Cheap, robust and low carbon: comparing district heating scenarios using stochastic ordering

03/09/2020 ∙ by Victoria Volodina, et al. ∙ 0

Strategies for meeting low carbon objectives in energy are likely to take greater account of the benefits of district heating. Currently, district heating schemes typically use combined heat and power (CHP) supplemented with heat pumps attached to low temperature waste heat sources, powered either by electricity from the CHP itself or from the National Grid. Schemes have competing objectives, of which we identify three: the need for inexpensive energy, meeting low carbon objectives and robustness against, particularly, variation in demand and electricity prices. This paper compares different system designs under three scenarios, using ideas from stochastic dominance close in spirit to traditional ideas of robust design. One conclusion is that, under all considered scenarios, a heat pump provides the most robust solution.



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1 Introduction

A comprehensive handling of risk is crucial in assessing the viability of infrastructure projects. From a commercial point of view, investors need to be reassured that they will receive a return on their investment, whilst the public sector needs to ensure that money is spent responsibly. The aim of this paper is to demonstrate a new approach to dealing with uncertainty for the planning of infrastructure projects in the form of stochastic orderings. The context is waste heat recovery, in which heat from industrial and urban sources is used as a zero carbon alternative to fossil fuels for district heating networks.

This paper grew out of two District Heating and Cooling (DHC) projects: CELSIUS 4 and ReUseHeat 32, along with a project entitled Managing Uncertainty in Government Modeling (MUGM), run by the Alan Turing Institute, London 24. The authors were fortuitously able to use experience gained in the former two projects to provide an exemplar for the third, in which the aim is to enhance the understanding and use of uncertainty methodology in major projects and policy making. Further influences have been the 2018 workshop on Uncertainty Quantification at the Isaac Newton institute 34 and the roughly contemporaneous UK networks on uncertainty: M2D 23, and CRUISSE 5. As if the ambitious aims of the above projects were not daunting enough, the European Commissions 2050 Roadmap for zero net carbon has become a force majeure, which is likely to have an impact on much European policy, but particularly energy policy.

Within the MUGM project, the adopted approach is, generically, model-based decision support. Our belief is that the decisions and policies made by national and local government, together with their private sector partners, should strongly influence the types of mathematical models used and the data capture necessary to calibrate the models. In addition, the metrics, or Key Performance Indicators (KPIs) used to assess the impact of the decisions should reflect uncertainty in some way. Here, we discuss Stochastic Key Performance Indicators (SKPIs) which help to capture uncertainty via probability distributions.

This paper is a modest approach to testing this bundle of ideas via the modeling of a district heating project. It is based on a real system in Brunswick (Braunschweig) in Germany, in which heat from a data centre will be used as an input to a district heating network for a newly constructed residential and commercial area in the city. The system is one of four demonstrators on the ReUseHeat project and the operators were recently awarded a prize for its innovative approach 31. Here, “based on” refers to the matching of variables, that is model inputs and outputs to the project, although the modeling results were not used in the design of the actual system. Rather, the model conclusions derived may provide assistance for the design of similar projects. Both CELSIUS and ReUseHeat have objectives to aid replication of projects and provide advice to other investors Lygnerud et al. [2019].

The area of Robust Engineering Design, which lies on the frontier between engineering and statistics, grew out of what may be called the “quality revolution” of the 1970s and 1980s. Broadly, it is an attempt to control the output variability of a component or system, while keeping the output optimum on target. It had a radical impact on the types of test-bed experiments performed on prototypes by including “noise factors” alongside “control factors” in the experiments Grove [1992]

. As if by osmosis, similar ideas arose in other areas. Thus, in portfolio theory, and portfolio design, the aim is, roughly, to achieve maximum yield with minimum volatility and the theory has been widened to cover real options. In mathematical finance, ARCH and GARCH models model both means and variances

Engle et al. [2012]. Mean-variance portfolio theory and stochastic optimisation are employed for the modeling of uncertainties and risk in energy system planning Ioannou et al. [2017]; Bhattacharya and Kojima [2012].

Although the above methods can be covered by some kind of utility theory, we are keen to preserve the notion of robustness, perhaps because of the physicality and clarity required to understand the impact of energy capital projects in the light of the 2050 targets. We will allow ourselves the freedom to choose one or more SKPIs but with a special methodological framework which is that of stochastic ordering. The details will be given in section 3

, but we can describe the framework heuristically now.

