Automatic Generation Control (AGC), Frequency Regulation, and Area Control Error

Unit 6, Lesson 1: State of charge and energy accounting

Course position: Energy storage, flexibility and demand response. 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

State of charge describes stored usable energy relative to a declared capacity and allowable window. Separate charging and discharging efficiencies matter because one megawatt-hour consumed while charging does not become one megawatt-hour delivered later. Power and energy limits apply simultaneously. A physically credible schedule should update energy sequentially, not treat each interval as independent.

Governing model, boundaries and assumptions

E_(t+1) = E_t + eta_charge P_charge Delta t − P_discharge Delta t/eta_discharge, using nonnegative charge/discharge powers

Worked quantitative example

A 10 MWh usable-capacity model starts at 6 MWh. Charge at 2 MW for 1 hour with eta_c=0.9, adding 1.8 MWh. Then discharge at 3 MW for 1 hour with eta_d=0.9, withdrawing 3.333 MWh from storage, leaving about 4.467 MWh. These assumptions exclude auxiliary loads and capacity degradation.

Interpretation and limiting cases

Do not confuse an efficiency value with a round-trip efficiency when each half-cycle has its own factor. At eta_c=eta_d=0.9 the ideal round-trip product is 0.81.

What could make this answer wrong?

A model that allows charging and discharging simultaneously without constraints can exploit numerical loops and report fictitious profits.

Applied comparison and design reasoning

Compare a high-power short-duration storage application with an overnight energy-shift application. Which one is limited by MW and which by MWh for a given duty cycle?

Guided problems

Problem A — first-principles check. Starting at 4 MWh, charge 2 MW for 2 h at ηc=0.85 and then discharge 1 MW for 1 h at ηd=0.9. Find final energy. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain why the round-trip efficiency is the product of one-way efficiencies in this definition. 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. With minimum allowed energy 2 MWh, determine the maximum deliverable MWh from a current 5 MWh state if eta_d=0.9 and disregard power constraints. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 6, Lesson 2: Peak shaving and arbitrage dispatch

Course position: Energy storage, flexibility and demand response. 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

Peak shaving reduces net demand during high load; arbitrage shifts grid purchases from high-price to lower-price intervals. They are related but not identical objectives: a demand tariff depends on maximum power, while energy arbitrage depends on price-weighted throughput. Calendar aging, cycle aging, network constraints and battery reserve commitments can turn an apparently profitable energy spread into a negative economic result.

Governing model, boundaries and assumptions

Two-interval gross spread: revenue = E_delivered price_sell − E_charged price_buy; E_delivered = E_charged eta_round_trip

Worked quantitative example

A battery buys 5 MWh at $40/MWh. With round-trip efficiency 0.80 it can sell 4 MWh later at $90/MWh, gross energy spread =4×90−5×40=$160. If estimated incremental degradation and operating costs are $45 for this cycle, the illustrative contribution becomes $115 before fixed costs and market charges.

Interpretation and limiting cases

A 50-dollar difference between sell and buy prices is not automatically a 50-dollar profit per delivered MWh: losses must be purchased and paid for.

What could make this answer wrong?

Using perfect future prices to backtest a battery dispatch gives an optimistic upper bound. A real controller requires forecasts and must handle price revisions.

Applied comparison and design reasoning

Compare a 1 MW/2 MWh and a 2 MW/2 MWh device under a 30-minute sharp demand peak. Which can provide higher power, and what energy remains for later use?

Guided problems

Problem A — first-principles check. If 4 MWh is purchased at $45/MWh and 3.2 MWh sold, compute the break-even sale price before fees. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Describe two reasons that ideal perfect-forecast revenue exaggerates realizable income. 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 minimum sale price needed to break even on a purchased 5 MWh block at 80% round trip and $45 cycle cost. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 6, Lesson 3: Demand response and rebound effects

Course position: Energy storage, flexibility and demand response. 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

Dispatchable demand changes may include shifting heating, cooling, pumps or flexible manufacturing. A load reduction today can create an increase later if the underlying energy service still must be delivered. Baseline estimation is therefore an evidence problem: what would the customer have consumed without the event? Behind-the-meter equipment limits, user comfort and production schedules constrain feasible calls.

