3. Cost Curve Definition

As a model input, technologies, that can reduce road vehicle CO2 emission or energy consumption, need to be identified and their reduction potentials and costs need to be quantified.

Starting from this data, JRC’s DIONE cost curve model develops CO2 or energy reduction cost curves, which describe the mathematical relationship between reduction potentials and related costs for different powertrains and vehicle segments. Depending on the powertrains and types of fuel or energy used by the vehicle, the cost curves can represent the costs of reducing of conventional energy consumption (liquid or gaseous fuels), or the total energy consumption, including also electric energy. This two types of cost curves are explained in detail in the DIONE model v2 report available on the website.

Likewise, the complete model description and mathematical documentation of the operations carried out can be found in the report, whereas instructions on how to use the model are provided below.

The diagram below shows the procedural steps that the tool is performing.

In the first step, optimization is carried out to identify cost-optimal packages of CO2 or energy reduction technologies. Then, several transformation steps are applied to transform the solutions found. Finally, a cost curve is fitted to the set of solutions. These steps are carried out by the DIONE model one after another, given that a suitable input data is provided to execute them.

3.1. Identifying Optimal Technology Packages

The aim of this first step is to identify cost-optimal technology packages to reduce CO2 emission as well as energy consumption. Given the set of available reduction technologies, the challenge consists in finding, among all feasible packages (i.e., combinations of compatible technologies or subsets of them) the set of optimal configurations with minimal total costs and maximum total reduction.

This step uses the input data in sheets “reduction_tech_cost”, “incompatibility_tech”, and “include_exclude_tech”. Firstly, the list of technologies available has to be provided along with their reduction potentials; secondly it has to be filled with information on which technologies are not mutually compatible, and thirdly it can be used to enforce or exclude some of the technologies for given constellations if required.

The large amount of possible optimization problems and combinations of technologies makes the problem computationally difficult. Moreover, available technologies are not always compatible, i.e., not all technologies can be combined with each other. For example, various engine downsizing technologies are available, but there is no package containing more than one, which means a simple combinatorial approach cannot be applied.

The DIONE cost curve model applies a multi-objective optimization algorithm to solve the problem efficiently and make the algorithm adaptable to the changes in input parameters. While, the algorithm to solve the optimization problem is a hybrid system that uses brute force optimization when there are less than ten technologies. Otherwise, it uses a Non-dominated Sorting Genetic Algorithm (i.e., NSGA-II implemented by the library pymoo) with a default termination criteria. This approach is both fast, about 20 seconds each optimization, and light, 200Mb of RAM per optimization process, making the code suitable to a normal personal computer.

3.2. Parameter Transformation

Once a set of Pareto-optimal technology packages has been found for a given year (i.e powertrain, vehicle segment, and cost scenario), some adjustments are made to each point. These transformations are needed for:

  1. Baseline adjustment: accounting for technologies that are already deployed in the reference year,

  2. Scaling for batteries: handling battery cost (or H2 storage cost) savings for xEV,

  3. Scaling for overlapping technologies: avoiding that potentials covered by different technologies are double-counted, and

  4. Re-baseline xEV: setting xEV energy and CO2 savings relative to reference year conventional vehicles.

These steps and their relative algorithms are described in a previous report.

Parameters for each of the post-processing steps should to be provided in the input file tab “reduction_tech_cost_params”.

The cost and reduction parameters of the post-processed technology bundles form the raw Pareto front. They are an output of the model, visible in the output file tab “reduction_tech_cost_pareto”.

3.3. Optimal Pareto Front

The raw Pareto cloud/front found in the previous step indicates the best technology combinations at the vehicle level. However, it is not the optimal solution used at the fleet level.

A reasonable solution to reduce technology cost is a linear combination of two technology packages for cost curves used at the fleet level. Hence, the optimal Pareto front (the optimal packages’ distribution) is defined in a discrete form by a series of segments.

The convex hull algorithm implemented by the scipy library is used for finding the optimal points that define the linear front from the raw Pareto cloud.

In the output file, the solutions forming the optimal Pareto front are marked as “optimal” in the output file tab “reduction_tech_cost_pareto”.

3.4. Fitting Cost Curves

A continuous analytical form of the cost curves can be fitted based on the raw Pareto cloud or the optimal Pareto front. The functional form of the cost curves is the following:

\[y = Ax^2 + Bx + C + \frac{c}{x-x0}\]

where A, B, C, c, and x0 are the unknown parameters to be fitted, and y and x represent the implementation technology cost and the relative energy reduction.

The DIONE model allows fixing some or all unknown parameters and defines cost curves distinguished by their fitting methods. The curves are:

  1. cloud cost curve: fitted using the raw Pareto cloud, and

  2. optimal cost curve: fitted using 1000 equally distributed points of the optimal Pareto front.

Note

The optimal cost curve is more stable because the raw Pareto cloud does not influence it.

Cost curve fitting parameters are the final output of the DIONE cost curve model. They can be found in the tab “reduction_tech_cost_curves” of the output file.

The fitting algorithm used for fitting the parameters of the curves is an adaptive multi-step algorithm that uses:

  1. the Nelder method to solve non-linear least squares problem fixing \(x0=1\) and using all other parameters equal to zero as initial guess,

  2. the Nelder method using the previous solution as initial guess,

  3. the Least-Squares minimization, using Trust Region Reflective method and the previous solution as initial guess.