A stochastic ordering is a partial ordering on distributions. Orderings provide a useful framework for comparing distributions in the following sense. Suppose we are aiming to minimise the cost of a project and have two choices to make regarding the design of that project (Design A and Design B). Both designs are uncertain and we can construct probability distributions of the cost in each case. If the distribution of choice A stochastically dominates that of choice B, for any given cost, the probability that the cost exceeds that particular value is lower for choice A than it is for choice B. In other words, if we are only interested in minimising the probability of the cost exceeding some chosen value, we should always choose Design A. This seems like a sensible and intuitive way to look at decision making and this is our approach here.

Stochastic orderings roughly divide into two classes: orderings that denote shift (in mean) and those that denote variability (dispersion). We have already mentioned high mean and low volatility. An ordering does not prescribe a KPI, SKPI or utility, but we claim that it provides a platform for uncertainty in the follow sense. Given an ordering, a suitable metric is one which is order-preserving with respect to that ordering. This means that it is a function whose expectation is ordered in the same direction as the stochastic underlying ordering. Thus both the mean and the median themselves are order preserving with respect to first order stochastic dominance and both the standard deviation and the Gini coefficients are ordered with respect to some well known dispersion orderings.

The versatility that stochastic ordering affords in extending the range of metrics for risk and uncertainty has been exploited in a number of fields. For example: Robust Design Cook and Jarrett [2018], Portfolio theory Annaert et al. [2009] and signal processing Tepedelenlioglu et al. [2011]. A review is given in Mosler and Scarsini [1993].

The study here is based on the use of an open source optimisation model for long-term energy planning called Open Source Energy Modeling System (OSeMOSYS)

Howells et al. [2011], described in detail in Section 4. To make the analysis as transparent as possible, we distinguish local and global sensitivity. Local will be with regard to recognisable sources of variability, such as operational costs and seasonal variation in demand, while global will be with respect to wider scenarios. Indeed, the definition and use of scenarios is a major issue in modeling the impact of measures to meet 2050 targets Wheatcroft et al. [2019].

Our case, as with most decision problems, deals with multiple objectives. We agree with the now quite old criticism of classical cost benefit analysis that criteria should not be reduced to monetary value alone Kelman [1981]. We shall, therefore, need multivariate stochastic orderings, for both shift and dispersion and our SKPI will be order preserving for one ordering or the other, or some combination of the two.

This paper is structured as follows. Section 2 covers background regarding district heating and the main objectives represented in the title of the paper. Section 3 provides a simple introduction to stochastic orderings. Section 4 describes a series of simple computer experiments on a specially selected set of system designs, choices of input variations and broader scenario-based alternatives. Section 5 draws conclusions based on a selection of stochastic orderings and SKPI. Section 6 gives tentative conclusions and points to the urgent need for further study.

2 Background

2.1 District heating

District heating is a system in which heat is produced by some centralised source and distributed to commercial or residential buildings via a network of insulated pipes. It is particularly well developed in northern Europe and Scandinavia. In Denmark, for example, over 60 percent of houses are connected to a district heating network, whilst rates of connection are over 50 percent in Sweden, Finland, and the Baltic states Agency [2018]. Historically, district heating has been powered by the burning of fossil fuels such as coal and gas. However, more recently, it has been seen as an opportunity to decarbonise the heating sector via the use of waste heat from industry and other sources. Traditionally, waste heat recovery has focused on high temperature heat from heavy industry which can be fed directly into the system. However, more recently, there has been a focus on the opportunities of recovering waste heat from low temperature urban sources such as, for example, metro stations Lygnerud et al. [2019] and underground rivers 20

. The prevalence of low temperature sources and their location near areas of high heat demand provides an opportunity in the wider agenda of carbon reduction. It is estimated that low temperature heat from metro stations, data centres, service sector buildings and waste water treatment plants could meet over 10 percent of heat demand in the European Union

Persson and Averfalk [2018].

A major difference between high and low temperature heat recovery is that, in the latter case, the heat typically needs to be upgraded before it is suitable for use in the network. This requires the use of a heat pump to increase the temperature to the required level. The installation of heat pumps poses additional technical challenges due to a lack of maturity in the technology and a lack of experience in installation and maintenance. Perhaps most importantly, heat pumps run on electricity and this creates additional operational costs and a vulnerability to increases in the price of electricity. This is considered a major risk in low temperature heat recovery Lygnerud et al. [2019].

At present, low temperature heat recovery is not widespread. There are a number of reasons for this, including a lack of political and commercial awareness, a lack of interest from heat ’owners’, the immaturity of the technology and a lack of a legal and regulatory framework Lygnerud et al. [2019]. The ReUseHeat project has been set up to demonstrate the use of urban waste heat recovery in a data centre, a hospital, a waste treatment plant and a metro station. The ambition is that lessons from current implementation can be disseminated, providing guidance, reducing risk and increasing the viability of future systems. The project also focuses on ‘soft’ issues such as design of contracts, business models and bankability.

