 197 160 305 215 307 717 493 557

Turku 0 142 334 563 907 1,163 1,425 752 1,134 436 1,036 647 413 951 797 453 308 143 622 162 1,000 491 511 474 599 505 590 581 647 806

Pori 142 0 191 434 767 1,024 1,285 612 994 309 897 508 286 813 658 409 263 186 577 111 459 632 654 617 742 647 864 438 508 667

Vaasa 334 191 0 246 579 835 1,097 424 806 121 709 320 98 624 470 377 267 321 367 240 686 817 948 911 927 691 674 1,000 1,000 477

Raahe 563 435 246 0 351 593 854 182 564 126 466 77 163 382 228 283 328 516 196 442 1,164 1,295 1426 1,389 1,405 1,169 1,152 595 795 903

Luleå 907 768 579 351 0 266 528 157 342 459 245 260 496 55 366 547 599 787 442 747 837 968 1,099 1,062 1,078 842 825 268 468 576

Umeå 1,163 1,024 835 591 265 0 264 413 598 1,000 501 516 752 213 622 803 1,000 441 698 360 572 702 833 797 812 577 560 128 203 363

Sundsvall 1,425 1,285 1,097 854 528 264 0 676 861 978 764 779 1,015 476 885 1,066 640 694 961 613 310 440 571 535 550 315 297 343 119 188

𝑫𝒍 , GWh

100 100 50 100 50 100 50 20.5 12.5 8 20.5 50 8 8 50 20.5 50 50 50 20.5 50 8 50 50 40 12.5 8 12.5 8 40

Nomenclature 𝑎 Truck availability, -

𝑎𝑘 Ship availability, -

𝐶𝑎 Price of the alternative fuel, €/MWh

𝐶𝑠𝐿 Price of LNG, €/MWh

𝐶𝑝 Port call cost, €

𝐶𝑓 Truck fuel consumption cost, €/km

𝐶𝑘𝑓 Ship propulsion cost, €/km

𝐶𝑘𝑟 Ship renting cost, €/month

𝑑𝑝,𝑚 Maritime distance, km

𝑑𝑝,𝑙𝑙 Road distance, km

𝐷 Set of inland customers 𝑑𝜖𝐷

𝐷𝑙 Energy demand, MWh

𝑓 fraction of the total capacity of the ship, -

𝑓s Fraction of storage capacity for LNG heel, -

𝐻 Time horizon, h

𝐼 Truck investment cost, €

𝐼𝑠 Tank storage investment cost, €/MWh

𝐼𝑤 Terminal fix investment cost, €

𝐽 Set of satellite terminals 𝑖𝜖𝐽

𝐾 Set of ship types 𝑘𝜖𝐾

𝐾𝑠 Set of ship types 𝑘𝜖𝐾𝑠 which cannot perform split delivery

𝐿 Set of customers 𝑙𝜖𝐿

LNG Liquefied natural gas

𝑀 Big-M parameter, -

MILP Mixed linear integer programming

NG Natural gas

𝑃 Set of ports 𝑝𝜖𝑃

𝑞𝑙𝑎 Amount of energy from alternative fuel, MWh

𝑞𝑝,𝑙 Amount of energy from LNG trucked, MWh

𝑄 Truck capacity, MWh

𝑄𝑘 Ship capacity, MWh

𝑄𝑖𝑈 Maximum ship size capacity allowed at the port, MWh

𝑄𝑠𝑈 Maximum amount of LNG available at supply ports, MWh

𝑟𝑘 Loading/unloading rate, MW

𝑆 Set of supply ports 𝑠𝜖𝑆

𝑠𝑖 Continuous variable indicating the size of the tank storage, MWh

𝑡𝑝 Berthing time, h

𝑣 Truck average speed, km/h

𝑣𝑘 Ship average cruising speed, km/h

𝑤𝑖 Binary variable is 1 if satellite terminal is activated, -

𝑥𝑝,𝑖,𝑘 Continuous variable indicating ship load transported, -

𝑦𝑝,𝑚,𝑘 Integer variable indicating number of time the route between p and m is travelled, -

𝑧𝑘 Integer variable indicating number of ship types

𝑧𝑝 Integer variable indicating number of trucks per port

𝑧𝑝,𝑙 Integer variable indicating number of truck trips between p and l

𝑍𝑝𝑈 Maximum number of truck’s departures from port p, -

γ Investment instalment factor, -

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