Governing model, boundaries and assumptions

Net event energy change = integral(P_actual−P_baseline) dt; rebound refers to positive post-event deviation relative to baseline

Worked quantitative example

An industrial demand program reduces a facility from a 5 MW baseline to 4 MW for 2 hours, saving 2 MWh over the event. During the next 2 hours it consumes 5.5 MW versus the 5 MW baseline, adding 1 MWh. Net energy reduction over 4 hours is 1 MWh, not 2 MWh; the power reduction at the event peak is still 1 MW.

Interpretation and limiting cases

Power reduction, deferred energy and permanently avoided energy are three different metrics. Programs can value the first without implying the third.

What could make this answer wrong?

A baseline derived only from recent high-use days may be biased. Participants can also change behavior before events, complicating measurement.

Applied comparison and design reasoning

An apartment complex offers controllable air-conditioning. Explain how weather, occupant comfort and post-event rebound would enter a realistic demand-response evaluation.

Guided problems

Problem A — first-principles check. Reduce load by 0.5 MW for two hours, then raise by 0.2 MW for three hours; find net energy change. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain why event-period peak reduction can still have value when net energy saving is small. 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. How could a randomized control group or suitable comparison days help establish whether a claimed 1 MW response actually occurred? Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 6, Lesson 4: Storage participation in ancillary services

Course position: Energy storage, flexibility and demand response. 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

Storage can combine energy shifting, reserve commitments and fast response, but the same headroom and energy cannot be promised independently to every product without a co-optimization constraint. A dispatch schedule must carry state of charge forward and consider uncertainty in activation. A resource able to inject 10 MW for one second may not be able to sustain 10 MW for fifteen minutes.

Governing model, boundaries and assumptions

Reserve duration bound: E_available ≥ P_reserve × T_requirement / eta_discharge; enforce current SOC and additional commitments

Worked quantitative example

A 10 MW battery has 3 MWh usable energy above its minimum and 0.9 discharge efficiency. The maximum continuous 10 MW export duration is 3×0.9/10=0.27 h, or 16.2 minutes, ignoring thermal limits. A 30-minute 10 MW reserve product requires at least 5/0.9≈5.56 MWh from storage.

Interpretation and limiting cases

Reserve headroom is a MW property while reserve endurance is an MWh property. Both must be specified, along with state-of-charge restoration and response timing.

What could make this answer wrong?

A co-located solar-plus-storage model that allocates the same solar output to both export and battery charge double-counts power.

Applied comparison and design reasoning

Suppose the plant commits to a 5 MW upward reserve and schedules a 7 MW export from a 10 MW inverter. Can it deliver both obligations simultaneously at full activation?

Guided problems

Problem A — first-principles check. A store can deliver 6 MW and has 1.5 MWh above the minimum with ηd=0.9. Find max continuous full-power duration. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain why one committed reserve product can conflict with a scheduled arbitrage discharge. 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. Design a minimum data table for a 24-hour schedule showing charge/discharge, reserve up, reserve down, energy state and infeasibility flags. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 6 project: Storage dispatch and reserve feasibility

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 a 10 MWh usable energy store starting at 5 MWh with charging/discharging efficiencies 0.90 and 0.92, respectively, and charge/discharge power limits 4 MW. Define four 1-hour intervals with energy prices $30, $40, $100 and $60/MWh. Try a schedule charging 3 MW in interval 1, discharging 3 MW in interval 3, and otherwise idle. Compute the energy state sequentially, cashflow before fixed costs, end state, and whether a simultaneous 2 MW upward reserve for one hour can be guaranteed at the start of interval 3.

Deliverables and acceptance checks

Create a time-series table for state, charge, discharge, purchased energy, sold energy and reserve headroom. Compare two alternative schedules and explain why price foreknowledge, degradation and market access make the simple profits optimistic.

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 6 assessment: Energy storage, flexibility and demand response

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

Question group 1: State of charge and energy accounting

Q1.1 — calculation. Starting at 4 MWh, charge 2 MW for 2 h at ηc=0.85 and then discharge 1 MW for 1 h at ηd=0.9. Find final energy. 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 the round-trip efficiency is the product of one-way efficiencies in this definition. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q1.3 — applied critique. With minimum allowed energy 2 MWh, determine the maximum deliverable MWh from a current 5 MWh state if eta_d=0.9 and disregard power constraints.