Perhaps the biggest barrier to the widespread rollout of low temperature waste heat recovery is a gulf between the risk assessment required by financial institutions and that which is typically provided by project developers. A common theme among investors who have expressed an interest in waste heat recovery is that they have money to invest but that the business case and risk analysis presented is simply not convincing enough to the institution. This is partly due to a lack of existing projects and experienced contractors and public money can help improve this situation. However, before low temperature waste heat recovery becomes widespread, it is clear that the gulf between the risk analysis that is typically provided and that which is required needs to be closed. It is hoped that the methodology presented in this paper will go some way towards closing this gap.

2.2 Heat recovery in Brunswick

We construct a simple model based loosely on the Brunswick demonstrator within the ReUseHeat project. The aim of the demonstrator is to supply heat to 400 newly built housing units, using waste heat from a nearby data centre which is upgraded to a suitable temperature with a heat pump. The new housing units will also be connected to the existing city-wide network which, at present, supplies 45 percent of residents in the city using a gas powered combined heat and power (CHP) unit. The intention is then that heat from the data centre will cover the baseline demand (that is demand that is present throughout the year such as for hot water) and heat from the CHP will cover seasonal demand. A diagram showing the layout of the Brunswick demonstrator is shown in Figure 1.

For the study described in this paper, the model is used to assess three different design options to meet the demand for heat. In the first, the entire heat demand is met by the CHP. In the second, the entire heat demand is met by waste heat from the data centre. Finally, following the setup of the demonstrator, baseline demand is met by heat from the data centre and seasonal demand is met by the CHP.

Figure 1: Diagram demonstrating the Brunswick demonstrator.


In order to build a model and run simulations, we employ an open source optimisation model for energy planning called Open Source Energy Modeling System (OSeMOSYS) [Howells et al., 2011]. OSeMOSYS is a deterministic, linear optimisation model that obtains the energy supply mix (generation capacity and energy delivery) that minimises the Net Present Cost (NPC), subject to meeting specified demand for all energy at each time step [14].

There are a number of other widely used medium to long term energy system models such as MARKAL [Kannan et al., 2007], TIMES [Loulou and Labriet, 2007], MESSAGE [LLC, 2009], LEAP [Heaps, 2008] and PRIMES [Mantzos, 2009]. However, we have chosen OSeMOSYS for two main reasons. Firstly, OSeMOSYS provides us with the flexibility to operate on a local (city) level and to define our own scenarios. Secondly, OSeMOSYS is an open source model and therefore freely available for comparative project modeling.

3 Uncertainty and Stochastic orderings

The basic machinery of uncertainty has traditionally been standard deviations, mean squared error, confidence intervals and their multivariate and Bayesian counterparts. But there is a demand for less probabilistic measures of uncertainty such as fuzzy set theory, Dempster-Shafer belief functions and various extensions of classical coverage theory. In addition, there is a recent interest in the use of scenarios to capture wider uncertainty issues.

A less prescriptive, but still probabilistic, methodology is suggested here, namely that of stochastic ordering. These are partial orderings of distributions often expressed as relationships between cumulative distribution functions (cdfs), written

or, for the corresponding random variables,

. Note that the partial ordering requirement gives transitivity:

It is suggested that the use of such orderings may be a useful way of describing uncertainty. The rationale is that stochastic orderings are weaker than a limited list of specific metrics whilst capturing the idea of more or less uncertainty.

For this, we need the idea of order preserving functions. A function is said to be order preserving with respect to a stochastic ordering, if, for their expectations (which may sometimes be called risks),

In many cases, we are able to characterise the ordering via the set of all order preserving functions. The best known ordering is first order stochastic dominance , which is defined when

and the inequality is strict for at least one value of .

Equivalently, we can define all order preserving functions which is the set of all non-decreasing functions in .

3.1 Dispersion orderings

For uncertainty, we can use so-called dispersion orderings. Dispersion (variability, scatter, or spread) measures the extent to which a distribution is stretched or squeezed. First, note the simple fact that, if we take two independent copies of a univariate random variable , with variance , then

The independent copy idea is very useful and leads to dispersion orderings for the multivariate case. Let and be two -dimensional random variables with (multivariate) cdfs and . Let be some distance function between and , which are both -dimensional. Then, as above, let and

be pairs of independent random vectors from the distributions

and , respectively. We define a dispersion ordering to be

if and only if

This was first introduced in Giovagnoli and Wynn [1995] for the special case in which

the Euclidean distance, or . This is named the “weak dispersion ordering”.

