High-Voltage Direct Current (HVDC) Transmission and Flexible AC Transmission Systems (FACTS)

Unit 8, Lesson 1: Probabilistic adequacy and capacity outage states

Course position: Planning, uncertainty, and integrated system studies. Study time: approximately 90–120 minutes including quantitative checks. Model domain: a synthetic, simplified calculation requiring interpretation. Safety classification: non-operational.

Physical idea and engineering context

Resource adequacy asks whether available capacity meets demand over a sequence of periods with weather and equipment uncertainty. A capacity outage probability table combines individual unit states under explicit availability assumptions. Loss-of-load probability and expected unserved energy are distinct: one records event probability while the other weights the shortage magnitude. Independence assumptions are particularly fragile during common-cause fuel or weather disruptions.

Governing model, boundaries and assumptions

For independent two-unit example, probability both online = p1 p2; expected unserved power = sum_s prob_s max(0, load−available_s)

Worked quantitative example

Two identical 60 MW generators each have independent availability 0.9; load is 80 MW for a study hour. Both are needed, so probability of any shortage is 1−0.9²=0.19. With both online unserved=0; with one online shortage=20 MW (probability 0.18); with neither online shortage=80 MW (probability 0.01). Expected unserved energy over this one hour is 0.18×20+0.01×80=4.4 MWh. The event probability is 19%, not 4.4%.

Interpretation and limiting cases

An availability percentage should refer to the modeled period and failure definition. Multiplying mean annual availability values does not prove independent outage behavior during an extreme event.

What could make this answer wrong?

A deterministic N−1 check and a probabilistic adequacy estimate answer different questions and need not agree on a resource choice.

Applied comparison and design reasoning

Test a common-cause outage model where both units fail together with some additional probability. Explain qualitatively why the independent estimate can understate tail risk.

Guided problems

Problem A — first-principles check. Two independent devices each have availability 0.8. Calculate probability both available and neither available. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain why a correlated weather event breaks the product of marginal availabilities. State which terms in the equation change and which remain fixed. If the problem lacks a unique numerical answer, identify the missing measurements rather than guessing.

Problem C — critical reading. Build the full four-state table and repeat the expected unserved energy for 50 MW load. Which states now contribute shortage? Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 8, Lesson 2: Security-constrained dispatch and contingency screening

Course position: Planning, uncertainty, and integrated system studies. Study time: approximately 90–120 minutes including quantitative checks. Model domain: a synthetic, simplified calculation requiring interpretation. Safety classification: non-operational.

Physical idea and engineering context

An economic dispatch minimizes cost subject to generator limits, demand balance and usually network constraints. A security-constrained formulation also imposes limits under selected credible contingencies, so a cheaper base-case dispatch can be rejected as insecure. Security margins reflect the studied contingency set and operating assumptions; a small study cannot certify all possible operating events.

Governing model, boundaries and assumptions

Minimize sum_i C_i(P_i) subject to sum_i P_i=P_demand+losses and constraints |F_line,k|≤F_max,k for each modeled state k

Worked quantitative example

A 100 MW load has two generators: G1 costs $20/MWh and G2 $40/MWh, both rated 100 MW. Ignoring losses and lines, cheapest dispatch is G1=100 MW. Suppose a transmission corridor permits at most 60 MW delivery from G1 in the base case and only 40 MW after a studied line outage. A security-constrained schedule limited by that outage may use G1=40, G2=60 MW, cost $3200/h versus $2000/h unconstrained. These are fabricated study numbers, not market prices.

Interpretation and limiting cases

The incremental cost of robustness is the difference between feasible optimization problems, not automatically “wasted” spending. A contingency set defines what risks are assessed.

What could make this answer wrong?

Assuming the same linear flow sensitivities for every topology can produce incorrect post-contingency power transfers. Full re-analysis may be necessary.

Applied comparison and design reasoning

If a lower-cost generator is behind a congested line, investigate whether storage on the load side, demand flexibility or transmission reinforcement changes the least-cost secure dispatch.

