solving multi-source pipe network layout problems
for hydrogen networks using dynamic programming
Master‘s Thesis
Jorick Wold, S1878913, j.g.wold@student.rug.nl Supervisors: Dr. X. Zhu & Dr. E. Ursavas
June 21, 2020
Abstract: Infrastructure that is able to transport hydrogen from producers to consumers is essential in the successful implementation of a hydrogen economy. This paper aims to extend the existing pipe network layout problem that can be used for the development of the least expensive layout for a pipe network. In order to make this model suitable for hydrogen networks, decentralization of production locations and the presence of storage capacity have been implemented. The complexity of this problem is in the assignment of pipe diameters from a set of commercially available diameters. Good solutions to this nonlinear model are found using a two-phase dynamic programming approach that is solved using Python. In the first phase, a layout is developed by applying a minimum cost flow algorithm to the problem that is relaxed by disregarding pipe diameters. Then by the use of a commercial solver, commercial pipe diameters are assigned to connections in the layout found in the first phase. By implementing multiple sources and the availability of storage into the pipe network layout problem, this model is able to demonstrate the effects of decentralization on the costs of network deployment.
The model is demonstrated with the HEAVENN project, a functional hydrogen economy under development in the Northern Netherlands. With input from industry experts, a case study has been developed which has been optimized. Four scenarios have been applied to test the robustness of the results against future events. Results confirmed that when deploying such infrastructure, costs can be minimized by meeting local demand with local production. A main aim in designing a hydrogen network should be to match production to local demand. This research is expected to contribute to the reduction of the lack of consensus on the layout of hydrogen networks.
Keywords: pipe network layout problem, dynamic programming, decentralization, storage, hydrogen
1
Introduction
Creating a functional hydrogen economy may contribute to the reduction of adverse effects that burning fossil fuels have on the environment. By burning fossil fu-els, greenhouse gases are released, causing heat to be trapped in the atmosphere (Andresen, Bode, & Schmitz, 2018). Besides gases, particles that may cause severe health problems are released in the form of ashes (Piko˜n, 2003; Turner et al., 2011). Awareness of these nega-tive effects has increased and problems originating from burning fuels have been acknowledged (Piko˜n, 2003; Liemberger, Halmschlager, Miltner, & Harasek, 2019). This awareness raised the need for alternative fuels, among which, hydrogen. In order to create a functional hydrogen economy, a transportation network will have to be deployed. Therefore, research on the feasibility of creating such a transportation network is essential to decelerate climate change.
Critical assessments on storage capability, transport manageability and toxicity, resulted in hydrogen being the ultimate chemical fuel (Hoffman, 1994). Hydrogen can be produced by water electrolysis. Advantages of hydrogen are that it can be used directly as a fuel and it can also be converted back into electrical energy by a fuel cell (Widera, 2019; Abdin et al., 2020). There is an imbalance between the energy produced by renewable energy sources and energy demand. The possibilities of storing energy in hydrogen and using it at a later time make hydrogen a very suitable fuel for solving this im-balance (Widera, 2019; Abdin et al., 2020; Haeseldonckx
& D’haeseleer, 2007). As hydrogen seems to be a very promising energy carrier, a large scale hydrogen project named HEAVENN will be deployed in the provinces of Groningen, Friesland, and Drenthe, in the Northern Netherlands. This project is financially and technically supported by the European Union and multiple partic-ipating companies. The area in which the project is deployed includes renewable energy sources, electrolyz-ers, residential areas, industry, and infrastructure used for transport (Hydrogen Valley, 2020).
Infrastructure for the transport of hydrogen is essen-tial to make the contribution of hydrogen to deceler-ation of climate problems a success. Locdeceler-ations where renewable energy is generated are often far apart. As electrolyzers are ideally built close to renewable en-ergy sources, there is a high degree of decentralization of hydrogen production sites (Abdin et al., 2020). A benefit of hydrogen over electric energy is its energy storing property. Due to the integration of intermittent renewable energy sources, the need for energy storage will increase (Stadler, 2008). Storing electrical energy in batteries could be an alternative for balancing, but it turns out to be too expensive when energy has to be stored for a longer period of time (Blanco & Faaij, 2018). Hydrogen can be stored in existing cavities for long periods of time at low costs (Widera, 2019). These two properties, decentralized production and energy storage, make hydrogen pipe networks different from other networks that have been modeled mathematically. By adding these properties to the existing pipe network layout problem, the model will be suitable for the design of hydrogen transport networks.
This research describes a model and solution method aiming to minimize the construction costs of a hydro-gen transport pipe network. The research question addressed in this paper is:
What are the minimum costs for the construc-tion of a pipe network, taking into account the degree of decentralization of production and the presence of storage sites?
Shiono et al. (2019) developed a dynamic programming approach to find an exact solution to the pipe network layout problem. The method developed by these re-searchers is limited to networks that contain one source and does not take storage into account. Therefore, these two properties are added to make the model suit-able for hydrogen networks. To make this extension, analytical quantitative research has been carried out. The extended model is applicable to find the optimal layout for the HEAVENN project. The results of the optimization serve to provide insight into the network and are of value for transmission system operator(s) and hydrogen (prod)using entities. The optimal layout, flows, expected costs, use of storage, and level of de-centralization, are the outputs of the optimization. As the model is generalizable, it can be applied to similar projects in other regions.
The structure of this research is as follows: the theo-retical background on subjects related to network model-ing for hydrogen pipe networks can be found in Chapter two. Chapter three contains the methodology stating the research question, the proposed model, and the proposed approach to solve the model. Chapter four contains numerical results. In this chapter, two network problems that were used in Shiono et al. (2019) are solved with the proposed approach in order to validate the method and demonstrate differences. In order to demonstrate the ability of the proposed method to find the optimal degree of decentralization, a third network has been designed. This third network contains all properties specific for hydrogen networks. Five
scenar-ios have been optimized to demonstrate the effects of realistic situations on the results of the optimization. The fifth chapter contains a case-study regarding the optimization of the HEAVENN project in the Northern Netherlands. This real-world network includes a storage facility and multiple locations where electrolyzers will be constructed. The results of this optimization were substantiated using four scenarios that were developed in collaboration with stakeholders. Chapter six contains general results in which the coherence of factors within the mathematical model are described. The seventh chapter contains the discussion. The last chapter, chap-ter eight, contains the conclusion regarding the model, the method, and the results of the optimizations.
2
Theoretical Background
The aim of this research is to develop a mathematical model to determine the optimal layout for a hydrogen transport networks and take properties specific for such networks into account. To underpin the relevance of this model, the importance of hydrogen for sustain-ability is illustrated first. Next, the different modes of transport for hydrogen are discussed. Then, existing mathematical models regarding network layout are ex-plained. Finally, the contribution of this research to existing literature is substantiated in section 2.4.
2.1
Hydrogen
As early as in 1975, hydrogen has been addressed as fuel of the future (Dell & Bridger, 1975). Hoffman (1994) analyzed all elements in the periodic table using nine criteria. These criteria were related to the energy storage capability, transport manageability, and toxicity. After extensive research into possible energy carriers, hydrogen proved to be the only suitable element fitting these nine criteria for a suitable chemical fuel.
A feature distinguishing hydrogen from fossil fuels is that it can be converted back into electrical energy by means of a fuel cell (Escher, 1994). As a result, hydrogen can be an energy carrier for the transport of electricity or provide direct energy by burning it. There are different ways to generate hydrogen. There is electrolysis of water, polymer electrolyte membrane (PEM) electrolyzers, partial oxidation, coal or coke based production, and it can be obtained from steam reforming (Ball & Wietschel, 2009; Dell & Bridger, 1975). Water and PEM electrolysis are suitable for converting renewable energy into hydrogen, in other processes hydrogen is often a by-product of industrial processes.