Question group 2: Peak shaving and arbitrage dispatch

Q2.1 — calculation. If 4 MWh is purchased at $45/MWh and 3.2 MWh sold, compute the break-even sale price before fees. 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. Describe two reasons that ideal perfect-forecast revenue exaggerates realizable income. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q2.3 — applied critique. Calculate the minimum sale price needed to break even on a purchased 5 MWh block at 80% round trip and $45 cycle cost.

Question group 3: Demand response and rebound effects

Q3.1 — calculation. Reduce load by 0.5 MW for two hours, then raise by 0.2 MW for three hours; find net energy change. 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. Explain why event-period peak reduction can still have value when net energy saving is small. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q3.3 — applied critique. How could a randomized control group or suitable comparison days help establish whether a claimed 1 MW response actually occurred?

Question group 4: Storage participation in ancillary services

Q4.1 — calculation. A store can deliver 6 MW and has 1.5 MWh above the minimum with ηd=0.9. Find max continuous full-power duration. 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. Explain why one committed reserve product can conflict with a scheduled arbitrage discharge. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q4.3 — applied critique. Design a minimum data table for a 24-hour schedule showing charge/discharge, reserve up, reserve down, energy state and infeasibility flags.

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 6: Energy storage, flexibility and demand response

Energy storage is best understood as constrained power over time, not simply a large number of megawatt-hours. Charging and discharging power, efficiencies, operating state of charge, capacity fade, reserve margins and degradation costs all affect feasible schedules. Demand response changes load profiles but may create rebound effects and baseline uncertainty. This unit develops reliable state equations, dispatch comparison and critical checks suitable for research models. It does not provide battery installation, fire protection or device-control instructions.

Deep reading — interacting limitations

Storage scheduling is a sequential constrained optimization: power moves energy state, energy state restricts future power and market obligations reserve a portion of both. Simultaneous charging and discharging should be prohibited or justified by device architecture; otherwise a solver can invent a continuous power loop. The word efficiency may mean inverter AC/DC conversion, battery coulombic efficiency, one-way delivered-energy efficiency or full round-trip energy efficiency. State the definition. A price spread only becomes revenue if market participation and operational permissions exist; degradation, fees and losses reduce contribution. Demand response requires an auditable baseline and must account for rebound. Engineers should not certify resilience hours from nameplate MWh without checking starting energy and critical-load trajectories.

Case synthesis: energy feasibility before revenue or resilience claims

A 10 MWh battery starts at 5 MWh usable energy. Suppose it charges 3 MW for one hour at a one-way charging efficiency of 90% and then discharges 4 MW for one hour at a 90% discharging efficiency. At the end of charging it holds 5+3×0.9=7.7 MWh. A 4 MWh delivered discharge removes 4/0.9≈4.444 MWh of stored energy, leaving approximately 3.256 MWh. The simultaneous restrictions on charge/discharge power, stored energy and state-of-charge margins make this schedule feasible only if the declared operating limits permit those values. A further 4 MW discharge for one hour might not be feasible if the minimum usable energy is above zero or if reserve energy has been committed elsewhere.

Now change the decision question. A tariff features a $35/MWh low-price interval and a $100/MWh high-price interval. Buying 3 MWh costs $105. From that purchased energy, the ideal delivered energy after the two one-way efficiencies is 3×0.9×0.9=2.43 MWh, gross high-price revenue $243. The spread contribution is therefore $138 before degradation, network charges, fixed costs and market access expenses. This is not an annual return forecast. Multiplying one perfect-price day by 365 would assume the same spread, available capacity, dispatch opportunity and cycle economics every day, all of which require evidence.

A second service may conflict with the first. Suppose the store promises a 4 MW upward reserve for half an hour. The energy commitment is 2 MWh delivered, requiring about 2.222 MWh of stored energy at 90% discharge efficiency. The available MW and available MWh after arbitrage cannot be counted twice; a co-optimization model should reserve both simultaneously. An energy-feasible plan may still violate inverter power, grid export or ramp constraints. The model also needs aging and temperature effects before any equipment procurement.

Demand response is subject to a comparable time-coupling issue. Shifting cooling load by 1 MW for one hour can create a rebound later, affecting energy economics even while reducing the event-period maximum. Both storage and demand response should be represented as trajectories with counterfactual baselines, not isolated favorable hours. A responsible decision summary shows state by interval, delivered services, uncertainty of future calls and a credible plan to restore flexibility afterward.