Guided problems

Problem A — first-principles check. Compare the dispatch cost of 50 MW at $20/MWh and 50 MW at $40/MWh against 80/20 MW dispatch. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain what makes a dispatch secure rather than just cost-minimizing. State which terms in the equation change and which remain fixed. If the problem lacks a unique numerical answer, identify the missing measurements rather than guessing.

Problem C — critical reading. What data would be required to decide whether planning for a second line outage is a reasonable criterion rather than unbounded overbuilding? Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 8, Lesson 3: Lifecycle cost and cost-of-service comparisons

Course position: Planning, uncertainty, and integrated system studies. Study time: approximately 90–120 minutes including quantitative checks. Model domain: a synthetic, simplified calculation requiring interpretation. Safety classification: non-operational.

Physical idea and engineering context

Capital cost, fixed operations, variable operating expense, losses, expected replacements, outage impact and discount rates contribute differently to decisions. Levelized cost is useful for comparing similar service profiles but can be misleading when a dispatchable firm asset and a weather-dependent generator provide different time-valued outputs. A planning comparison should specify study horizon and the service being purchased.

Governing model, boundaries and assumptions

Present value PV = sum_t cashflow_t/(1+r)^t; annualized equivalent = PV × [r(1+r)^n/((1+r)^n−1)] when r>0

Worked quantitative example

A toy project spends $1,000 at t=0 and saves $300 at each year-end for four years. At r=10%, present value of savings is 300×(1−1/1.1^4)/0.10≈$950.96; net present value is about −$49.04. At a lower discount rate the result may change sign. All values are made-up dollars for an arithmetic exercise.

Interpretation and limiting cases

Real and nominal discount rates must match real and nominal cashflows. A project’s interest rate is not automatically the social or utility planning discount rate.

What could make this answer wrong?

A storage option with low upfront cost can be more expensive if frequent replacements are omitted. A line with high capex can reduce losses and outage exposure for decades.

Applied comparison and design reasoning

Compare reinforcing an overloaded transformer with contracted local demand response. List what duration, firm availability and customer participation evidence must enter the comparison.

Guided problems

Problem A — first-principles check. Discount $110 received one year later at a 10% rate and compare it with $100 paid today. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. State why real and nominal cashflows cannot be mixed with mismatched discount rates. State which terms in the equation change and which remain fixed. If the problem lacks a unique numerical answer, identify the missing measurements rather than guessing.

Problem C — critical reading. Calculate the undiscounted total savings in the toy project and explain why that result differs from the present-value result. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 8, Lesson 4: Model validation, provenance and decision communication

Course position: Planning, uncertainty, and integrated system studies. Study time: approximately 90–120 minutes including quantitative checks. Model domain: a synthetic, simplified calculation requiring interpretation. Safety classification: non-operational.

Physical idea and engineering context

A model is validated for a specific question, operating regime and input dataset. Reproducibility requires the software version, equipment data provenance, assumptions, scenario definitions, reference directions and checks against independent computations. Good uncertainty communication distinguishes scenario variation from probabilistic confidence and avoids claiming “verified accuracy” from a single converged solver run. Evidence should determine where greater model fidelity matters most.

Governing model, boundaries and assumptions

A validation residual can be defined e_k = y_measured,k − y_model,k; diagnostics examine bias, dispersion and dependence on operating condition

Worked quantitative example

Suppose a feeder model predicts 0.980, 0.970 and 0.960 pu while measurements are 0.977, 0.971 and 0.951 pu. Residuals are −0.003, +0.001 and −0.009 pu. Average residual is −0.00367 pu, but three observations cannot establish general accuracy. The negative error at the heaviest load hints that the model may understate voltage drop, yet measurement location and instrument calibration require checking.

Interpretation and limiting cases

A low overall mean error can hide a systematic failure in the most critical operating regime. Report errors by load level, topology and weather where relevant.

What could make this answer wrong?

Tuning a model on the same dataset used for its validation can overstate predictive performance. Independent operating periods are preferable when available.

Applied comparison and design reasoning

An engineer proposes an expensive capacitor bank from one power-flow snapshot. Construct the minimum validation and sensitivity evidence needed before choosing a procurement option.

