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Dovi M.-S.,
Yenching Academy,
University of Peking, Beijing, China
Meshalkin V.P.,
memaer of the RAS,
Dmitry Mendeleev University of Chemical
Technology of Russia, & N.S. Kurnakov
Institute General and Inorganic Chemistry RAS
Kantiukov R.R.,
Dmitry Mendeleev University
of Chemical Technology of Russia
A previously considered general and simplified mathematical models for the economic optima
structure design and operation management of natural gas (NG) transportation networks (NGTN)
in accordance with economic objective goal is further developed to include financial market mechanisms and environmental constraints. It is shown that introducing these constraints leads to more complex mathematical models resulting in multiobjective optimization stochastic algorithms.
The goal of this article is to formulate a general mathematical model capable of including both financial and environmental constaints and to show how suitable algorithms can be employed to carry out the resulting economic and ecological optimization problem. Indeed, the high computational load required in case of complex gas networks and supply chains makes it imperative to use the most advanced and efficient numerical algorithms of decision making.
Keywords: supply chain management, energy efficiency, financial constraints, impact of methane leakages.
Introduction
It was shown in a previous paper [1] that additional features to the mathematical models presently used for the optimal structure design and operation management ofnaturalgas networks (NGTN) can considerably add to their significance, thanks to the availability of increased computational power. In particular, we considered the possible delivery of natural gas (NG) or Liquid natural gas (LNG) at multiple feed-in points, which can be a key issue in many optimal configurations of methane supply chains.
In the last years, the liberalization of natural gas markets has introduced additional variables in the general design and operation. Indeed, the requirement of optimizing financial portfolios makes it imperative to take into account not only economic factors, but also financial considerations including the market of futures and options related to the natural gas commodity trading.
Thus, while Wu et al. [2] developed the mathematical framework for cost minimization in gas pipelines,a general introduction of the role of spot and futures prices in the commodities markets (with particular reference to the oil and gas market) was provided by Pindyck [3]. The significance of competitive markets has been highlighted by Cremer et al. [4] and the issue of infrastructure maintenance under different ownership conditions was analyzed by Papadakis and Kleindorfer [5]. Contesse et al. [6] provided a general mixed-integer linear programming (MILP) algorithm for problems with financial terms. Storage valuation was considered by Chen and Forsyth [7], whereas the optimization of short-term portfolios was analyzed by Chen and Baldick [8]. A model for optimizing investments in natural gas networks was presented by Davidson et al. [9]. A complete model, including both physical and financial constraints and objective functions, was developed by Midthun [10] and by Tomasgard et al. [11].
The optimization of the NGTN and supply chain portfolios implies stochastic models generally based on a two-stage (or multiple stages) program with relatively complete recourse [12].
The stochasticity is present on the right-hand side of the constraints and in the objective function, which foresees the maximization of expected profits including cash flows and shortfall costs.
Similarly, the incorporation of environmental impacts has given rise to a new approach to traditional NGTN and supply chain networks. Some fundamental concepts have been laid out by Hugo and Pistikopoulos [13] and an extensive review on this subject has been published by Srivastava [14]. The particular nature of chemical supply chains has been considered by Guillén-Gosálbez and Grossmann [15], [16], as well as by Ruiz-Femenia et al. [17], whereas an LCA approach for the description of a holistic supply chains model is proposed by Bojarski et al. [18]. An algorithm for green gas (typically biogas from farms) supply chains optimization is presented by Bekkering et al. [19]. Zhakadiakin and Meshalkin [20] illustrate the role of sustainable supply chain modelling for establishing a trade-off between profitability and environmental concerns.
The role of carbon emissions in general supply chain network design is considered by Elhedhli and Merrick [21] and by Brandenburg et al. [22]. While the model of the latter two authors is focussed on CO2 emissions generated by vehicles used for general supply chains, the recent White Paper by Balcombe et al. [23] considers emissions specially related to the whole network of natural gas supply systems.