2.1.1 Energy balancing
over fossil fuels (Chen, Kumar, Wong, Chiu, & Wang, 2019).
With high market penetration of renewable energy sources, only 20-25 percent of the energy generated can be integrated into current electricity systems (Stadler, 2008). This is caused by the intermittent production of renewable energy sources and emphasizes the impor-tance of energy balancing techniques. It is expected that a storage capacity of 600 TW per hour will be needed in the future due to fluctuating production from renewable energy sources (Blanco & Faaij, 2018). Transforming electricity into hydrogen offers great potential for the transformation from fossil fuel use to increasing inte-gration of renewable energy sources and the reduction of carbon emissions in several sectors (Widera, 2019; Walker, van Lanen, Fowler, & Mukherjee, 2016). The possibility of storing electrical energy will allow almost all excess renewable energy to be used productively in 2050, assuming 80 percent of the energy demand gener-ated is from renewables (Widera, 2019; Lewandowska-Bernat & Desideri, 2018).
Energy storage is essential in order to increase the integration of renewable energy into the current elec-tricity system. Storing electric energy in batteries is an example of a technique that could be suitable for short term storage. When energy has to be stored for a longer period of time, batteries have shown to be too expensive (Blanco & Faaij, 2018). Storing electric energy in hydrogen may be a better solution for long term storage. Various ways of storing hydrogen have been developed. Especially underground cavities may be suitable for long-term and low-cost mass storage (Andersson & Gr¨onkvist, 2019; Caglayan et al., 2020; Michalski et al., 2017). Storage in cavities is a very region-specific solution and is therefore not applicable in every hydrogen network. An alternative to storage in cavities is storage by compression in the pipe network, known as line-packing (R´ıos-Mercado & Borraz-S´anchez, 2015). This technique is particularly suitable for absorb-ing short-term fluctuations in the supply of renewable energy. A disadvantage of line-packing is the respon-sibility of the gas transporter to keep sufficient gas in pipes during a given planning horizon.
2.2
Transport
A major social acceptance barrier for hydrogen is the lack of transport infrastructure (Johnson & Ogden, 2012). Transport of hydrogen is possible in different ways, e.g. in liquid or gaseous form by truck, by train, with vessels, or through pipelines (Yang & Ogden, 2007; Guandalini, Colbertaldo, & Campanari, 2017; Lahnaoui, Wulf, & Dalmazzone, 2017).
Andresen et al. (2018) have executed dynamic simu-lations of three different transport options, i.e. using the natural gas grid, dedicated pipelines, and trailer transport. These transport options have been assessed on efficiency, costs, and CO2 emissions. It is found that transport using the existing gas grid is the least efficient, but most environmentally friendly and least expensive, solution. The results of this study may not be representative as they consider a case study which
is region-specific.
The study of Demir & Dincer (2018) assessed the costs of pressurized tanks, cryogenic liquid hydrogen tankers, and gas pipelines for transport. They found that the third option is up to three times cheaper than the first. They emphasize the financial benefits of stor-ing hydrogen in existstor-ing caverns over liquefaction for storage.
The optimal mode for delivery depends on the flow rate and distance (Andr´e et al., 2013; R´ıos-Mercado & Borraz-S´anchez, 2015). Costs will increase for both, but not at an equal rate. There will be a need for a transportation network largely fulfilled by pipelines for networks with significant quantities of flows over medium to long distances. There is a lack of consensus on the structure of such a network and the effects of deployment over time. The effects of energy losses due to friction on the internal wall is generally known as the pressure drop equation. This should not be overlooked when designing such networks (Andr´e et al., 2013).
Multiple researches show the possibility of injecting up to 20 percent hydrogen into the existing low-pressure natural gas network (Pellegrino, Lanzini, & Leone, 2017; Guandalini et al., 2017; Liemberger et al., 2019; Quar-ton & Samsatli, 2018). These injections are possible without using techniques such as turbines or compres-sors. This is interesting as the ’fuel cost minimization problem’, considering the costs of turbines and com-pressors amongst others, is of extreme importance for gas operators (R´ıos-Mercado & Borraz-S´anchez, 2015).
A risk of transporting hydrogen through pipelines is the fact that hydrogen embrittlement is complicated to predict as it depends on the material, the pipeline’s history, and past pressure fluctuations (Haeseldonckx & D’haeseleer, 2007). Another risk is that hydrogen, being a very small molecule, can leak through the small-est openings (Liemberger et al., 2019). However, if all pipelines are properly inspected and maintained, it is technically possible to transport hydrogen through existing pipelines (Liemberger et al., 2019).
2.3
Optimization models
Science has already studied the design of water, electric-ity, gas, and road networks. These designs have often been developed by means of optimization models. In these models, the key characteristics of the business problem are used to determine the best possible solu-tion. First, different mathematical approaches for this type of model are discussed, then pipe network layout problems suitable for the extension in this research are explained.
2.3.1 Mathematical modelling
models, research has only been done from a descrip-tive perspecdescrip-tive. These consider the effects of network changes over time and can take seasonal influences into account. However, these have shown limitations in the ability to find global optima for the minimum costs (Osiadacz & Bell, 1986).
Dynamic programming is a steady-state approach that has proven itself in optimizing problems related to designing pipe networks (Shiono et al., 2019; Andr´e et al., 2013). It considers breaking down the problem in sub-problems that are solved consecutively. An ad-vantage is that non-linearity can easily be handled and an exact solution is guaranteed. A disadvantage of this technique is the exponential increasing computation time with increasing dimension of the vector of state variables (R´ıos-Mercado & Borraz-S´anchez, 2015).
Other methods like gradient search and linearization approaches proved not to be successful in designing pipe networks. Gradient search concerns optimizing by moving the solution in the direction of the greatest change. Linearization approaches are known to linearize the non-linear problem before solving it. Both these models are known not to guarantee an exact solution (Flores-Villarreal & R´ıos-Mercado, 2003).
Mixed-integer linear programming concerns an ap-proach where some variables are constrained to integer values. This is a widely adopted approach in pipe net-work layout modeling as it has proven to be effective. The main disadvantages of this approach are that global optima cannot be guaranteed to be found, as well as that it is not possible to take non-linear effects into account (Urbanucci, 2018; Sahinidis, 2019).
2.3.2 Pipe network layout problems
The objective of the pipe network layout problem (PNLP) is to design a network that is able to satisfy all demand while minimizing the network construction costs. The three most important constraints in PNLP models are the flow-balance equation, the pressure-loss constraint, and the allocation of diameters. The flow-balance equation states that the flow to a location is equal to the sum of the flow out and demand at that location. The pressure-loss constraint accounts for the allowable pressure loss over the pipe length from pro-ducer to consumer. The diameters of pipes need to be determined because they affect the amount of pressure loss and the possible amount of flow through the pipe. Andr´e et al. (2013) developed an approach for design-ing pipe networks. Their approach is able to calculate the total costs of a hydrogen network while taking into account the network characteristics, operating condi-tions and costs. A three-stage dynamic approach is ap-plied to solve the problem. First, a minimum spanning tree is calculated. Second, the diameters of pipelines are determined. Finally, heuristics are applied to optimize the solution. The capacities of their pipe networks are explained by the nonlinear relation of the flow and the pressure at the two ends of a pipe (Andr´e et al., 2013). In this model nonlinear relations between flows and pipe diameter are left out of consideration.