For any given distance , the class of order preserving functions comprises all non-decreasing functions, of . By choice of distance and function, , we can therefore cover a large range of dispersion metrics (SKPI).

3.1.1 ordering

We define the ordering to be the case in which

and therefore is defined to be the distance. Note that, when , the ordering is preserved under since is an increasing function of . This is not the case for , however, since is not an increasing function of .

3.1.2 Generalised simplex ordering

A natural extension of the independent copies idea is to take copies : and define a function that describes the separation between them. In Pronzato et al. [2017], it is shown that if we take

where is the dimension simplex (in dimensions) whose vertices are then is a function of the variance covariance matrix of the underlying distribution.

This prompts a dispersion ordering: defined by

where are independent values from the distribution. The case in which was introduced by Oja Oja [1983] and discussed in Giovagnoli and Wynn [1995] and is referred to as the ‘Simplex ordering’. Here, we remove the requirement that and refer to the approach as the Generalised simplex ordering. Note, again, that whenever we see the ordering , we can write down the class of order preserving functions using non-decreasing functions, in the present case of . Dispersion orderings based on Hausdorff distance López-Díaz [2006] and Mahalanobis Pronzato et al. [2018] distance are possible alternatives to the simplex ordering.

Various versions of the above orderings will be used in the following, typically to provide plots showing the stochastic partial ordering dominance of one configuration (design) with respect to another.

3.1.3 Quantifying closeness

It is useful to note that, if for some stochastic ordering such as , or , it may still be the case that the distributions of or are ‘close’, and we shall see some examples in which this is true. In that case, we can use some standard distance between the relevant distributions to quantify the similarity. A well known example, and the approach taken in this paper, is the maximum deviation or Kolmogorov-Smirnov (KS) distance which is defined as follows. Let the random variables and have cdfs and respectively. The KS distance between and is given by

This is a portmanteau measure of distance used as a formal test statistic in statistical hypothesis testing, but used here simply to compare the closeness of scenarios and designs; formal statistical tests are not appropriate if only because of the artificiality of the simulation sample size. Our tables will give

showing a percentage of the range of the cdfs . The value of induces bounds on other metrics such as tail areas of distributions and risks (expected losses), under suitable conditions.

3.1.4 Scaling and dispersion orderings

It is important to consider the impact of scaling on our dispersion orderings. Variables like NPC and carbon emissions have natural units of measurement associated with them, providing a natural scaling. However, it is important to consider that such scalings are arbitrary. We therefore argue that, ideally, the choice of unit should not affect the ordering. Here, we investigate the effect of scaling on the and Generalised simplex dispersion orderings respectively.

First, consider the distance ordering. Here, the scaling of the variables impacts the relative contribution of each one in the calculation of the distance. To illustrate, consider the case in which and therefore in which is given by

Suppose that and are on the range whilst and are on the range where . Define and to be normalised values of and , scaled so that they are on the range. The distance can therefore be written as

From the expression above, it is clear that we place a higher weighting on the first term due to scaling by a factor of and therefore the random variables are not considered equally. As a result, the ordering is sensitive to scale. In this paper, we take a fairly ad-hoc solution to the question of scaling and pre-process the data to ensure that they operate in the same range across all dimensions. This is done by simply dividing each variable through by its range.

We now consider the impact of scaling on the Generalised simplex dispersion ordering. To illustrate, we consider the case in which and compute the area of triangles formed with three distinct points and , i.e.

where and are on the ranges and respectively. We define and as the normalised values of and and re-write the expression for the area of the triangles as:

The effect of scaling is therefore to multiply the area of the triangle by . The rescaling must therefore preserve the simplex ordering and we conclude that the simplex volume is a scale-free, homogeneous measure. We argue that this is a major advantage of the Generalised simplex ordering since no arbitrary pre-processing is required to put the variables on comparable scales.

3.2 Interpreting stochastic orderings

As described in the previous section, we are able to embed standard measures of uncertainty such as variance and MSEs within a theory of stochastic orderings. Thus, to look for first order stochastic dominance, we can simply plot the empirical cdfs. If one always lies above or on top of the other, then we can claim that one dominates the other, with respect to any increasing function. An example is a (right) tail probability, which is the expectation of the indicator function of the tail area. Such metrics are related to Value At Risk metrics in finance.