Including environmental targets in addition to the traditional profit maximization goal quite naturally leads to bi-objective or multi-objective optimization algorithms. An application of mixed-integer linear programming to general environmental problems is proposed by Guillén-Gosálbez [24], whereas special applications to sustainable supply chains are considered by Wang et al. [25] and by Saffar et al. [26], [27]. The multi-objective algorithm addresses the problem of constructing the Relevant Pareto frontier by solving for each point of the frontier an MILP task. One of the best-known algorithms for the solution of the overall problem is the one proposed by Messac et al. [28]. An approach used by Pishvaee et al. [29], [30], [31] and based on multi-objective fuzzy mathematical programming seems to be hardly applicable, due to the prohibitively high computational time necessary to the solution of real-life problems in
gas network systems, as opposed to small general supply chains [32].
While safety considerations and risk analysis are conceptually different from environmental protection measures, there are evident links whenever failures produce disastrous accidents, as in case of explosions [33]. Furthermore, safety can be considered one further goal to be pursued in the multi-objective optimization procedure.
Thus, while considering market mechanisms makes it necessary to introduce stochastic programming, multi-objective optimization must be considered when the environmental sustainability of natural gas supply chains has to be taken into account.
The aim of this paper is precisely to combine these two approaches for the development of a comprehensive model and to examine suitable algorithms for the generation of optimal strategies.
Furthermore, the inclusion of methane fugitive leaks as suggested by Balcombe et al. [23] into the general optimization problem will be considered; indeed, the increasing concern over the use of natural gas, due to its high global warming potential (GWP) makes it imperative to include methane fugitive leaks in the overall greenhouse gases (GHG) balances of natural gas supply chains.
To our knowledge, it is the first model that considers these two issues in natural gas networks.
General mathematical models
As anticipated in the introduction, one of the goals of this article is to combine financial and environmental constraints with the traditional optimization mathematical models of natural gas networks. The resulting algorithm is a multi-objective stochastic programming problem.
To take into account market features (both in the objective function and in the constraints), the approach suggested by Tomasgard et al. [11] will be employed whereas the strategy and the resulting algorithm proposed by Wang et al. [25] for sustainable gas networks and supply chains and for the construction of the corresponding Pareto frontier will be used.
A further additional feature of this article is the inclusion of methane fugitive leaks in the GHG balance of natural gas supply chains.
The financial constraints
The simplest model including market mechanisms considers the possibility of spot transactions at the delivery nodes in addition to the contractual deliveries. Obviously, the resulting model can only provide optimal operating conditions but it cannot be used during the design phase.
Since transactions are considered at delivery nodes only, upstream trading is not taken into account. This could be a minor restriction if natural monopolistic gas providers (or “national champions” with a limited scope for upstream negotiations) are analyzed.
The absence of gas futures trading implies assuming risk neutrality and rules out any hedging strategy. On the other hand, omitting futures transactions dispenses with the necessity of including no-arbitrage conditions in the model.
The model requires the maximization of the overall profit over the number of intervals that make up the horizon considered. However, decisions are to be made under uncertainty, because only a statistical information about the distributions of the values that the random variables contained in the model will take in the future is available.
Tomasgard et al. [11] consider a two-period horizon  each of the periods containing a certain number of intervals (typically six month intervals in two semester periods over a one year horizon). Generalizations of the resulting model to multistage problems can be implemented following Kall and Wallace [34].
The mass balances at delivery nodes are to be rewritten as:
where  are purchases and sales of natural gas at time t through spot transactions.
The optimization algorithm requires the objective function to include the profit originated from contract sales and spot transactions in each time interval. If demand levels, contractual prices and spot prices are subject to random fluctuations, the resulting profit has to be maximized using appropriate stochastic programming techniques.
In particular, let  stand for the demand level, contractual prices and spot prices at node m. The corresponding profit in the time interval t is given by:
In the last interval of the horizon variations in the value of storage gas (EGV, the expected gas value function) has to be taken into account [10].
The overall optimization task can be cast into the form:
If a two-period horizon is considered and uncertainties are limited to intervals in the second period, the objective function can be written as:
Constraints (1) can be regarded as providing the necessary stochastic recourse to the fulfillment of mass balances. Thus the optimization of (2) subject to (1), as well as to the equations 
reported in the previous article [1], has a two stage recourse. Furthermore, since the functional structure of constraints is constant, it can be regarded as a stochastic programming problem with fixed recourse. Additionally, due to the structure of the remaining constraints and the nature of probability distributions, it can be shown that it is also relatively complete [35].