Shiono et al. (2019) developed a pipe network layout problem that was solved using dynamic programming. This method is able to find an exact solution that is an approximate of the global optimal solution. Their objective is to find the minimum construction costs, stated as a function of the length and diameters of pipes in the layout. First a relaxation that assumes continuous diameters for pipes is applied. Then, the Dreyfus-Wagner algorithm is applied to find a Steiner tree for the layout of the network. The second phase is an optimization able to decide on diameters for the layout found (Shiono et al., 2019). They found major improvements in accuracy and reductions in calculation time compared to commercial solvers. This network design is limited to planar networks with only one source and without storage locations. By ’planar graph’, a graph is meant that can be drawn on the plane in such a way, intersections of edges are only found at their endpoints (Beezer, 2008).
¨
Uster & Dilavero˘glu (2014) minimized the total in-vestment and operating costs for the construction or expansion of a transmission network. The method ap-plied concerns a mixed-integer nonlinear optimization model. They determined locations, capacities, connec-tions, compressor stations and gas pressure require-ments. Further, the expected increases and decreases for the next few years have been taken into account. They left length, the highly nonlinear diameter of pipes, and storage locations out of scope and used existing data to estimate costs.
Johnson & Ogden (2012) developed a network layout model optimizing costs in order to demonstrate the fea-sibility of hydrogen network application. Their aim was to develop indications for interconnected transmission infrastructure within large geographic regions linking multiple producer networks. A mixed-integer linear programming model was developed in order to solve the model using the General Algebraic Modeling System (GAMS). The model focuses on deploying a network system in a multiple-stage approach. It assumes capac-ity in pipes cannot be reused in subsequent calculation steps. This model is lacking the functionality to cope with storage within the network.
The models addressed before assume the costs of a hydrogen network are, amongst others, related to its to-tal length (Andr´e et al., 2013; Shiono et al., 2019; Chow, Kopp, & Portney, 2003). There are differences in cost factors taken into account with regard to the inclusion or exclusion of compressors, pipe diameters, or construc-tion of sources. These papers differentiate on capital expenditures and operational expenditures depending on the interest in building or using the network.
2.4
Contributions
energy from intermittent renewable sources (Chen et al., 2019; Walker et al., 2016). In order to be able to use hydrogen as alternative for fossil fuels, efficient pipe infrastructure must first be in place for transport.
Models for hydrogen pipe network layouts should be able to take decentralization into account. This need is caused by the distances between renewable energy sources and the fact that electrolyzers should ideally be placed close to sources. The importance of local pro-duction in minimizing costs is emphasized by (Andr´e et al., 2013; George, 2010). However, opening several different hydrogen production locations may contribute to increases in the total costs of the network. In re-search conducted by Balta-Ozkan & Baldwin (2013), the importance of the increasing size of hydrogen net-works in relation to economies of scale is highlighted. This underlines the importance of the model taking into account the effects of decentralization. Consensus on the layout of such networks is expected to increase with the results found in this research.
Storage, transport, and stationary applications must be facilitated for successful implementation of a hydro-gen economy. The mathematical model constructed by Shiono et al. (2019) is extended to match these require-ments. By modeling the possibility of having multiple sources as a decision variable, the model is able to decide on the degree of decentralization. By the addition of a storage location to where the surplus of the hydrogen produced can be transported, the network can be used for grid-balancing. By the ability of the model to de-velop the minimum cost layouts for hydrogen networks, it will contribute to consensus on the layout that is suitable for such networks. The combination of all hy-drogen network specific properties can be found in the region addressed by the HEAVENN project. Therefore, this project is used for the demonstration of the model developed in this research. The theoretical framework in Figure 1 describes the process in which the PLNP for hydrogen networks with multiple production locations and the availability of storage is structured.
Figure 1: Theoretical framework
3
Methodology
In this chapter, an outline of the research methodology is provided. The research context is addressed first, this consists of the general methodology and research question. Then, the research model is formulated, and a solution approach is developed.
3.1
Research context
This research concerns the extension of the pipe network layout problem and the corresponding approach to solve the problem proposed by Shiono et al. (2019). The extensions to the existing model are aimed at making the PNLP suitable for hydrogen networks. The two most significant differences from the existing pipe network layout problem, used by Shiono et al. (2019), are:
• The ability to take decentralization of production locations into account instead of using a single source.
• The addition of a storage facility within the net-work.
The research goal is to develop a mathematical model and solution approach that can be used to determine an exact solution for the extended PLNP. The aim is to minimize the costs associated with capital invest-ments regarding hydrogen transport network deploy-ment. More specifically, the costs considered are the costs related to construction of the pipe connections and electrolyzers. The adaptations of the existing model and the aim to develop a network at minimum costs lead to the following research question:
What are the minimum costs for the construction of a pipe network, taking into account the degree of de-centralization of production and the presence of storage sites?
Two different sources of data are used in order to develop, validate, and test the extension to model. The first data source concerns the paper of Shiono et al. (2019). The model in this paper was selected as the starting point for this research. Besides the model, two networks were used to validate the solution approach by comparing solutions to two other existing methods. In this research, these two networks are reused to validate the solutions found by this extension for hydrogen net-works. The second data source considers data supplied in the form of a grant agreement and is supplemented by public and expert information on the layout of ex-isting gas pipe networks (New Energy Coalition, 2020). This concerns data on pipelines, capacities, pressures, and locations of the hydrogen sites. First, an imaginary network will be optimized in order to demonstrate the functioning of the solution approach developed and the effects different parameters have on these optimizations. Finally, data related to the HEAVENN project will be used as input to develop a hydrogen transport network for this specific project.
3.2
Model formulation
objective function and constraints of the optimization. The function used to evaluate the quantitative criterion of importance (costs) is the objective function. This function is given in section 3.2.4. Functions containing (in)equalities limiting decisions are called constraints. These are depicted in section 3.2.5.
To formulate the PNLP model, Shiono et al. (2019) defined a directed graph D = (N (D)), (A(D)) that has been constructed from the provided graph, G = (N (G), E(G)). In the provided graph, N (G) represents the set of nodes, and E(G) is the set of edges. An example of a provided graph is depicted in Figure 2. In the directed graph, arcs (i, j) ∈ A(D) and (j, i) ∈ A(D) are associated to connection (i, j) ∈ E(G). This directed graph can be used in the PNLP.
The networks observed in this study are restricted to planar graphs. A planar graph is considered a graph that can be drawn in such a way on a plane that its edges intersect only at their endpoints. Faces within a planar graph are regions bounded by edges, including the outer outer infinitely large region. These properties can be observed in Figure 2.
Figure 2: Example of provided graph, G = (N (G), E(G))
3.2.1 Assumptions
To define the model and solution approach, assumptions made by Shiono et al. (2019) have been supplemented with assumptions relating to the extension of the model. These assumptions have been adopted from the existing model:
1. A planar graph is considered for the layout of the network.
2. Possible pipe diameter must be in a pre-determined set of commercial pipeline diameters.
3. Costs are a function of the length and diameter of the pipeline connecting two nodes for each connec-tion where a pipe is deployed.
4. A layout is a sub-graph in which all demand is met.
5. Utility companies are required to supply to cus-tomers within a pre-determined range of pressure Shiono & Suzuki (2016).
6. For gas networks 1.8 ≤ α ≤ 2 and 4.8 ≤ β ≤ 5.3, µ is a positive constant related to α and β. These constants have been physically observed (Hansen, Madsen, & Nielsen, 1991). The constants represent the characteristics of the flow and can be obtained from different known pressure drop equations. For example, if the flows in the network are described by the Cox equation, α = 2, β = 5, and µ = 0.224.
