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A fuel price, fuel requirement and product rate can all be uncertain at once—and they may move together. Monte Carlo analysis carries those joint assumptions through the cost balance, but its output is only as honest as the input model.
Open the chapter contents · Thermodynamics root · Gold Standard of Education · How X Works
Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The teaching chapters remain open; these grouped routes form the index.
Route 1 · Build the uncertainty model
Route 2 · Work the simulation
Route 3 · Stress the dependence
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · BUILD THE UNCERTAINTY MODEL
1. One cost number hides a distribution of possible states
An exergoeconomic cost balance combines physical exergy flows with fuel, capital, operation and maintenance cost rates. A simple one-product balance is c_P = (c_F F + Z)/P, where c_F is fuel unit cost, F is fuel-exergy rate, Z is the non-fuel cost rate and P is product-exergy rate.
Each input can be uncertain. Fuel price may move with demand, equipment condition changes F, output demand changes P, and levelised cost Z depends on financing and operating assumptions. Monte Carlo analysis draws coherent input states, solves the cost model for each state and studies the resulting output distribution.
This guide extends the Thermodynamics root, exergoeconomic sensitivity and shadow costs and thermoeconomic cost formation. Local derivatives ask what happens near one state; Monte Carlo asks what a stated joint uncertainty model implies across many states.
CHAPTER 2 OF 12 · BUILD THE UNCERTAINTY MODEL
2. Build the joint input model before drawing
For every uncertain input, state the quantity, unit, central value, distribution, bounds, evidence and time horizon. Positive prices may suit lognormal or bounded distributions better than an unbounded normal. Efficiencies must remain physically admissible. Shares that sum to one need a joint construction that preserves that identity.
Dependence is part of the model. Fuel price and fuel requirement may rise together during high-load operation, while efficiency and fuel requirement may move oppositely. Sampling each marginal independently can create physically inconsistent combinations and distort output uncertainty.
Separate aleatory variation from epistemic uncertainty when the decision needs that distinction. A frequency distribution of operating states is not the same as uncertainty about a poorly known parameter.
CHAPTER 3 OF 12 · WORK THE SIMULATION
3. Worked audit: five transparent joint draws
Before a large simulation, run a few hand-checkable states through the balance. Set Z = 0.80 and P = 80 in every row while allowing fuel price and fuel requirement to rise together.
| Draw | Fuel unit cost c_F | Fuel rate F | Product unit cost (c_F F + 0.80)/80 |
|---|---|---|---|
| 1 | 0.045 | 96 | 0.0640 |
| 2 | 0.055 | 98 | 0.0774 |
| 3 | 0.060 | 100 | 0.0850 |
| 4 | 0.070 | 104 | 0.1010 |
| 5 | 0.085 | 110 | 0.1269 |
The five outputs average 0.0909. That number is not a Monte Carlo estimate worth reporting; the ledger exists to confirm units, direction and code. A production simulation should use enough draws for the requested mean, quantile or decision probability to stabilize.
A simulation is repeated accounting, not magic
For each draw, sample one valid joint input vector, solve the thermodynamic state if it is uncertain, calculate exergy flows, close component cost balances, then store the decision outputs. Rejecting impossible draws after the fact can bias results; it is better to sample from a distribution that respects constraints.
Record failures rather than silently dropping them. Solver non-convergence may cluster in difficult operating states and therefore contain decision-relevant information. Distinguish numerical failure from a physically infeasible design.
Use independent random streams or controlled seeds when comparing designs so that common random numbers reduce simulation noise in the difference.
CHAPTER 4 OF 12 · WORK THE SIMULATION
4. Correlation changes output uncertainty
At c_F = 0.060, F = 100, P = 80 and Z = 0.80, suppose the standard deviations are 0.010 for c_F, 8 for F and 0.10 for Z. A first-order variance approximation includes a covariance term between c_F and F.
| Assumption | Variance numerator before dividing by P squared | Approximate SD of c_P | Relative to independent case |
|---|---|---|---|
| Independent c_F and F | 1.2404 | 0.01392 | 1.000 |
| Correlation +0.70 | 1.9124 | 0.01729 | 1.242 |
The correlated numerator adds 2 F c_F Cov(c_F,F), with covariance 0.70 × 0.010 × 8 = 0.056. Negative correlation would reduce this local variance. The check is valuable because it can expose a Monte Carlo implementation that ignored dependence.
CHAPTER 5 OF 12 · WORK THE SIMULATION
5. Summaries must match the decision
Report mean and median when skewness matters, intervals or quantiles for spread, and probabilities tied to a real threshold such as P(c_P exceeds the contract ceiling). A 95th percentile is not a worst case and says nothing beyond the input model’s support.