Guided problems

Problem A — first-principles check. For measured/predicted voltage pairs (0.99,0.992) and (0.96,0.970), calculate signed residuals. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Describe the extra measurements needed before claiming the model is reliable at stressed voltage. State which terms in the equation change and which remain fixed. If the problem lacks a unique numerical answer, identify the missing measurements rather than guessing.

Problem C — critical reading. Produce a one-page decision memo distinguishing observed facts, model results, untested assumptions and recommendations that require further measurements. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 8 project: Probabilistic adequacy and secure planning case

Question to answer

How much of an engineering conclusion is justified by the following small study, and what further data would be required to make a real operational or investment decision? Explain the difference between a physically interpretable calculation and a certified design or utility-approved action. All data below are synthetic, safe to explore in a spreadsheet or local script.

Case and fixed assumptions

Use two hypothetical units of 60 MW each with independent availability 90% and constant study load of 80 MW for four representative hours. Enumerate the four outage states and calculate loss-of-load probability and expected unserved energy at 80 MW. Recalculate when load falls to 50 MW. Then compare a $15M reinforcement with a $5M flexible-load program using a clearly declared 5-year study horizon and 8% discount rate; do not invent annual outage costs without labeling scenario assumptions.

Deliverables and acceptance checks

Deliver the complete state table with probabilities summing to one, one sensitivity for correlated failures, NPV formulae for options, and a decision summary with measured facts distinct from hypothetical cost inputs.

Suggested grading rubric

• 20%: physical model boundary and unit/sign correctness.
• 25%: correct quantitative computation and explicit verification.
• 20%: feasible state/limit handling over the full study interval.
• 20%: meaningful sensitivity, comparison and uncertainty.
• 15%: clear evidence-based conclusion and ethical/safety limitations.

Unit 8 assessment: Planning, uncertainty, and integrated system studies

Submission rule: Show dimensional steps, declare all bases and state which missing data prevent an operational conclusion. Problems use artificial scenarios.

Question group 1: Probabilistic adequacy and capacity outage states

Q1.1 — calculation. Two independent devices each have availability 0.8. Calculate probability both available and neither available. Present each quantity with units and report whether the answer is steady-state, transient, probability, energy or power, as appropriate.

Q1.2 — assumption stress test. Explain why a correlated weather event breaks the product of marginal availabilities. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q1.3 — applied critique. Build the full four-state table and repeat the expected unserved energy for 50 MW load. Which states now contribute shortage?

Question group 2: Security-constrained dispatch and contingency screening

Q2.1 — calculation. Compare the dispatch cost of 50 MW at $20/MWh and 50 MW at $40/MWh against 80/20 MW dispatch. Present each quantity with units and report whether the answer is steady-state, transient, probability, energy or power, as appropriate.

Q2.2 — assumption stress test. Explain what makes a dispatch secure rather than just cost-minimizing. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q2.3 — applied critique. What data would be required to decide whether planning for a second line outage is a reasonable criterion rather than unbounded overbuilding?

Question group 3: Lifecycle cost and cost-of-service comparisons

Q3.1 — calculation. Discount $110 received one year later at a 10% rate and compare it with $100 paid today. Present each quantity with units and report whether the answer is steady-state, transient, probability, energy or power, as appropriate.

Q3.2 — assumption stress test. State why real and nominal cashflows cannot be mixed with mismatched discount rates. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q3.3 — applied critique. Calculate the undiscounted total savings in the toy project and explain why that result differs from the present-value result.

Question group 4: Model validation, provenance and decision communication

Q4.1 — calculation. For measured/predicted voltage pairs (0.99,0.992) and (0.96,0.970), calculate signed residuals. Present each quantity with units and report whether the answer is steady-state, transient, probability, energy or power, as appropriate.

Q4.2 — assumption stress test. Describe the extra measurements needed before claiming the model is reliable at stressed voltage. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q4.3 — applied critique. Produce a one-page decision memo distinguishing observed facts, model results, untested assumptions and recommendations that require further measurements.