Assumptions that have been added with the extension of the model:
7. There is no active storing of gas by compression inside pipelines (line-packing).
8. At junction nodes, pressure loss due to the shape of the junction is negligible. This is justified as these losses are negligible compared to the pressure drop along the pipe lengths (Brown, Mahgerefteh, Martynov, Sundara, & Dowell, 2015)
9. A flow passing an active source node, is increased in pressure to the pressure level of that source.
10. Source nodes are able to produce the expected amount of hydrogen, and pressure provided by different source nodes is at the same level.
11. The pooling problem, regarding blending different sources to ensure gas composition, is of no concern.
3.2.2 Decision variables
The decision variables in Table 1 are the variables that can be controlled by the decision-maker or solver. Typ-ically these variables are controlled in such way that an optimal value for the objective function is obtained.
The first decision variable is the flowrate qij. With
this variable, a flow can be assigned from node i to j. By assigning a flow, a connection is created between two nodes. The costs of connections assigned by the optimization are summarized in the objective function.
The decision variable dij represents the diameter of
the pipe connecting nodes i and j. By assigning a flow, a connection is created between the two nodes. The pipe connecting these two nodes affects the amount of flow, lengths, and pressure loss over the connection. The costs of the connections assigned by the optimization increase with the length and diameter of the pipe. The optimization aims to minimize total costs by selecting the smallest diameters possible.
The third decision variable xidescribes the decision
whether or not to open a particular location that can serve as a source.
The fourth decision variable τirepresents the amount
of hydrogen stored at a node. Surplus production results in flows towards storage, this affects the diameter of pipes towards this node.
The last decision variable pi represents the pressure
at node i, This variable affects the diameter of the pipes connected to this node.
Table 1: Decision variables Variables
qij = flowrate from i to j (m3/hour) dij =
diameter of pipe connecting i to j (mm)
xi =
boolean variable, 1 if node i is producing, else: 0 τi = storage at i (m3/hour)
3.2.3 Parameters
In order to model the PNLP optimization problem, a set of parameters has been defined. Table 2 describes the parameters used in the objective function and con-straints.
Table 2: Model parameters Parameter
node i, (i=1,..G) = node in the network pmin = minimum pressure (MPa)
ps = source pressure (MPa)
δi =
customer demand at i (m3/hour)
τi,max =
maximum storage capacity at i (m3) εi = production at i (m3/hour) lij = length of an edge (m) ci =
costs of opening source i (thousands euros)
Ω =
set of diameters (mm) dk : k = 1, 2, .., M
0 < d1< d2< ... < dM
h = total time period covered by the optimization (hours) ϕ = total costs (thousands euros) α = constant, 1.8 ≤ α ≤ 2 β = constant, 4.8 ≤ β ≤ 5.3 a = costs per meter
(thousands euros) γ = constant, depending on
burial depth and material µ = constant, related to α and β
3.2.4 Objective function
Equation 1 represents the objective function. This function is aimed at minimizing the total construction costs of the network. Total costs are a function of costs for the pipe layout and costs for electrolyzers constructed. The costs of pipes that have to be laid are calculated based on their length and diameter.
ϕ = min X (i,j)∈A(D) fij(dij) + X i∈A(D) (xi∗ ci) (1) 3.2.5 Constraints
In these constraints, relations are mathematically de-fined in the form of restrictions and limitations for variables and parameters.
εi∗ xi+ X (j|(j,i)∈A(D)) qji− X (k|(i,k)∈A(D)) qik− τi= δi ∀(i) ∈ N (D) (2)
Equation 2 is the flow-balancing equation, stating that the sum of all flows to the node and production at the node is
equal to the sum of the demand, storage, and flows leaving that node. X i∈A(D) (εi∗ xi) − X i∈A(D) δi= X i∈A(D) τi ∀(i) ∈ N (D) (3)
According to equation 3 the sum of all hydrogen produced minus the amount consumed is the total amount stored in the network.
0 ≤ τi≤ τi,max/h ∀(i) ∈ N (D) (4)
In equation 4, the amount of gas stored in a node is limited to the capacity of the node and is non-negative. The maximum capacity is divided by hours as flows are in m3/hour and storage capacity is given in m3
qij≥ 0, ∀(i, j) ∈ A(D) (5)
Equation 5 ensures flows are non-negative.
dij∈ Ω, if qij> 0, ∀(i, j) ∈ A(D) (6)
dij= 0, if qij= 0, ∀(i, j) ∈ A(D) (7)
Equations 6 and 7 assign a pipe diameter to an edge if a flow is assigned.
pmin≤ pi≤ ps ∀(i) ∈ N (D) (8)
pi≥ 0, ∀(i) ∈ N (D) (9)
Equations 8 and 9) are related to the pressure within the network. The pressure must never become negative, and pressure within the network is always between the minimum and source pressure limits.
(p2i− p 2 j)d
β
ij− µlijqijα = 0, if qij> 0, ∀(i, j) ∈ A(D), (10)
where for any (i, j) ∈ A(D)
fij(dij) =
(
0 if qij= 0,
alijdγij if qij> 0.
(11) Andr´e et al. (2013) state that the effects of energy losses due to friction on the internal wall of the pipelines should not be overlooked. The amount of energy loss depends on the gas composition and flow properties. This is generally known as the pressure drop equation (Eq. 10) (Shiono et al., 2019). This equation has to be satisfied when there is a flow from i to j. The final constraint (Eq. 11) represents the construction costs of a connection in case a flow is assigned.
3.3
Solution approach
Commercial solvers have not been able to solve the pipe network layout problem in acceptable time. Dynamic pro-gramming has repeatedly demonstrated to be capable of finding a solution for the PNLP by making decisions one at a time.
of applying the relaxation on diameters in the first phase is that the method cannot guarantee a global solution but finds an approximate of the optimal solution.
The aim of optimization is to minimize the costs of the network that has to be deployed in order to supply all demand. As multiple combinations of electrolyzers may be able to supply all demand, the optimization should decide on the set of electrolyzers that has to be constructed. A feasible set of electrolyzers is defined as a set that can meet the total demand in the network but does not exceed the sum of the total demand and storage capacity. The layout of the network and diameters of pipes are affected by the locations of electrolyzers and the quantities they produce. Therefore, the total costs of the network differ per combination of electrolyzers. As a result, the method for determining the optimal cost for the layout has to be performed for each feasible set of electrolyzers to obtain the best solution.
In Paragraph 3.3.1, first a relaxation is applied to the PNLP proposed by Shiono et al. (2019). By means of this relaxation, in Paragraph 3.3.2, it is demonstrated that the Steiner tree problem and minimum cost network flow problem are suitable for finding a layout for the relaxed problem. Than, the application of the method used to find a layout is described. Finally, Paragraph 3.3.3 contains the mixed integer problem that is used for the allocation of diameters to the pipes that have flows allocated in the layout.
3.3.1 Relaxation of the PNLP
Relaxations are a modeling strategy in optimization prob-lems. Using relaxations, the solution to a difficult problem is approximated by solving a similar problem that is eas-ier to solve. The solution to the relaxed problem provides information on the original problem.
The relaxation applied to this PNLP is that the diameter of pipe connections can be selected arbitrarily, with dij≥ 0
((i, j) ∈ A(D)) (Shiono et al., 2019). Using this relaxation, equations 6 and 7 are relaxed to dij≥ 0 ((i, j) ∈ A(D)). The
relaxed PLNP referred to as a pipe network layout problem with a continuous diameter (PNLPC). Feasible solutions to this PNLPC do exist. An example of a feasible solution is a directed spanning tree connecting a source in a single-source network to all demand locations, where all arcs on the tree are pipes with sufficiently large diameters.