For comparing two designs, simulate their cost difference under shared uncertain states. P(design A costs less than B) can be more useful than two separate overlapping intervals. Preserve other criteria such as exergy efficiency, emissions, reliability and capacity.
Expected cost alone can reward a design with an unacceptable upper tail. Decision robustness asks whether the preferred choice persists across plausible states and alternative defensible input models.
CHAPTER 6 OF 12 · STRESS THE DEPENDENCE
6. Convergence is output-specific
The sample mean often stabilizes faster than a high quantile or a rare exceedance probability. Plot the running estimate and Monte Carlo standard error for each reported metric. Repeat with several seeds or use batch means.
Doubling the number of draws should change conclusions less than the decision margin. If it does not, run longer, improve sampling or report the unresolved simulation uncertainty. Quasi-Monte Carlo or Latin hypercube methods can improve efficiency, but dependence and constraints must still be preserved.
Do not choose the run length solely because a round number looks large.
CHAPTER 7 OF 12 · STRESS THE DEPENDENCE
7. Changed context: price changes the operating state
If higher fuel prices trigger dispatch changes, c_F cannot be varied while F and P remain frozen. Each draw must solve the operational or optimization model under that price. The resulting cost distribution combines market uncertainty with endogenous plant response.
Switching surfaces can make the response discontinuous: one unit turns off, a backup fuel starts or a capacity constraint binds. Local derivatives may fail there, which is exactly where simulation becomes useful.
Check whether optimization noise or solver tolerances are being mistaken for physical uncertainty.
CHAPTER 8 OF 12 · STRESS THE DEPENDENCE
8. Changed context: multiple products share joint cost
In combined heat, power and cooling systems, auxiliary equations allocate shared cost among products. Sampling uncertain inputs while holding one arbitrary allocation rule fixed produces conditional product-cost distributions, not allocation-free truth.
Repeat the analysis across defensible allocation rules or report total system cost separately from allocated product costs. Correlations can also arise because product shares must sum to one. Independent draws of electricity, heat and cooling fractions may violate the identity.
The multi-product exergy allocation guide provides the necessary boundary and auxiliary-equation checks.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Monte Carlo does not create knowledge about unknown distributions. Thousands of precise draws from weak assumptions can produce a narrow-looking but misleading result. Correlation coefficients alone may not capture nonlinear or tail dependence.
Sensitivity ranking with correlated inputs is not the same as ranking independent variables. Shared variation can make attribution non-unique; use methods designed for dependent inputs and state the interpretation.
Simulation uncertainty, parameter uncertainty, model-form uncertainty and future scenario uncertainty are different layers. Keep them labelled rather than merging every unknown into one probability distribution.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Evidence and reproducibility route
Momen and Behbahaninia couple Monte Carlo uncertainty with exergoeconomic evaluation in A New Approach for Exergoeconomics Evaluation by Considering Uncertainty with Monte Carlo Method. The cost-balance foundation is the SPECO method in SPECO: A systematic and general methodology for calculating efficiencies and costs in thermal systems. For dependent-input sensitivity, see Kucherenko, Tarantola and Annoni’s Estimation of global sensitivity indices for models with dependent variables.
A reproducible package contains system boundary, reference environment, process equations, cost balances, auxiliary rules, every marginal distribution and evidence source, dependence model, constraint transformation, sample generator and seed, draw count, solver tolerances, failure log, convergence plots, stored summary code and scenario alternatives.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
How many Monte Carlo draws are enough?
Enough for the specific reported metric to converge within a decision-relevant tolerance. Tail probabilities usually need more than means.
Can I impose correlation with a covariance matrix?
For compatible multivariate distributions, yes. Non-normal marginals, bounds and nonlinear dependence may require copulas, latent variables or direct process-based sampling.
Should impossible draws be deleted?
Prefer a joint generator that cannot create them. If a solver fails, report the rate and location before deciding how to handle it.
Does the widest input range identify the most important variable?
No. Importance depends on both uncertainty and model response, including interactions and dependence.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Final transfer check and connected reading
Before using a Monte Carlo cost result, answer six questions: Are marginal distributions evidence-based? Are dependence and physical constraints preserved? Does every draw close the exergy and cost balances? Are solver failures visible? Have the reported tail metrics converged? Does the preferred decision survive alternative plausible input models and allocation rules?
Continue through the Thermodynamics root, exergoeconomic sensitivity and shadow costs, thermoeconomic cost formation, the Gold Standard of Education and the How X Works hub. The transferable skill is to read a simulated distribution as a transparent consequence of joint assumptions, not as a crystal ball.
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