Mathematical Formulation¶
We separate the financial and conversion-efficiency aspects of a production process, which are generic across all technologies, from the physical and technical aspects, which are necessarily specific to the particular process. The motivation for this is that the financial and waste computations can be done uniformly for any technology (even for disparate ones such as PV cells and biofuels) and that different experts may be required to assess the cost, waste, and techno-physical aspects of technological progress. Table 13 defines the indices that are used for the variables that are defined in Table 14.
Set |
Description |
Examples |
|---|---|---|
\(c \in \mathcal{C}\) |
capital |
equipment |
\(f \in \mathcal{F}\) |
fixed cost |
rent, insurance |
\(i \in \mathcal{I}\) |
input |
feedstock, labor |
\(o \in \mathcal{O}\) |
output |
product, co-product, waste |
\(m \in \mathcal{M}\) |
metric |
cost, jobs, carbon footprint, efficiency, lifetime |
\(p \in \mathcal{P}\) |
technical parameter |
temperature, pressure |
\(\nu \in N\) |
technology type |
electrolysis, PV cell |
\(\theta \in \Theta\) |
design |
the result of a particular investment |
\(\chi \in X\) |
investment category |
investment alternatives |
\(\phi \in \Phi_\chi\) |
investment |
a particular investment |
\(\omega \in \Omega\) |
portfolio |
a basket of investments |
Variable |
Type |
Description |
Units |
|---|---|---|---|
\(K\) |
calculated |
unit cost |
USD/unit |
\(C_c\) |
function |
capital cost |
USD |
\(\tau_c\) |
cost |
lifetime of capital |
year |
\(S\) |
cost |
scale of operation |
unit/year |
\(F_f\) |
function |
fixed cost |
USD/year |
\(I_i\) |
input |
input quantity |
input/unit |
\(I^*_i\) |
calculated |
ideal input quantity |
input/unit |
\(\eta_i\) |
waste |
input efficiency |
input/input |
\(p_i\) |
cost |
input price |
USD/input |
\(O_o\) |
calculated |
output quantity |
output/unit |
\(O^*_o\) |
calculated |
ideal output quantity |
output/unit |
\(\eta^\prime_o\) |
waste |
output efficiency |
output/output |
\(p^\prime_o\) |
cost |
output price (+/-) |
USD/output |
\(\mu_m\) |
calculated |
metric |
metric/unit |
\(P_o\) |
function |
production function |
output/unit |
\(M_m\) |
function |
metric function |
metric/unit |
\(\alpha_p\) |
parameter |
technical parameter |
(mixed) |
\(\xi_\theta\) |
variable |
design inputs |
(mixed) |
\(\zeta_\theta\) |
variable |
design outputs |
(mixed) |
\(\psi\) |
function |
design evaluation |
(mixed) |
\(\sigma_\phi\) |
function |
design probability |
1 |
\(q_\phi\) |
variable |
investment cost |
USD |
\(\mathbf{\zeta}_\phi\) |
random variable |
investment outcome |
(mixed) |
\(\mathbf{Z}(\omega)\) |
random variable |
portfolio outcome |
(mixed) |
\(Q(\omega)\) |
calculated |
portfolio cost |
USD |
\(Q^\mathrm{min}\) |
parameter |
minimum portfolio cost |
USD |
\(Q^\mathrm{max}\) |
parameter |
maximum portfolio cost |
USD |
\(q^\mathrm{min}_\phi\) |
parameter |
minimum category cost |
USD |
\(q^\mathrm{max}_\phi\) |
parameter |
maximum category cost |
USD |
\(Z^\mathrm{min}\) |
parameter |
minimum output/metric |
(mixed) |
\(Z^\mathrm{max}\) |
parameter |
maximum output/metric |
(mixed) |
\(\mathbb{F}\), \(\mathbb{G}\) |
operator |
evaluate probabilities |
(mixed) |
Cost¶
The cost characterizations (capital and fixed costs) are represented as functions of the scale of operations and of the technical parameters in the design:
Capital cost: \(C_c(S, \alpha_p)\).
Fixed cost: \(F_f(S, \alpha_p)\).
The per-unit cost is computed using a simple levelization formula:
\(K = \left( \sum_c C_c / \tau_c + \sum_f F_f \right) / S + \sum_i p_i \cdot I_i - \sum_o p^\prime_o \cdot O_o\)
Waste¶
The waste relative to the idealized production process is captured by the \(\eta\) parameters. Expert elicitation might estimate how the \(\eta\)s would change in response to R&D investment.