In order to omit the continuous dijfrom the PNLP,
equa-tion 10 has been rearranged. After rearrangements, equaequa-tion 10 is represented by: dij= µlijqijα p2 i − p2j 1β , if qij> 0, ∀(i, j) ∈ A(D) (12)
This equation can now be substituted in equation 11 of the model in order to omit dij. The substitution leads to
the new PNLPC below:
ϕc= min X (i,j)∈A(D) gij(qij, pi, pj)+ X i∈A(D) (xi∗ ci) ∀(i, j) ∈ A(D), (13) s.t. εi∗ xi+ X (j|(j,i)∈A(D)) qji− X (k|(i,k)∈A(D)) qik− τi= δi ∀i ∈ N (D) (14) X i∈A(D) (εi∗ xi) − X i∈A(D) δi= X i∈A(D) τi ∀(i) ∈ N (D) (15) 0 ≤ τi≤ τi,max/h ∀(i) ∈ N (D) (16) qij≥ 0, ∀(i, j) ∈ A(D) (17) pmin≤ pi≤ ps ∀(i) ∈ N (D) (18) pi≥ 0, ∀(i) ∈ N (D) (19)
where for any (i, j) ∈ A(D)
gij(qij, pi, pj) = 0, if qij= 0, aµγβl β+γ β ij (p 2 i− p2j) −γ βq αγ β ij , if qij> 0. (20)
As stated by Shiono et al. (2019), it should be noted from equation 20 that 0 < αγ/β < 1 and 0 < γ/β < 1 in the PNLPC. An arbitrary feasible solution to the PNLPC can be found with a set pi and qij and a single source
node providing all demand. Then, fixing the pressure at each node at pimakes this a minimum costs network flow
problem. A graph including flows sent over cycles cannot be the optimal solution for the minimum cost flow algorithm due to unnecessarily increasing costs through these cycles (Zangwill, 1968). Recent research compared layout finding methods for multi-source multi-sink networks. The results confirmed that optimal layouts should never contain cycles. Therefore, the optimal solution to a MCNFP is a tree-shaped layout (Heijnen, Chappin, & Herder, 2020). As with the arbitrary feasible solution the PNLPC can become a MCNFP. The optimal layout for a PNLPC is also a tree. An example of a tree-shaped feasible solution with one source providing all demand is depicted in Figure 3.
Figure 3: Example of a feasible solution
3.3.2 Network layout
It has been shown that the optimal layout of the PNLPC is a tree-shaped graph. Two known optimization problems have demonstrated to find a tree-shape graph for pipe network layout problems. These optimization problems are known as the Steiner tree problem and the minimum cost network flow problem (MCNFP). Both these methods have been applied successfully in the past (Walters & Smith, 1995; Walters & Lohbeck, 1993; Shiono et al., 2019).
a unit along an edge with the number of units sent over that edge. In pipe network layout problems, the cost of a unit flow over an edge is related to the length of the edge. As these two optimization models differ in objective function, the resulting layouts found by these models may differ. The methods are applied to find solutions to the relaxed problem. This causes resulting layouts to be approximations of the global optimal solution.
In the method developed by Shiono et al. (2019), the Steiner tree problem is applied to obtain a minimum span-ning tree for the layout. Their research aimed to improve the layout found by the existing Steiner tree method by including continuous diameters in the Steiner tree problem. Shiono et al. (2019) then applied their solution approach to various networks to demonstrate that the results found had improved compared to existing Steiner-tree approaches. The main disadvantages of their method are that it is only suitable for networks with one source and that time increases exponentially with increasing graph complexity.
The aim of this extension is to decide on the level of decentralization for which the total construction costs are minimal. The disadvantages of the method developed by Shiono et al. (2019) make their method unsuitable for this extension. First of all, there are multiple sources present within the hydrogen network. Second, since this extension considers a problem where the interest lies in the degree of decentralization, layouts will have to be found for every set of feasible production locations. This causes the time it would take to solve the extended model using the Steiner tree approach developed by Shiono et al. (2019) to be unac-ceptable. An approach that is suitable for coping with these hydrogen network specific problems has to be used in order to reduce the calculation time.
The minimum cost flow network problem is known to handle multiple sources and sinks within a network (Heijnen et al., 2020). The layout resulting from this problem is tree shaped (Zangwill, 1968). Because of the differences in objective function between the Steiner tree problem and the MCNFP described earlier, it is expected that changing the method will cause the resulting layouts to differ. The MCNFP is applied for finding the network layout as this model seems a very suitable approach.
3.3.3 Diameter assignment
In this section, discrete diameters are allocated for edges of the layout found for the PNLPC. Empirical results showed that a commercial solver could be used efficiently in order to obtain results for this step within reasonable time (Shiono & Suzuki, 2016).
Similar to the original model, let L∗= (N (L∗), A(L∗)) represent a directed graph. In this graph, A(L∗) is the set of arcs for which a flow is assigned in the layout resulting from phase one (q∗ij > 0). The nodes that represent the
endpoints for all (i, j) ∈ A(L∗) are represented by N (L∗). The problem can be simplified under the assumption that the layout resulting from phase one is a tree spanning all production and demand nodes. The set of leaves in L∗is represented by V (⊆ N (L∗)), and the path from source to a leaf v ∈ V is represented by Psv. Every s − v path, from a
source to a leaf, is unique and qij∗ represents the total flow on
an edge. Using decision variable dij (diameter), the PNLP
can be represented as an integer programming problem:
min X (i,j)∈A(L∗) alijdγij, (21) s.t. X (i,j)∈A(Psv) µlijq∗ij dβij ≤ πmax, ∀v ∈ V (22) dij∈ Ω ∀(i, j) ∈ A(L ∗ ), (23)
In the objective function (Eq. 21), the aim is to minimize the costs by optimizing the decision variable dij. Choosing
smaller diameters will result in lower total costs. However, choosing smaller diameters results in increasing pressure loss. As the pressure constraint (Eq. 22) has to be satisfied, this choice of diameter is a trade-off in the optimization, here p2
s− p2min= πmax. The final constraint (Eq. 23) states
that any value chosen for dijwill have to be chosen from
a set of commercially available diameters. In the objective function a is expressed in thousand euros per meter with constant γ as exponent that can be adapted for burial depth. For example, if construction costs are 10 euros per meter, a = 0.010γ.
Results published by Shiono & Suzuki (2016) demon-strated a commercial solver is able to solve this problem within reasonable time. Therefore, in this study the second phase is solved using a commercially available solver.
4
Numerical results
This chapter concerns the demonstration and validation of the extended model and associated solution approach. In Paragraph 4.1, the experimental settings are described. These concern the software used and data that are used as input. Next, two networks developed by Shiono et al. (2019) are used to validate the proposed solution approach in Paragraph 4.2. The extensions regarding multiple sources and the use of a storage location are demonstrated by means of a third network in Paragraph 4.3. For this third network, experiments in the form of scenarios were performed to demonstrate effects of realistic situations on the results of the optimization.
4.1
Experimental settings
To find a solution to the proposed model and implement the proposed solution approach, Python was chosen as pro-gramming language. Advantages of using Python are the many available modules that are available as freeware and the flexibility in communication with other software and file formats.
To assign diameters to the pipes on which flows are as-signed in the layout, the set of commercial diameters dkin
Table 3 has been used. This set of commercial diameters was obtained from the research of Shiono et al. (2019).
With regard to the pressure drop equation, the Cox equa-tion has been applied (α = 2, β = 5, and µ = 0.224). This equation is a common application for medium and high pres-sure networks and a common assignment for gas distribution systems (Shiono et al., 2019).