Final integrative question

Imagine the physical asset is operated under a new but plausible loading, controller or contingency condition. Identify two competing objectives, formulate a consistent indicator for each, and explain a trade-off. Provide a reproducible table with at least four rows and one sensitivity. Explicitly separate observed data from assumption-driven results.

Solution framework: Select quantities that share the same physical boundary and time reference, prove unit consistency, locate binding constraints and state the domain of validity. A single “correct” design does not follow without a stated objective and validated site data.

Unit 8: Planning, uncertainty, and integrated system studies

System investment decisions require a chain from defensible assumptions to options, constraints, sensitivities and credible decision criteria. An apparently precise single-number simulation can be worse than an honest range if it conceals uncertainty in load growth, fuel, weather, equipment failure or construction. Planning includes reliability and resilience, but these are not synonyms: reliable average performance does not establish resilience during exceptional disruptive events. This unit closes the sequence with probabilistic adequacy, scenario engineering, lifecycle cost and model governance.

Deep reading — interacting limitations

Planning quality is limited by data provenance and scenario design, not merely solver sophistication. A capacity-outage probability calculation becomes optimistic if failures are correlated by weather, maintenance, fuel constraints or common substations. An N−1 screen and a probabilistic resource-adequacy estimate are complementary, not equivalent. Economic comparisons must put cashflows on a consistent time and price basis and include the reliability service provided. A model is a decision aid with a specified region of validity. If it reproduces three voltage points, that does not mean it accurately predicts a severe fault or decade-long peak load growth. Report what was independently checked, what was assumed, and what decision would change under a plausible alternative.

Case synthesis: decision thresholds are conditional on what fails together

A toy adequacy model contains two 60 MW generators, each available with probability 0.9 in the studied hour, serving a steady 80 MW demand. Under an explicit independence assumption, the probability both are available is 0.81. A shortage occurs whenever at least one is unavailable, so loss-of-load probability is 0.19. A single outage leaves 60 MW capacity and a 20 MW shortage; its probability is 0.18. A double outage leaves 80 MW unmet with probability 0.01. Expected unserved energy in that hour equals 0.18×20+0.01×80=4.4 MWh. The expected value is not a prediction that exactly 4.4 MWh will be interrupted in any observed hour: individual scenarios have 0, 20 or 80 MWh of shortage.

This small model can fail in several ways. An extreme weather event may disable both units simultaneously. A shared fuel supplier or substation can create common-cause failures. A solar generator’s availability can correlate with demand and with other solar plants; an annual average output fraction does not establish capacity during a peak-hour event. A fixed demand assumption ignores peak-load uncertainty. The resulting uncertainty is structural, not just rounding error. A useful sensitivity scenario changes independence, common-cause probability or demand conditions explicitly; it does not assign a fabricated confidence interval.

Planning also includes network security. A least-cost dispatch that uses a cheap remote generator may violate a line’s post-contingency rating even when total generation exceeds total demand. Security-constrained scheduling or network reinforcement can require a more expensive base-case choice to preserve service during a credible outage. What counts as a credible contingency depends on planning rules and engineering evidence. Probabilistic adequacy and N−1 security address different dimensions and neither replaces the other.

Finally, convert technical benefits to a consistent economic frame. A discounted cashflow comparison should name price year, inflation basis, discount rate, horizon, service equivalence, end-of-life treatment and scenario uncertainty. If annual outage cost is unknown, write it as an unknown variable and calculate the break-even value that would justify the investment. This is more useful than guessing a market-wide benefit. The final recommendation should distinguish demonstrated model constraints, estimated operating risks, stakeholder requirements and unresolved facts that could change the preferred option.

Unit 8, Lesson 1: Probabilistic adequacy and capacity outage states

Course position: Planning, uncertainty, and integrated system studies. Study time: approximately 90–120 minutes including quantitative checks. Model domain: a synthetic, simplified calculation requiring interpretation. Safety classification: non-operational.