Unit 6, Lesson 1: State of charge and energy accounting

Course position: Energy storage, flexibility and demand response. 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

State of charge describes stored usable energy relative to a declared capacity and allowable window. Separate charging and discharging efficiencies matter because one megawatt-hour consumed while charging does not become one megawatt-hour delivered later. Power and energy limits apply simultaneously. A physically credible schedule should update energy sequentially, not treat each interval as independent.

Governing model, boundaries and assumptions

E_(t+1) = E_t + eta_charge P_charge Delta t − P_discharge Delta t/eta_discharge, using nonnegative charge/discharge powers

Worked quantitative example

A 10 MWh usable-capacity model starts at 6 MWh. Charge at 2 MW for 1 hour with eta_c=0.9, adding 1.8 MWh. Then discharge at 3 MW for 1 hour with eta_d=0.9, withdrawing 3.333 MWh from storage, leaving about 4.467 MWh. These assumptions exclude auxiliary loads and capacity degradation.

Interpretation and limiting cases

Do not confuse an efficiency value with a round-trip efficiency when each half-cycle has its own factor. At eta_c=eta_d=0.9 the ideal round-trip product is 0.81.

What could make this answer wrong?

A model that allows charging and discharging simultaneously without constraints can exploit numerical loops and report fictitious profits.

Applied comparison and design reasoning

Compare a high-power short-duration storage application with an overnight energy-shift application. Which one is limited by MW and which by MWh for a given duty cycle?

Guided problems

Problem A — first-principles check. Starting at 4 MWh, charge 2 MW for 2 h at ηc=0.85 and then discharge 1 MW for 1 h at ηd=0.9. Find final energy. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain why the round-trip efficiency is the product of one-way efficiencies in this definition. 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. With minimum allowed energy 2 MWh, determine the maximum deliverable MWh from a current 5 MWh state if eta_d=0.9 and disregard power constraints. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 6, Lesson 2: Peak shaving and arbitrage dispatch

Course position: Energy storage, flexibility and demand response. 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

Peak shaving reduces net demand during high load; arbitrage shifts grid purchases from high-price to lower-price intervals. They are related but not identical objectives: a demand tariff depends on maximum power, while energy arbitrage depends on price-weighted throughput. Calendar aging, cycle aging, network constraints and battery reserve commitments can turn an apparently profitable energy spread into a negative economic result.

Governing model, boundaries and assumptions

Two-interval gross spread: revenue = E_delivered price_sell − E_charged price_buy; E_delivered = E_charged eta_round_trip

Worked quantitative example

A battery buys 5 MWh at $40/MWh. With round-trip efficiency 0.80 it can sell 4 MWh later at $90/MWh, gross energy spread =4×90−5×40=$160. If estimated incremental degradation and operating costs are $45 for this cycle, the illustrative contribution becomes $115 before fixed costs and market charges.

Interpretation and limiting cases

A 50-dollar difference between sell and buy prices is not automatically a 50-dollar profit per delivered MWh: losses must be purchased and paid for.

What could make this answer wrong?

Using perfect future prices to backtest a battery dispatch gives an optimistic upper bound. A real controller requires forecasts and must handle price revisions.

Applied comparison and design reasoning

Compare a 1 MW/2 MWh and a 2 MW/2 MWh device under a 30-minute sharp demand peak. Which can provide higher power, and what energy remains for later use?

Guided problems

Problem A — first-principles check. If 4 MWh is purchased at $45/MWh and 3.2 MWh sold, compute the break-even sale price before fees. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Describe two reasons that ideal perfect-forecast revenue exaggerates realizable income. 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 minimum sale price needed to break even on a purchased 5 MWh block at 80% round trip and $45 cycle cost. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 6, Lesson 3: Demand response and rebound effects

Course position: Energy storage, flexibility and demand response. 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

Dispatchable demand changes may include shifting heating, cooling, pumps or flexible manufacturing. A load reduction today can create an increase later if the underlying energy service still must be delivered. Baseline estimation is therefore an evidence problem: what would the customer have consumed without the event? Behind-the-meter equipment limits, user comfort and production schedules constrain feasible calls.

Governing model, boundaries and assumptions

Net event energy change = integral(P_actual−P_baseline) dt; rebound refers to positive post-event deviation relative to baseline

Worked quantitative example

An industrial demand program reduces a facility from a 5 MW baseline to 4 MW for 2 hours, saving 2 MWh over the event. During the next 2 hours it consumes 5.5 MW versus the 5 MW baseline, adding 1 MWh. Net energy reduction over 4 hours is 1 MWh, not 2 MWh; the power reduction at the event peak is still 1 MW.