Waste of input: \(I^*_i = \eta_i I_i\).
Waste of output: \(O_o = \eta^\prime_o O^*_o\).
Production¶
The production function idealizes production by ignoring waste, but accounting for physical and technical processes (e.g., stoichiometry). This requires a technical model or a tabulation/fit of the results of technical modeling.
\(O^*_o = P_o(S, C_c, \tau_c, F_f, I^*_i, \alpha_p)\)
Metrics¶
Metrics such as efficiency, lifetime, or carbon footprint are also compute based on the physical and technical characteristics of the process. This requires a technical model or a tabulation/fit of the results of technical modeling. We use the convention that higher values are worse and lower values are better.
\(\mu_m = M_m(S, C_c, \tau_c, F_f, I_i, I^*_i, O^*_o, O_o, K, \alpha_p)\)
Designs¶
A design represents a state of affairs for a technology \(\nu\). If we denote the design as \(\theta\), we have the tuple of input variables
\(\xi_\theta = \left(S, C_c, \tau_c, F_f, I_i, \eta_i, \eta^\prime_o, \alpha_p, p_i, p^\prime_o\middle) \right|_\theta\)
and the tuple of output variables
\(\zeta_\theta = \left(K, I^*_i, O^*_o, O_o, \mu_m\middle) \right|_\theta\)
and their relationship
\(\zeta_\theta = \psi_\nu\left(\xi_\theta\middle) \right|_{\nu = \nu(\theta)}\)
given the tuple of functions
\(\psi_\nu = \left(P_o, M_m\middle) \right|_\nu\)
for the technology.
Investments¶
An investment \(\phi\) assigns a probability distribution to designs:
\(\sigma_\phi(\theta) = P\left(\theta \middle| \phi\right)\).
such that
\(\int d\theta \sigma_\phi(\theta) = 1\) or \(\sum_\theta \sigma_\phi(\theta) = 1\),
depending upon whether one is performing the computations discretely or continuously. Expectations and other measures on probability distributions can be computed from the \(\sigma_\phi(\theta)\). We treat the outcome \(\mathbf{\zeta}_\phi\) as a random variable for the outcomes \(\zeta_\theta\) according to the distribution \(\sigma_\phi(\theta)\).
Because investment options may be mutually exclusive, as is the case for investing in the same R&D at different funding levels, we say \(\Phi_\chi\) is the set of mutually exclusive investments (i.e., only one can occur simultaneously) in investment category \(\chi\): investments in different categories \(\chi\) can be combined arbitrarily, but just one investment from each \(\Phi_\chi\) may be chosen.
Thus the universe of all portfolios is \(\Omega = \prod_\chi \Phi_\chi\), so a particular portfolio \(\omega \in \Omega\) has components \(\phi = \omega_\chi \in \Phi_\chi\). The overall outcome of a portfolio is a random variable:
\(\mathbf{Z}(\omega) = \sum_\chi \mathbf{\zeta}_\phi \mid_{\phi = \omega_\chi}\)
The cost of an investment in one of the constituents \(\phi\) is \(q_\phi\), so the cost of a porfolio is:
\(Q(\omega) = \sum_\chi q_\phi \mid_{\phi = \omega_\chi}\)
Decision problem¶
The multi-objective decision problem is
\(\min_{\omega \in \Omega} \ \mathbb{F} \ \mathbf{Z}(\omega)\)
such that
\(Q^\mathrm{min} \leq Q(\omega) \leq Q^\mathrm{max}\) ,
\(q^\mathrm{min}_\phi \leq q_{\phi=\omega_\chi} \leq q^\mathrm{max}_\phi\) ,
\(Z^\mathrm{min} \leq \mathbb{G} \ \mathbf{Z}(\omega) \leq Z^\mathrm{max}\) ,
where \(\mathbb{F}\) and \(\mathbb{G}\) are the expectation operator \(\mathbb{E}\), the value-at-risk, or another operator on probability spaces. Recall that \(\mathbf{Z}\) is a vector with components for cost \(K\) and each metric \(\mu_m\), so this is a multi-objective problem.
The two-stage decision problem is a special case of the general problem outlined here: Each design \(\theta\) can be considers as a composite of one or more stages.
Experts¶
Each expert elicitation takes the form of an assessment of the probability and range (e.g., 10th to 90th percentile) of change in the cost or waste parameters or the production or metric functions. In essence, the expert elicitation defines \(\sigma_\phi(\theta)\) for each potential design \(\theta\) resulting from each investment \(\phi\).