For ease of expression, we concentrate on two dimensional cases. This is both instructive and useful since we have concentrated on two output variables, that are NPC and emissions, which we label and respectively. Since the objective is to minimise these values, lower values are always preferred. Design option has first-order stochastic dominance (FSD) over design option if:

where and are cdfs of for design options and , respectively. Broadly, design option dominates design option when the cdf of the former is always greater than or equal to that of the latter and therefore the two cdfs do not cross.

4 Experimental design

We perform an experiment to demonstrate the use of stochastic orderings with waste heat recovery as an example. Here, we compare three different designs for supplying heat to a newly constructed housing development in Brunswick, Germany (described in section 2.2). The three designs differ according to the technology mix employed for supplying domestic heat and are outlined in Table 1. Each design is evaluated in terms of its Net Present Cost (€) and -equivalent emissions (in metric tonnes). The three designs are evaluated in the context of both local (sensitivity analysis) and global (scenarios) variability in the model inputs.

We are interested in the effects of varying four different inputs to the model. These are

  1. Operational costs.

  2. Discount rate.

  3. Coefficient of Performance (COP) for the heat pump (heat delivered per unit of electricity).

  4. Emission Activity Factor (the emissions produced (in metric tonnes) from operating a particular technology in the energy system).

Variations in the four input variables are expected to impact both the Net Present Cost (NPC) and the level of emissions. For each input variable, we specify three levels: low, medium and high. We then perform simulations with a full factorial design (often known as a fully crossed design) so that all possible combinations across the model inputs are considered [George et al., 2005]. Each of the three design options are represented in the form of a Reference Energy System (RES) in Figure 2.

Design type Description
Design Option 1 Combined Heat and Power (CHP) is employed to meet both the baseload and seasonal heat demand.
Design Option 2 A heat pump is employed to meet baseload Heat Demand and CHP is used to meet seasonal heat demand.
Design Option 3 A heat pump is employed with a small amount of storage to meet both the baseload and seasonal heat demand.
Table 1: Description of design options in the study.

We define three scenarios that differ in terms of selected elements of government climate policy and consumer engagement with green technology. A description of the three scenarios is given in Table 2 and details of the numbers used are given in the appendix. The operational lifetime of a typical heat pump is around twenty years and thus this number is used as the time horizon for the study.

Under each scenario, we produce model simulations that, for a given set of model inputs, generate the volume of emissions and NPC as model outputs. First, we compare the empirical cdfs of the model outputs under different scenarios and design options. We then compare different scenarios and design options using empirical cdfs of the and Generalised simplex dispersion metrics. Whenever one empirical cdf dominates another, we calculate the KS distance to quantify the distance between them.

Figure 2: Reference energy system (RES) for Design Option 1 (first row), Design Option 2 (second row) and Design Option 3 (third row). All technologies are represented as ‘blocks’ and energy carriers as ‘lines’.
Scenario Description
Green scenario Penalty per metric tonne of emissions: 100 € per metric tonne. Annual change in baseload and seasonal heat demand: -1%. Increasing gas prices and decreasing electricity prices (see Section 7.2 for more details).
Neutral scenario Penalty per Mton of emissions (40 € per Mton). Baseload and Seasonal Heat Demand fluctuate around the central projections Gas and electricity prices stay within the central projected values (see Section 7.2 for more details).
Market scenario Penalty per metric tonne of emissions: zero. Annual change in baseload and seasonal heat demand: 1%. Decreasing gas prices and increasing electricity prices (see Section 7.2 for more details).
Table 2: Details of scenarios.

5 Results

The two model outputs are plotted against each other for each of the three design options and under each scenario in Figure 3. As expected, due to the use of natural gas, the highest emissions are produced by design option 1, followed by design option 2 and then design option 3. The second row of plots in Figure 3 demonstrate that the different scenarios have only a small impact on levels of emissions under each of the design options. This, of course, is not surprising since government interventions typically aim at changing behaviour through cost.

Differences in the three scenarios have a much larger impact on Net Present Costs (NPC). Under the market and neutral government scenarios, which assume limited intervention and only small changes in customer behaviour, design option 1 achieves a lower NPC than both design options 2 and 3. This is mostly due to the high investment cost of the Heat Pump. However, under the green government scenario, which assumes that major policy will be put in place aimed at reducing climate change, the NPC for design option 1 tends to be higher than for design options 2 and 3 due to a high carbon tax. Therefore, under the green government scenario, design options 2 and 3 become more attractive alternatives, thus demonstrating the value of considering different scenarios when making planning decisions.

Figure 3: First row: Net Present Costs against carbon emissions for all three design options under the three scenarios. Second row: Net Present Costs against emissions for all three design options, plotted separately for each scenario.