The environmental constraints
The inclusion of environmental constraints in the gas supply chain affects investments decisions in the design phase. Since economic and sustainability goals are potentially conflicting, a multi-objective optimization has to be considered with a view to obtaining a desirable compromise between them.
An optimal trade-off can be achieved by setting up a second objective function that provides an estimation of the overall greenhouse effect generated by production, transmission and delivery in the network system. Unlike previous analyses that focussed on the generation of  in the overall supply chain (mainly related to the amount of energy used), this article considers also the global warming potential (GWP) of methane (i.e. its warming effect relative to  Indeed, when released as uncombusted natural gas into the atmosphere, the GWP of methane ranges between 28 and 32, depending on the time horizon considered.
Methane can be released at various points of the supply chain and consequently the total GHG effect of natural gas use must include the fugitive emissions during feedstock extraction, transmission and delivery of the fuel. In other words a life-cycle analysis (LCA) has to be carried out [35] to estimate its full impact and to identify the coefficients and the parameters contained in the second objective function.
The link between environmental investment decisions and the economic assessment (both in the planning and in the operation phases) can be established through the introduction of additional decision variables whose values correspond to different environmental protection levels. Since they consider only  emissions, Wang et al. [25] use environmental decision levels  only for the facilities (the nodes) and disregard emissions in the pipelines. However, methane fugitive leaks in the pipelines can be significant [23] and depend on length, material quality and depth of cover (i.e. distance from the pipeline to the surface of the ground above the pipeline). Since these three features can change in different arcs of the pipeline and carry considerable economic costs, additional sustainability variables  must be considered.
To each value  there correspond cost functions  that are increasing functions of their arguments, as well as environmental impact functions  that are decreasing functions of their arguments. Typically, all functions
are expressed as piecewise linear functions using the procedure for special ordered sets of type 2 (SOS2) used in modelling storage inflows and outflows.
The resulting optimization task can be written as a trade-off between a financial or economic objective function – such as equation (2) modified so as to include the terms containing   – and an environmental objective function based on 
For instance the multi-objective problem including the economic goal (2) can be reshaped as
The negative sign in  is related to the necessity of considering a minimum (or maximum) problem for both objective functions.
The constraints remain unchanged, unless environmental regulations are to be enforced, in which case suitable constraints on  (and consequently on  are introduced.
Algorithms and programm complexes
The stochastic programming algorithm required by the introduction of market transactions can also be cast into the form of MILP. However, specialized algorithms (such as the EMP environment, which is an extension for GAMS, [36]) can greatly reduce the relevant necessary burden.
The averaging operator (which depends on the market features) has to be provided in any case.
The multi-objective optimization problem resulting from the inclusion of green technologies for the limitation of GHG emissions can be tackled using Messac’s technique [28] for constructing the relevant Pareto frontier.
After optimizing the two objective functions separately and normalizing them, so as to avoid scaling problems the two points are reported in a plane whose axes represent the values of the objective functions. Uniting the two points and tracing the perpendicular to a set of points lying on this line makes it possible to solve a sequence of minimization problems with respect to the objective function  by including one additional linear constraint that eliminates the area shaded in blue. It is evident from the figure that the minimum point of each of these optimization tasks lies on the Pareto frontier, which consequently can be traced accurately.
Conclusions
The simultaneous optimization of financial portfolios and green goals, the inclusion of fugitive methane leaks into the overall environmental assessment, or the possibility of considering safety issues as a third objective function, considerably enlarge the range of problems that can be tackled by gas network systems optimization. However, this increased model accuracy comes at the price of considerably higher computational times. Even if the present power of digital computing systems (including parallel processing) has greatly increased the horizon of practical feasibility for many numeric algorithms, multi-objective stochastic programming can still be beyond the technical capability of most computing systems
if continental gas supply networks are examined. In this case some simplifications and/or approximations may become necessary.
Thus, multi-objective optimizations can be simplified into single-objective optimizations by comparing a limited number of scenarios. In other words, the variables characterizing environmental issues can be discretized to provide different scenarios, which are then compared to identify the optimal solution.
Approximations can be introduced by analyzing the deviations introduced by market mechanisms through first-order perturbation methods applied to the solution obtained without considering financial transactions.
As already mentioned in the previous article, technical expertise, numerical skill and ecomic insight are still required for large gas network systems to be realistically described and successfully optimized.
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