Table 3: Commercial pipe diameters Set Diameters (millimeters)
dk 50, 100, 150, 200, 300, 400, 500
4.2
Method validation
The approach proposed by Shiono et al. (2019) was assessed by comparing their results to two existing methods. As their approach is an extension to the Steiner tree problem, they assessed their approach against the existing Steiner tree problem for a network based on the Tokyo area. Results showed that, with including the diameters in the Steiner tree algorithm, Shiono et al. (2019) improved existing Steiner tree approaches. To demonstrate the time complexity of their method, a second modified network representing a residential area in the Tokyo area has been used. The increase in faces for different scenarios caused the calculation time of their algorithm to increase exponentially too. The results they found for this second network were compared to results found by the commercial solver ’SCIP’. Results have shown
that SCIP did not guarantee to find a solution after six hours of calculating. The method proposed by Shiono et al. (2019) found exact solutions in acceptable time on the same computer.
The proposed solution approach in this extension concerns the use of the MCNFP to find a layout for the PNLPC. The networks concerning the Tokyo area, used for validation purposes by Shiono et al. (2019) can be reused to validate the proposed model and solution approach. The quality of the results found by the proposed method will be assessed by comparing these results to the results found by Shiono et al. (2019). Although, due to the differences in the objective function of the methods used to find the layout for the relaxed PNLPC, the resulting layouts are expected differ. In the paper of Shiono et al. (2019), the focus is mainly on improving the exact approximate solutions with respect to previous work. This extension considered the possibility that multiple sources are present within the network and the ability to use a storage location. These two extensions result in the need to optimize multiple different occasions of the network. This causes the time it takes to arrive at a good solution to be the most important factor for this extension.
4.2.1 Network 1, Tokyo area network
The network shown in Figure 4 is a network in which a single production location is present. There are 22 demand nodes, 23 junction nodes, and 54 edges. The parameters that have been used in the optimizations are in Table 4.
Shiono et al. (2019) compared the layouts resulting from the existing Steiner tree problem to their own algorithm.
Figure 4: Tokyo area network, (Shiono et al., 2019)
Figure 6: Tokyo area network with assigned diameters (millimeters) case B, (γ = 1.3) (Shiono et al., 2019).
Figure 7: Tokyo area network with assigned diameters (millimeters) case C, (γ = 1.3)(Shiono et al., 2019).
Table 4: Parameters Tokyo area network parameter value Ps 0.2 MPa Pmin 0.1 MPa a 0.01γ γ 0.7, 1.0, and 1.3 ε1 9000 m3/hour
The method proposed in this extension, using the MCNFP to find a layout, has optimized this same network for validation purposes. The solutions found by the three methods are referred to as ’A’ for the proposed MCNFP method, ’B’ for the method developed by Shiono et al. (2019), and ’C’ for the existing Steiner tree method. The optimizations resulted in the figures shown in Figures 5, 6, and 7.
The methods are compared with respect to calculation time and resulting costs. The numerical results are depicted in Table 5 where costs are represented by ϕ. The time the calculation took to find results is displayed in seconds.
The network is optimized for three different values for γ (burial depth). As γ only applies to the second phase of the dynamic algorithms, it has no effect on the layouts found. Due to the small contribution of the second phase to the total calculation time, and the fact the layouts resulting from the first phase are not affected by γ, the calculation time of the proposed method was not affected by different γ values. The proposed method has shown that it was able to solve the problem in significantly less time than the two existing Steiner tree methods.
Table 5: Results for Tokyo area network
Obtained using the three different methods for PNLP solution A: proposed MCNFP method
B: Steiner tree method from (Shiono et al., 2019) C: existing Steiner tree method
γ A (ϕA∗) B (ϕB∗) C (ϕC∗) B−A A Computation time (s) A B C 0.7 22,482 22,655 23,018 0.8% 4 72.8 14.3 1.0 24,700 25,900 26,900 4.8% 4 71.1 14.4 1.3 27,419 29,788 31,901 8.6% 4 71.0 14.5
A costs difference of 8.6% has been observed between the Steiner tree method of Shiono et al. (2019) and the proposed MCNFP approach. This is caused by differences in layouts resulting from the Steiner tree problem and the MCNFP and the exponential effect of γ in the second phase of the algorithm. The difference in the layouts found for the two Steiner tree methods (Figures 6 and 7) is caused by whether or not a continuous diameter is considered while finding a layout. The differences in layout found by method A are caused by the objective function of the MCNFP which is different from the objective function of the Steiner tree problem.
Figure 8: Residential network, (scenario 5) (Shiono et al., 2019)
The results, B and C, have been found by Shiono et al. (2019), and were computed using MATLAB 2012a (MAT-LAB, 2012) and Gurobi 5.6.2 (Gurobi Optimization, 2020) on a Lenovo IdeaPad U310 (CPU:1.7 gigahertz, RAM: 4 gigabytes). For results for solution approach A, calcula-tions are found using Python 3.7, and Gurobi, computed on a Microsoft surface pro 5 (CPU: 2.5 gigahertz, RAM: 16 gigabytes).
4.2.2 Network 2, Residential network
This second network represents a modified version of a network in a residential Tokyo area (Figure 8). Shiono et al. (2019) have used this network to demonstrate the effects of network complexity on calculation time. The complexity for finding a Steiner tree in a provided network is related to the number of faces in the graph. By setting up a series of scenarios with increasing numbers of faces, it was demonstrated that the time it took for their algorithm to find the solution increased exponentially with increasing network complexity. They compared results found by their Steiner tree algorithm to results found by a commercial MINLP solver. It was concluded that the method using the Steiner tree approach outperformed this commercial solver ’SCIP’ (The SCIP Optimization Suite 7.0 , 2020). The MCNFP solution approach is validated by optimizing this network and comparing the results with the results found by Shiono et al. (2019).
Table 7: Parameters residential network parameter value Ps 1 MPa Pmin 0.3 MPa γ 1 a 0.01γ γ 1.0
Table 8: Residential network, scenarios (Shiono et al., 2019)
The residential network in Figure 8 consisted of 60 bi-directional arcs, 35 junction- or demand-nodes, and one source. The parameters used for the optimizations of this network are in Table 7. Five different scenarios have been de-fined for this network, all different in the amount of demand nodes (Table 8). For example: in scenario two, there is one production node, there are eight demand nodes (2-9), and 27 junction nodes (10-36). Every next scenario for which the layout of this network has been determined concerned an increased in complexity.
The results found by the three different optimization methods are in Table 6. Layouts corresponding to the results found by the Steiner tree extension of Shiono et al. (2019) and the proposed MCNFP approach are in Figure 9 and 10. For the optimizations performed by SCIP, in case an exact solution was not found after 6 hours of calculating, the lower (LB) and upper (UB) bounds are presented where ’N/A’ stands for not available.
The results show that with increasing graph complexity, calculation times increased exponentially for the Steiner tree method developed by Shiono et al. (2019). The proposed MCNFP approach succeeded to find solutions to any scenario in one second. This confirmed that the calculation time of the proposed method does not to depend on the amount of faces in the graph. This success in finding a layout in a very short time is of great importance for optimizing networks with large numbers of feasible production location sets.
Table 6: Results residential network
SCIP Shiono Proposed
method Computation time (s)
Figure 9: Residential network, Optimal layout for each scenario (Shiono et al., 2019)
Figure 10: Residential network, Optimal layout for each scenario, proposed solution
In terms of resulting cost, all results found for the pro-posed method are within the bounds found by SCIP. There-fore, it can be concluded that the proposed method performs better than this commercially available solver. Comparing the resulting costs of the proposed method to the results found by the Steiner tree method of Shiono et al. (2019), differences have been observed. The maximum difference between the Steiner tree method and the proposed method is found to be 10%. These differences in results are ex-plained by the differences in layout resulting from phase one where different objective functions are applied. These can be observed in Figures 9 and 10.