Physical idea and engineering context

Resource adequacy asks whether available capacity meets demand over a sequence of periods with weather and equipment uncertainty. A capacity outage probability table combines individual unit states under explicit availability assumptions. Loss-of-load probability and expected unserved energy are distinct: one records event probability while the other weights the shortage magnitude. Independence assumptions are particularly fragile during common-cause fuel or weather disruptions.

Governing model, boundaries and assumptions

For independent two-unit example, probability both online = p1 p2; expected unserved power = sum_s prob_s max(0, load−available_s)

Worked quantitative example

Two identical 60 MW generators each have independent availability 0.9; load is 80 MW for a study hour. Both are needed, so probability of any shortage is 1−0.9²=0.19. With both online unserved=0; with one online shortage=20 MW (probability 0.18); with neither online shortage=80 MW (probability 0.01). Expected unserved energy over this one hour is 0.18×20+0.01×80=4.4 MWh. The event probability is 19%, not 4.4%.

Interpretation and limiting cases

An availability percentage should refer to the modeled period and failure definition. Multiplying mean annual availability values does not prove independent outage behavior during an extreme event.

What could make this answer wrong?

A deterministic N−1 check and a probabilistic adequacy estimate answer different questions and need not agree on a resource choice.

Applied comparison and design reasoning

Test a common-cause outage model where both units fail together with some additional probability. Explain qualitatively why the independent estimate can understate tail risk.

Guided problems

Problem A — first-principles check. Two independent devices each have availability 0.8. Calculate probability both available and neither available. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain why a correlated weather event breaks the product of marginal availabilities. State which terms in the equation change and which remain fixed. If the problem lacks a unique numerical answer, identify the missing measurements rather than guessing.

Problem C — critical reading. Build the full four-state table and repeat the expected unserved energy for 50 MW load. Which states now contribute shortage? Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 8, Lesson 2: Security-constrained dispatch and contingency screening

Course position: Planning, uncertainty, and integrated system studies. Study time: approximately 90–120 minutes including quantitative checks. Model domain: a synthetic, simplified calculation requiring interpretation. Safety classification: non-operational.

Physical idea and engineering context

An economic dispatch minimizes cost subject to generator limits, demand balance and usually network constraints. A security-constrained formulation also imposes limits under selected credible contingencies, so a cheaper base-case dispatch can be rejected as insecure. Security margins reflect the studied contingency set and operating assumptions; a small study cannot certify all possible operating events.

Governing model, boundaries and assumptions

Minimize sum_i C_i(P_i) subject to sum_i P_i=P_demand+losses and constraints |F_line,k|≤F_max,k for each modeled state k

Worked quantitative example

A 100 MW load has two generators: G1 costs $20/MWh and G2 $40/MWh, both rated 100 MW. Ignoring losses and lines, cheapest dispatch is G1=100 MW. Suppose a transmission corridor permits at most 60 MW delivery from G1 in the base case and only 40 MW after a studied line outage. A security-constrained schedule limited by that outage may use G1=40, G2=60 MW, cost $3200/h versus $2000/h unconstrained. These are fabricated study numbers, not market prices.

Interpretation and limiting cases

The incremental cost of robustness is the difference between feasible optimization problems, not automatically “wasted” spending. A contingency set defines what risks are assessed.

What could make this answer wrong?

Assuming the same linear flow sensitivities for every topology can produce incorrect post-contingency power transfers. Full re-analysis may be necessary.

Applied comparison and design reasoning

If a lower-cost generator is behind a congested line, investigate whether storage on the load side, demand flexibility or transmission reinforcement changes the least-cost secure dispatch.

Guided problems

Problem A — first-principles check. Compare the dispatch cost of 50 MW at $20/MWh and 50 MW at $40/MWh against 80/20 MW dispatch. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain what makes a dispatch secure rather than just cost-minimizing. State which terms in the equation change and which remain fixed. If the problem lacks a unique numerical answer, identify the missing measurements rather than guessing.