5.1 Orderings in the mean

Empirical cdfs corresponding to NPC are shown in Figure 4. In the first row, cdfs for each design option under the three scenarios are presented. The second row of plots shows the cdfs of NPC for each scenario under the three design options. In the third row, for clarity, these are repeated but shown with common axis limits. Focusing first on the top row, under all three scenarios there are clear stochastic orderings between the three design options. In both the neutral and market scenarios, design option 1 dominates design option 2 which dominates design option 3 whilst, under the green government scenario, the ordering is reversed due to the increased financial support from policy makers towards renewable sources. Therefore, if the sole aim is to minimise NPC, under the green scenario design option 3 would be the preferred option whilst design option 1 would be the preferred option under the neutral and market scenarios. The reason for this difference is that, in the green government scenario, the government introduces high penalties for generation of emissions, which makes design option 1 an unattractive option for providing heat (see Section 4 for more details).

Focusing now on the second and third rows of Figure 4, there are, again, clear orderings. Under design option 1, the market scenario dominates the neutral scenario which dominates the green scenario. On the contrary, for design option 3, the green government scenario dominates the neutral scenario which dominates the market scenario. This difference in ordering may lead a planner to think carefully about their choice of design option. If, for example, they consider policy featured in the green government scenario to be likely, they may choose to build design option 2 or 3 to mitigate that risk.

Under design option 1, the cdfs differ substantially compared to the cdfs produced for design option 3, demonstrating the impact of the different scenarios. We calculate the KS distance to quantify the difference in the empirical cdfs of the green and neutral scenarios under each design option and these are shown in table 3. Here, since the distances between the cdfs tend to be large, the effect on cost of different scenarios is high.

Empirical cdfs corresponding to emissions are shown in Figure 5. Similarly to figure 4, the empirical cdfs of emissions for each design option under each of the three scenarios are shown in the first row. The second and third row plots show the empirical cdfs of emissions for each scenario under the three designs. From the top row, it is clear that, under all three scenarios, design option 3 dominates design option 2, which dominates design option 1. This is not surprising and confirms our expectation that waste heat recovery is a carbon reducing technology under most reasonable policy decisions.

Figure 4: Empirical cdfs for NPC for (i) all three design options plotted together for each individual scenario (first row), (ii) all three scenarios plotted together for each individual design option (second row) and (iii) empirical cdfs for all three scenarios for each individual design option plotted on the same scale (third row).
Figure 5: Empirical cdfs for emissions for (i) all three design options plotted together for each individual scenario (first row), (ii) all three scenarios plotted together for each individual design option (second row) and (iii) empirical cdfs for all three scenarios for each individual design option plotted on the same scale (third row).

From the second and third rows of plots, we can see that the green government scenario dominates the neutral scenario which dominates the market scenario under each design option. Again, this is not surprising since, under the green government scenario it is assumed that heat demand will fall due to improved insulation. The KS distances, shown in table 3, are, again, large, implying that the effect on emissions from choosing one design option over another is high. The KS distances between different pairs of design options under different scenarios are shown in table 4.

Green vs Market Green vs Neutral Neutral vs Market
Variable NPC NPC NPC
Design 1
Design 2
Design 3
Table 3: Kolmogorov-Smirnov distances between empirical cdfs of NPC and for pairs of scenarios under different designs.
design 1 vs 2 design 1 vs 3 design 2 vs 3
Variable NPC NPC NPC
Market Scenario
Neutral Scenario
Green Scenario
Table 4: Kolmogorov-Smirnov distances between empirical cdfs of NPC and for pairs of designs under different scenarios.

In Section 1 we discussed the benefits of design options which, on average, show low NPC and emissions. Until now, we have considered orderings which denote dominance in the mean for NPC and emissions separately. In addition to the mean, however, we are interested in variability and thus we now consider orderings which reflect dispersion.

5.2 dispersion ordering

The first row of Figure 6 shows empirical cdfs of the metric for the three design options under each of the three different scenarios. Here, the empirical cdfs tend to cross and thus, in those cases, there is no ordering. The exception is the green government scenario in which design option 1 is dominated by both design options 2 and 3. This implies that design option 1, in which the entire heat demand is met by CHP, has higher dispersion, and therefore is less robust, than the other two design options.

Focusing now on the second row of Figure 6, under design option 1 there is clearly an ordering in which the market scenario dominates the neutral scenario which dominates the green government scenario. This means that, the more green minded the government, the less robust the design. Under design option 3, the green scenario dominates the neutral scenario which dominates the market scenario meaning that the robustness increases with more green minded policy.