The computations from the paper of Shiono et al. (2019) were performed on a Fujitsu LIFEBOOK WA3/B1 (CPU: 2.8 gigahertz, RAM: 16 gigabytes). Computations for the proposed method were performed on a Microsoft surface pro 5 (CPU: 2.5 gigahertz, RAM: 16 gigabytes).
4.2.3 Comparing solutions
This paragraph discusses important findings from the two validation networks. First, exact results for the minimum total costs found using the MCNFP solution approach are compared with the results of previously used methods. Next, the effect of different γ values on the results of the optimiza-tion is discussed. Finally, the time aspect of the calculaoptimiza-tions is discussed.
Different exact results have been found with the proposed MCNFP solution approach and the Steiner tree approach de-veloped by Shiono et al. (2019). These differences originate from the layouts found for the PNLPC in phase one. As the methods applied to optimize these layouts have different objectives, they may result in slightly different layouts. Be-sides this, the methods are not applied to the original PNLP but to the relaxed PNLPC. Therefore, the methods cannot guarantee to find the global optimum for the PNLP. As a result of the non-linearity in this second phase, it can differ per network which of the two solution approaches finds the best solution.
In the first network, ’Tokyo area network’, the effect of changing γ values on optimization results was demonstrated. The differences in cost found by the proposed MCNFP
approach and the Steiner tree approach of Shiono et al. (2019) increased with increasing γ while the layout did not change. This is a result of the exponential effect of γ in the assignment of pipe diameters. These results were in accordance with the results found when Shiono et al. (2019) compared their extended Steiner tree approach to existing Steiner tree solutions. It could be concluded that the results of the proposed model and solution approach were as expected.
A noticeable result is that for the proposed method the calculation time was strongly reduced compared to existing methods. In the network regarding the residential area, it can be seen that calculation time for this proposed approach does not depend on the number of faces within the graph. As a result of this reduced calculation time, this method demonstrated to be very suitable for optimizing multiple layouts to approximate the optimal level of decentralization for a real-world hydrogen network.
4.3
Numerical example
In Section 4.2, the model has been validated by comparing the results and performance of the proposed method with results found by Shiono et al. (2019). The aim of this section is to demonstrate the performance of the extension by study-ing the effects of decentralization on the construction costs of a network. The proposed solution approach is applied to a network that includes multiple sources, consumers, and a storage location. This network contains all the properties of a hydrogen network and can therefore be used to demon-strate that the proposed algorithm is suitable for application on hydrogen networks.
4.3.1 Hydrogen network
The imaginary hydrogen network is depicted in Figure 11. Properties of the seven locations in this network are de-picted in Table 9. As large industrial consumers often have electrolyzers on site, demand has been assigned to some production locations. The large quantity produced at lo-cation 0 was deliberately chosen in order to obtain a large number of feasible solution combinations for demonstration purposes. Random coordinates have been selected for ge-ographic locations. These lie within a circle with a radius of 50 kilometers. To estimate the length of pipes between these locations, euclidean distances have been obtained. Pa-rameters describing the properties of this network are in Table 11. The costs of purchasing, installing, and starting an electrolyzer are related to the size of the plant. In 2020, the capital expenditure for electrolyzers is 2 million euros per ton produced per day (Buttler & Spliethoff, 2018). The input used for the optimization is depicted in Appendix B.
Table 11: Parameters Hydrogen network Input parameters Ps 0.2 Pmin 0.1 γ 1 a 0.01γ hours 8,765.81 m3hydrogen per kg 11.988
Figure 11: Hydrogen network (distance in meters)
4.3.2 Hydrogen network results
After modelling the data for this network, the optimization was carried out for each set of feasible electrolyzers. The resulting layouts and diameters for pipes that have been assigned are in Figure 12. The costs related to the deploy-ment of pipes and construction of electrolyzers are in Table 10. The output of the calculations performed by the Python
script is depicted in Appendix C. It took 5 seconds to obtain these results with the proposed solution approach.
Eight combinations of production locations have proven to form feasible solution sets. No solution exists without including location 0. This shows that a network can be highly dependent on a single large production location. The optimal layout and degree of decentralization are found by opening all four production sites. The corresponding total costs are 254 million euros. This amount consists of 67 million euros for construction of electrolyzers and 187 million euros for deploying the pipes. The difference in costs for electrolyzers between the best and worst solutions is 1.8 million euros, while the difference in costs for pipes between these solutions is almost 185 million euros. These results show that the layout is most important, whereas the costs of electrolyzers are secondary in this optimization problem.
The resulting layouts for four different sets of production locations have the same connections. These are shown in Figures 12a, 12c, 12d, and 12g. The maximum difference observed for costs regarding pipe construction for these four layouts is almost 138 million euros. The only change in these four optimizations is whether or not production locations are open, thus the amount of hydrogen that is produced. By opening production locations close to demand, less transport is necessary and pipe diameters may reduce. This effect that reducing pipe diameters has on the total cost is emphasized by two other solutions represented by Figures 12b and 12e. The only difference between these two layouts is the diameter of the pipe connecting locations 0 and 2. This pipe has a length of 27.43 km, the effect of the 50 millimeter increase in diameter is almost 14 million euros.
A noticeable insight is that location 3 can only be reached by means of a pipe of 28.5 km length from location 4. The difference in demand at this location (240 tpy) and the surplus that is produced when it is active (60 tpy) results in a reduction in the diameter of the pipe connecting this location. The reduction in diameter is 50 millimeters, this saves 14 million euros. Activating this electrolyzer costs 1.6 million euros, the total reduction by activating location 3 is 12.5 million euros. It can be concluded that production at location 3 must be initiated at all times for this network.
Lengths of connections are another important contribu-tion to the costs of the network. The total graph lengths of the layouts are in Table 10. The total lengths of networks (Figures 12b and 12e) are the longest with over 200 km of pipes. This total length increased due to the distance between locations 1 and 4 where a flow is assigned. For this network it shows to be disadvantageous to start production at location 1, this is caused by the lack of demand near this location. The effect of local production on pipe length is not as significant as the effect of diameters. Local produc-tion almost never meets local demand exactly. Therefore, small quantities have to be transported and pipes will be deployed. The length of pipes is not affected by the quantity of hydrogen that is transported. These results emphasize the importance of balanced production and demand.
Table 10: Optimization for sets of feasible production locations for the hydrogen network and results of scenarios Active production locations Storage amounts Graph length
Costs initial model (thousands euros)
Total costs per scenario (thousands euros)
(tpy) (meters) Electrolyzers Pipes Total 1 2 3 4 5
0 3,785 154,786 64,109 333,057 397,167 397,167 438,313 372,038 244,631 494,949 0,1 3,929 205,218 64,898 372,164 437,062 437,062 478,209 417,596 295,852 531,254 0,2 3,821 154.786 64,306 229,277 383,648 383,648 411,079 365,377 231,113 440,874 0,3 4,085 154,786 65,753 209,161 274,914 N/E 316,061 269,656 195,462 405,647 0,1,2 3,965 205,218 65,095 358,448 423,544 423,544 450,975 410,936 282,334 517,312 0,1,3 4,229 135,484 66,542 201,182 267,725 N/E 308,871 259,533 186,599 445,543 0,2,3 4,121 154,786 65,950 195,445 261,396 N/E 288,827 262,996 181,944 391,705 0,1,2,3 4,265 135,484 66,739 187,467 254,207 N/E 281,638 252,872 173,081 431,601 4.3.3 Experimental scenarios
This section describes five realistic scenarios that highlight the importance and coherence of specific factors within the PNLP model. The results found for these scenarios will be presented and discussed.