Problem C — critical reading. What data would be required to decide whether planning for a second line outage is a reasonable criterion rather than unbounded overbuilding? Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 8, Lesson 3: Lifecycle cost and cost-of-service comparisons

Course position: Planning, uncertainty, and integrated system studies. Study time: approximately 90–120 minutes including quantitative checks. Model domain: a synthetic, simplified calculation requiring interpretation. Safety classification: non-operational.

Physical idea and engineering context

Capital cost, fixed operations, variable operating expense, losses, expected replacements, outage impact and discount rates contribute differently to decisions. Levelized cost is useful for comparing similar service profiles but can be misleading when a dispatchable firm asset and a weather-dependent generator provide different time-valued outputs. A planning comparison should specify study horizon and the service being purchased.

Governing model, boundaries and assumptions

Present value PV = sum_t cashflow_t/(1+r)^t; annualized equivalent = PV × [r(1+r)^n/((1+r)^n−1)] when r>0

Worked quantitative example

A toy project spends $1,000 at t=0 and saves $300 at each year-end for four years. At r=10%, present value of savings is 300×(1−1/1.1^4)/0.10≈$950.96; net present value is about −$49.04. At a lower discount rate the result may change sign. All values are made-up dollars for an arithmetic exercise.

Interpretation and limiting cases

Real and nominal discount rates must match real and nominal cashflows. A project’s interest rate is not automatically the social or utility planning discount rate.

What could make this answer wrong?

A storage option with low upfront cost can be more expensive if frequent replacements are omitted. A line with high capex can reduce losses and outage exposure for decades.

Applied comparison and design reasoning

Compare reinforcing an overloaded transformer with contracted local demand response. List what duration, firm availability and customer participation evidence must enter the comparison.

Guided problems

Problem A — first-principles check. Discount $110 received one year later at a 10% rate and compare it with $100 paid today. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. State why real and nominal cashflows cannot be mixed with mismatched discount rates. State which terms in the equation change and which remain fixed. If the problem lacks a unique numerical answer, identify the missing measurements rather than guessing.

Problem C — critical reading. Calculate the undiscounted total savings in the toy project and explain why that result differs from the present-value result. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 8, Lesson 4: Model validation, provenance and decision communication

Course position: Planning, uncertainty, and integrated system studies. Study time: approximately 90–120 minutes including quantitative checks. Model domain: a synthetic, simplified calculation requiring interpretation. Safety classification: non-operational.

Physical idea and engineering context

A model is validated for a specific question, operating regime and input dataset. Reproducibility requires the software version, equipment data provenance, assumptions, scenario definitions, reference directions and checks against independent computations. Good uncertainty communication distinguishes scenario variation from probabilistic confidence and avoids claiming “verified accuracy” from a single converged solver run. Evidence should determine where greater model fidelity matters most.

Governing model, boundaries and assumptions

A validation residual can be defined e_k = y_measured,k − y_model,k; diagnostics examine bias, dispersion and dependence on operating condition

Worked quantitative example

Suppose a feeder model predicts 0.980, 0.970 and 0.960 pu while measurements are 0.977, 0.971 and 0.951 pu. Residuals are −0.003, +0.001 and −0.009 pu. Average residual is −0.00367 pu, but three observations cannot establish general accuracy. The negative error at the heaviest load hints that the model may understate voltage drop, yet measurement location and instrument calibration require checking.

Interpretation and limiting cases

A low overall mean error can hide a systematic failure in the most critical operating regime. Report errors by load level, topology and weather where relevant.

What could make this answer wrong?

Tuning a model on the same dataset used for its validation can overstate predictive performance. Independent operating periods are preferable when available.

Applied comparison and design reasoning

An engineer proposes an expensive capacitor bank from one power-flow snapshot. Construct the minimum validation and sensitivity evidence needed before choosing a procurement option.

Guided problems

Problem A — first-principles check. For measured/predicted voltage pairs (0.99,0.992) and (0.96,0.970), calculate signed residuals. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Describe the extra measurements needed before claiming the model is reliable at stressed voltage. State which terms in the equation change and which remain fixed. If the problem lacks a unique numerical answer, identify the missing measurements rather than guessing.