KS distances under each design option for different pairs of scenarios are shown in Table 5. Here, under design options 2 and 3, although there is stochastic dominance, the distances are small and therefore there is little difference in the robustness of each one. Under design option 1, on the other hand, the distances are bigger meaning that the scenarios have a large impact on robustness.

KS distances between pairs of design options under each scenario are show in Table 6. Under the Market and Neutral scenarios, the empirical cdfs cross so we do not calculate the distance. Under the green scenario, the distance is large showing that the difference in robustness between the design options is high.

Figure 6: First row: empirical cdfs of the distance for all three design options plotted together for each individual scenario. Second row: empirical cdfs of the distance for all three scenarios plotted together for each individual design option.
Design Green vs Market Green vs Neutral Neutral vs Market
Design 1
Design 2
Design 3
Table 5: Kolmogorov-Smirnov distances for the ordering between pairs of scenarios for each design.
Scenario design 1 vs design 2 design 1 vs design 3 design 2 vs design 3
Market - - -
Neutral - - -
Table 6: Kolmogorov-Smirnov distances for the ordering between pairs of designs under each scenario. A dash denotes that the cdfs cross and therefore there is no ordering.

5.3 Generalised simplex dispersion ordering

Figure 7 is the same as Figure 6 but for the Generalised Simplex ordering rather than the ordering. Focusing first on the top row, in which each panel shows empirical cdfs of each design option under a particular scenario, contrary to the case there is a clear ordering. In all three scenarios, the empirical cdf for design option 3 dominates that of design options 1 and 2. Design option 3 also dominates design option 1 in the green and neutral government scenarios. Design option 3 is therefore shown to be the most robust option in all three cases.

From the second row of Figure 7, we can see that, for design option 1, the market scenario dominates and is thus the scenario in which the design is most robust. Design options 2 and 3, on the other hand, are most robust in the green scenario.

Table 7 shows the Kolmogov-Smirnov distance between the cdfs of different pairs of scenarios under each design and Table 8 shows the distance between the cdfs of different pairs of design options under each scenario.

Design Green vs Market Green vs Neutral Neutral vs Market
design 1
design 2
design 3
Table 7: Kolmogorov-Smirnov distances for the generalised simplex ordering between pairs of scenarios for each design.
Figure 7: First row: empirical cdfs of the Generalised simplex metric for all three design options plotted together for each individual scenario. Second row: empirical cdfs of the Generalised simplex for all three scenarios plotted together for each individual design option.
Scenario Design 1 vs Design 2 Design 1 vs Design 3 Design 2 vs Design 3
Table 8: Kolmogorov-Smirnov distances for the generalised simplex ordering between pairs of designs under each scenario.

6 Discussion

There are three main components of the simple methodology presented here. The first, which is close in spirit to ideas from robust engineering design, is that different combinations of technology have different levels and types of response in sensitivity analysis. Second, it is recommended that this sensitivity analysis should be at two levels (at least): normal response to input variability but also to more general scenarios. The latter may, for example, be the result of international changes in the commercial and political environment. The third component is to widen the portfolio of risk measures and we are recommending the use of metrics which are order preserving with respect to some underlying stochastic ordering. This includes many familiar metrics.

We have applied methods of stochastic dominance in the context of local energy planning to compare different options for providing residential heat in the face of uncertainty. We have considered stochastic orderings both in shift and dispersion. In particular, two dispersion orderings, the and Generalised Simplex dispersion orderings, have been deployed. We have demonstrated the effects of scaling both for the distance ordering and the Generalised simplex ordering. Crucially, whilst the former is sensitive to changes in scale, this is not true of the latter. Whilst an ad-hoc rescaling approach was proposed for the ordering, we consider insensitivity to scale to be an attractive feature.

We have demonstrated that, under a green government scenario, which assumes active government policy to meet the 2050 net zero carbon target, lower equivalent emissions can be produced at a lower cost by employing a heat pump with heat from a data centre. In addition, we have found this this is more robust and less volatile compared to the other design options under all three scenarios. We argue that, if it can be shown to be true in general, this robustness is an attractive feature of low temperature waste heat recovery.

District Heating has traditionally been associated with local generation of hot water using gas or oil fired CHP, but, as policy adapts to the low carbon agenda, better use of waste heat is already being given greater priority. Combined with a heat pump, powered with renewable electricity, it is potentially carbon free. The use of gas or oil fired CHP can be used as a backup for a period but will eventually cease when regulations prohibit the use of fossil fuels. Thus, robust heat pump technology, fed by waste heat, in addition to more heat-source, ground-source and other sources is the future. It deserves considerable research effort including innovative modeling methods. This paper is a modest contribution to this effort. It is probably a little early to draw such substantial policy implications, but this work points to support for larger heat pumps giving more robustness in normal and abnormal conditions.