Scenario 1: Limited storage capacity. The capacity of the storage location is limited to 4.000 tpy, this is expected to reduce the amount of feasible production location combinations.
Scenario 2: Locally increased demand. An unexpected growth in demand by the sudden popularity of fuel cell electric vehicles might occur. To simulate such increases in demand, demand at location 2 is increased to 925 tpy. Scenario 3: Different pipe diameters. The new set of diameters includes the following sizes: (40, 60, 80, 100, 125, 150, 175, 200, 300, and 400 mm). By being able to choose diameters from a set with reduced intervals, costs are expected to reduce.
Scenario 4: Transport network pressure. In this sce-nario, the allowable pressure loss is increased to the pa-rameters for network 2, the papa-rameters of the Japanese gas network. The source pressure (Ps) is set to 1.0 MPa,
the minimum pressure at demand locations (Pmin) is 0.3
MPa.
Scenario 5: Construction problem. Problems may arise during construction, problems of a network. Connec-tions from node 0 to 2, and from node 5 to 6, could not be made.
The resulting costs of these scenarios are depicted in Table 10. In these results, combinations of production locations that are not feasible for specific scenarios are indicated with N/E (non-existent). The optimal solution for each scenario is framed. The differences in results are caused by pipe diameters and layouts that have changed.
As a result of the limited storage capacity in scenario 1, the amount of sets with feasible production locations has reduced. For sets of electrolyzers that were feasible, results did not change. The optimal solution is now represented by Figure 12c. As a result of the reduction in sets of feasible production locations less demand is met by local produc-tion. This may result in the need for large quantities of gas transported over longer distances. Therefore, reducing the storage capacity may increase costs.
By increasing demand at location 2, local production was no longer able to meet demand. This caused pipes in the direction of this location to increase in diameter (Figure 13a). The results in Table 10 show the effect of this increase in diameter on the costs to be over 26 million euros. This emphasized that local production, and thus decentralization, contributes to reducing the total cost of the network as long as production matches demand.
In the third scenario, a new set of possible pipe diameters was defined. A decrease in construction costs due to the availability of pipes with diameters matching the flows has been observed in Figure 13b and Table 10. The decrease in costs resulting from the new set of diameters for the optimal layout of this network was less than 1%. It can be concluded
(a) (0) (b) (0,1) (c) (0,2) (d) (0,3)
(e) (0,1,2) (f ) (0,1,3) (g) (0,2,3) (h) (0,1,2,3)
that the existing set of commercial diameters is a suitable set for this network. However, as pipe diameter significantly contributes to the total costs, it is relevant to consider this scenario when deploying a network.
The fourth scenario considered an increase in the allow-able pressure loss. By increasing this allowallow-able pressure loss (πmaxin Constraint 22), different diameters have been
allocated (Figure 13c). This emphasized the positive effect of increasing the allowable pressure loss on the costs of the network to be constructed.
The fifth scenario demonstrated the effect of availability of connections on the total cost of a network. The results showed changes in the optimal degree of decentralization for this network (Figure 13d and Table 10). A more expensive layout had to be designed where node 4 has a vital role as junction. In this solution, it turned out that it was no longer efficient to open production site 1. This emphasized the importance of proper analysis beforehand.
From these results it can be concluded that storage ca-pacity is a strong determinant in the sets of feasible combi-nations of production locations. By having a high capacity storage location available within a network, storage does not have to be a limiting factor in finding the least expensive network. The importance of local production for the cost of a network have been emphasized. The benefit of local pro-duction increases if the amount of propro-duction corresponds to local demand. For this network it proved irrelevant to use a different set of diameters because the savings found were less than 1%. This could yield different results for other networks and given the high costs for this type of infrastruc-tural project, it is relevant to consider. Although the length of a layout will remain the same by increasing the allowable pressure loss. The diameter of pipes will decrease, this will also greatly reduce the costs. For the costs of the pipes in such projects it is advantageous to make the allowable pressure difference as large as possible.
(a) scenario 2
(b) scenario 3
(c) scenario 4
(d) scenario 5
5
HEAVENN project
In Chapter 3, the proposed model and solution approach have been validated using two networks derived from the Tokyo area. The functionality of the extension to the PNLP model has been demonstrated through an imaginary hydro-gen network in Chapter 4. After demonstrating the success of the method that was proposed, it is applied to a real-world case study in this chapter. This case study concerns the HEAVENN project, a hydrogen infrastructure project in northern Netherlands (Hydrogen Valley , 2020). This region was the first to receive a grant from the Fuel Cells and Hydrogen Joint Undertaking (FCH JU) of the European Commission. The grant awarded for this project is 20 million euros, public-private co-financing of 70 million euros brings the project to 90 million euros. The development of a func-tional green hydrogen economy in the Northern Netherlands is partly financed from this subsidy. The project covers a period of 6 years and started in January 2020.
Within this project’s geographical area, the provinces of Friesland, Groningen, and Drenthe, all facilities relevant for a hydrogen network are present (Figure 14). Demand for hydrogen in this region consists of industry, residential heating projects, and mobility. There is large industry that includes chemical industry, production facilities, power plants, and a port. There are natural cavities that are expected to be suitable for the storage of hydrogen. Possible infrastructure for transport of hydrogen is present in gas and water pipe networks that are no longer used.
Stakeholders taking part in this project include, amongst others, Gasunie, NAM/Shell, New Energy Coalition, and Groningen Seaports. A researcher at Royal Dutch Shell has been interviewed for informational and validation purposes. A second interview was conducted with a program manager working at Groningen Seaports. This company is respon-sible for a hydrogen network in Delfzijl and Eemshaven, the construction of which started in 2016. Transcripts of interviews with these experts are in Appendix E.
The current projected demand within the project is less than the amount of hydrogen production that is planned. According to Shell, the aim of the project is not to meet demand but to develop a functional hydrogen economy. As the researcher at Shell stated, without production, there will be no demand and there is no need for production. Therefore, a market-push approach has been applied, expectations are that demand will follow. The HEAVENN project is initiated to demonstrate and encourage the implementation of hydrogen economies.
The joint interest of all stakeholders is to develop and construct the first large-scale functional hydrogen network. Besides the common interest, the different stakeholders will also have their specific interests. For example, Gasunie, the transmission system owner, aims to minimize the costs of constructing the network. The optimization of the imagi-nary hydrogen network in Chapter 4 showed that opening electrolyzers close to demand and matching production to local demand reduces the cost of the network. Therefore, Gasunie will aim to encourage producers to meet local de-mand. Another stakeholder is Shell, their responsibilities include the construction of electrolyzers. They aim to maxi-mize the return on investment, which might result in using the plant to its full capacity. However, if this production exceeds local demand, there will be a need for thicker trans-port pipes, which has negative consequences for Gasunie. The interests of different stakeholders and the consequences these interests have on the results of the optimization are discussed in this case study.
Figure 14: HEAVENN region (shaded)
The model proposed in this research applies to the HEAVENN project. With use of the proposed MCNFP solution approach, the optimal layout and degree of decen-tralization can be determined. Considering the expectation that the network will expand in the future, it should be developed in such way it is robust to future changes. A series of different scenarios simulating possible future events will be applied to demonstrate the robustness of the solution. From a business perspective, these scenarios are essential to gain insights into the effects of future situations and possible conflicting stakes.
In this case study, the data needed as input for optimizing the model is collected in Section 5.1. Processing this data in the model is explained in Section 5.2. The results of this optimization are then depicted in Section 5.3. After discussing the results, in section 5.4, scenarios are applied to test the solutions of the first optimization. Section 5.5 discusses the results and effects of scenarios on the developed network.