Problem C — critical reading. Produce a one-page decision memo distinguishing observed facts, model results, untested assumptions and recommendations that require further measurements. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 8 project: Probabilistic adequacy and secure planning case

Question to answer

How much of an engineering conclusion is justified by the following small study, and what further data would be required to make a real operational or investment decision? Explain the difference between a physically interpretable calculation and a certified design or utility-approved action. All data below are synthetic, safe to explore in a spreadsheet or local script.

Case and fixed assumptions

Use two hypothetical units of 60 MW each with independent availability 90% and constant study load of 80 MW for four representative hours. Enumerate the four outage states and calculate loss-of-load probability and expected unserved energy at 80 MW. Recalculate when load falls to 50 MW. Then compare a $15M reinforcement with a $5M flexible-load program using a clearly declared 5-year study horizon and 8% discount rate; do not invent annual outage costs without labeling scenario assumptions.

Deliverables and acceptance checks

Deliver the complete state table with probabilities summing to one, one sensitivity for correlated failures, NPV formulae for options, and a decision summary with measured facts distinct from hypothetical cost inputs.

Suggested grading rubric

• 20%: physical model boundary and unit/sign correctness.
• 25%: correct quantitative computation and explicit verification.
• 20%: feasible state/limit handling over the full study interval.
• 20%: meaningful sensitivity, comparison and uncertainty.
• 15%: clear evidence-based conclusion and ethical/safety limitations.

Unit 8 assessment: Planning, uncertainty, and integrated system studies

Submission rule: Show dimensional steps, declare all bases and state which missing data prevent an operational conclusion. Problems use artificial scenarios.

Question group 1: Probabilistic adequacy and capacity outage states

Q1.1 — calculation. Two independent devices each have availability 0.8. Calculate probability both available and neither available. Present each quantity with units and report whether the answer is steady-state, transient, probability, energy or power, as appropriate.

Q1.2 — assumption stress test. Explain why a correlated weather event breaks the product of marginal availabilities. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q1.3 — applied critique. Build the full four-state table and repeat the expected unserved energy for 50 MW load. Which states now contribute shortage?

Question group 2: Security-constrained dispatch and contingency screening

Q2.1 — calculation. Compare the dispatch cost of 50 MW at $20/MWh and 50 MW at $40/MWh against 80/20 MW dispatch. Present each quantity with units and report whether the answer is steady-state, transient, probability, energy or power, as appropriate.

Q2.2 — assumption stress test. Explain what makes a dispatch secure rather than just cost-minimizing. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q2.3 — applied critique. What data would be required to decide whether planning for a second line outage is a reasonable criterion rather than unbounded overbuilding?

Question group 3: Lifecycle cost and cost-of-service comparisons

Q3.1 — calculation. Discount $110 received one year later at a 10% rate and compare it with $100 paid today. Present each quantity with units and report whether the answer is steady-state, transient, probability, energy or power, as appropriate.

Q3.2 — assumption stress test. State why real and nominal cashflows cannot be mixed with mismatched discount rates. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q3.3 — applied critique. Calculate the undiscounted total savings in the toy project and explain why that result differs from the present-value result.

Question group 4: Model validation, provenance and decision communication

Q4.1 — calculation. For measured/predicted voltage pairs (0.99,0.992) and (0.96,0.970), calculate signed residuals. Present each quantity with units and report whether the answer is steady-state, transient, probability, energy or power, as appropriate.

Q4.2 — assumption stress test. Describe the extra measurements needed before claiming the model is reliable at stressed voltage. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q4.3 — applied critique. Produce a one-page decision memo distinguishing observed facts, model results, untested assumptions and recommendations that require further measurements.

Final integrative question

Imagine the physical asset is operated under a new but plausible loading, controller or contingency condition. Identify two competing objectives, formulate a consistent indicator for each, and explain a trade-off. Provide a reproducible table with at least four rows and one sensitivity. Explicitly separate observed data from assumption-driven results.

Solution framework: Select quantities that share the same physical boundary and time reference, prove unit consistency, locate binding constraints and state the domain of validity. A single “correct” design does not follow without a stated objective and validated site data.