This paper was funded by the Managing Uncertainty in Government Modelling (MUGM) project (R-EDI-002) at the Alan Turing Institute, London. This paper is supported by European Union’s Horizon 2020 research and innovation programme under grant agreement No 767429, project ReUseHeat.

7 Appendix

For OSeMOSYS, there are 4 units that needs to be specified carefully and we provide the specification in Table 9.

Choice of units
Energy MWh
Power MW
Cost Millions €
Emissions Mton (metric tonne)
Table 9: Units specification for OSeMOSYS.

7.1 Parameter values for local sensitivity analysis

High Medium Low
Discount rate 0.08 0.05 0.02
Table 10: The values of discount rate considered in experiments
High Medium Low
Heat Pump (Wholesale Heat) 5 3.6 3
CHP (Electricity) 0.33 0.3 0.27
CHP (Wholesale Heat) 0.53 0.51 0.49
Table 11: The values of coefficient of performance (COP) of technologies for generating either wholesale heat or electricity considered in experiments.
High Medium Low
CHP (CO2) 0.5865 0.5100 0.4335
CHP (NOx) 0.1656 0.1440 0.1224
Heat Pump (CO2) 0.5831 0.5070 0.4309
Gas Import (CO2) 0.0578 0.0503 0.0428
Table 12: The values of Emission Activity Factor of individual technology in generating either CO2 or NOx as part of operation. Emission Activity Factor corresponds to the amount of emission (CO2-equivalent) in Mton per energy produced in MWh.
Year High Medium Low
2020 0.0360 0.0360 0.0360
2021 0.0365 0.0360 0.0355
2022 0.0371 0.0360 0.0349
2023 0.0376 0.0360 0.0344
2024 0.0382 0.0360 0.0339
2025 0.0388 0.0360 0.0334
2026 0.0394 0.0360 0.0329
2027 0.0400 0.0360 0.0324
2028 0.0406 0.0360 0.0319
2029 0.0412 0.0360 0.0314
2030 0.0418 0.0360 0.0310
2031 0.0424 0.0360 0.0305
2032 0.0430 0.0360 0.0300
2033 0.0437 0.0360 0.0296
2034 0.0443 0.0360 0.0291
2035 0.0450 0.0360 0.0287
2036 0.0457 0.0360 0.0283
2037 0.0464 0.0360 0.0278
2038 0.0471 0.0360 0.0274
2039 0.0478 0.0360 0.0270
Table 13: Operational costs (Fixed costs) for CHP and Heat Pump (Millions €/MWh/a).

7.2 Parameter values for global sensitivity analysis

The electricity and gas price projections employed in our global sensitivity analysis are similar to the ones used in Foster et al. [2016].

Year Green scenario Neutral scenario Market scenario
2020 113.07 125.26 147.43
2021 119.72 131.91 154.08
2022 120.83 135.24 155.19
2023 125.26 140.78 157.41
2024 129.70 145.22 161.84
2025 134.13 148.54 161.84
2026 136.35 150.76 165.17
2027 136.35 148.54 162.95
2028 136.35 147.43 161.84
2029 139.67 151.87 166.28
2030 138.56 151.87 171.82
2031 138.56 151.87 171.82
2032 138.56 151.87 171.82
2033 138.56 151.87 171.82
2034 138.56 151.87 171.82
2035 138.56 151.87 171.82
2036 138.56 151.87 171.82
2037 138.56 151.87 171.82
2038 138.56 151.87 171.82
2039 138.56 151.87 171.82
Table 14: Electricity prices in € per MWh for each scenario.
Year Green scenario Neutral scenario Market scenario
2020 42.12 29.93 22.5
2021 43.23 31.04 22.5
2022 44.34 31.04 22.5
2023 44.34 32.15 22.5
2024 45.45 33.25 22.5
2025 46.56 34.36 22.5
2026 47.67 34.36 22.5
2027 49.88 35.47 22.5
2028 49.88 35.47 22.5
2029 49.88 35.47 22.5
2030 49.88 36.58 22.5
2031 49.88 36.58 22.5
2032 49.88 36.58 22.5
2033 49.88 36.58 22.5
2034 49.88 36.58 22.5
2035 49.88 36.58 22.5
2036 49.88 36.58 22.5
2037 49.88 36.58 22.5
2038 49.88 36.58 22.5
2039 49.88 36.58 22.5
Table 15: Gas prices in € per MWh for each scenario.


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