Interpretation and limiting cases

Power reduction, deferred energy and permanently avoided energy are three different metrics. Programs can value the first without implying the third.

What could make this answer wrong?

A baseline derived only from recent high-use days may be biased. Participants can also change behavior before events, complicating measurement.

Applied comparison and design reasoning

An apartment complex offers controllable air-conditioning. Explain how weather, occupant comfort and post-event rebound would enter a realistic demand-response evaluation.

Guided problems

Problem A — first-principles check. Reduce load by 0.5 MW for two hours, then raise by 0.2 MW for three hours; find net energy change. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain why event-period peak reduction can still have value when net energy saving is small. 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. How could a randomized control group or suitable comparison days help establish whether a claimed 1 MW response actually occurred? Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 6, Lesson 4: Storage participation in ancillary services

Course position: Energy storage, flexibility and demand response. 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

Storage can combine energy shifting, reserve commitments and fast response, but the same headroom and energy cannot be promised independently to every product without a co-optimization constraint. A dispatch schedule must carry state of charge forward and consider uncertainty in activation. A resource able to inject 10 MW for one second may not be able to sustain 10 MW for fifteen minutes.

Governing model, boundaries and assumptions

Reserve duration bound: E_available ≥ P_reserve × T_requirement / eta_discharge; enforce current SOC and additional commitments

Worked quantitative example

A 10 MW battery has 3 MWh usable energy above its minimum and 0.9 discharge efficiency. The maximum continuous 10 MW export duration is 3×0.9/10=0.27 h, or 16.2 minutes, ignoring thermal limits. A 30-minute 10 MW reserve product requires at least 5/0.9≈5.56 MWh from storage.

Interpretation and limiting cases

Reserve headroom is a MW property while reserve endurance is an MWh property. Both must be specified, along with state-of-charge restoration and response timing.

What could make this answer wrong?

A co-located solar-plus-storage model that allocates the same solar output to both export and battery charge double-counts power.

Applied comparison and design reasoning

Suppose the plant commits to a 5 MW upward reserve and schedules a 7 MW export from a 10 MW inverter. Can it deliver both obligations simultaneously at full activation?

Guided problems

Problem A — first-principles check. A store can deliver 6 MW and has 1.5 MWh above the minimum with ηd=0.9. Find max continuous full-power duration. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain why one committed reserve product can conflict with a scheduled arbitrage discharge. 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. Design a minimum data table for a 24-hour schedule showing charge/discharge, reserve up, reserve down, energy state and infeasibility flags. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 6 project: Storage dispatch and reserve feasibility

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 a 10 MWh usable energy store starting at 5 MWh with charging/discharging efficiencies 0.90 and 0.92, respectively, and charge/discharge power limits 4 MW. Define four 1-hour intervals with energy prices $30, $40, $100 and $60/MWh. Try a schedule charging 3 MW in interval 1, discharging 3 MW in interval 3, and otherwise idle. Compute the energy state sequentially, cashflow before fixed costs, end state, and whether a simultaneous 2 MW upward reserve for one hour can be guaranteed at the start of interval 3.

Deliverables and acceptance checks

Create a time-series table for state, charge, discharge, purchased energy, sold energy and reserve headroom. Compare two alternative schedules and explain why price foreknowledge, degradation and market access make the simple profits optimistic.

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 6 assessment: Energy storage, flexibility and demand response

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

Question group 1: State of charge and energy accounting

Q1.1 — calculation. Starting at 4 MWh, charge 2 MW for 2 h at ηc=0.85 and then discharge 1 MW for 1 h at ηd=0.9. Find final energy. 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 the round-trip efficiency is the product of one-way efficiencies in this definition. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q1.3 — applied critique. With minimum allowed energy 2 MWh, determine the maximum deliverable MWh from a current 5 MWh state if eta_d=0.9 and disregard power constraints.

Question group 2: Peak shaving and arbitrage dispatch

Q2.1 — calculation. If 4 MWh is purchased at $45/MWh and 3.2 MWh sold, compute the break-even sale price before fees. 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. Describe two reasons that ideal perfect-forecast revenue exaggerates realizable income. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q2.3 — applied critique. Calculate the minimum sale price needed to break even on a purchased 5 MWh block at 80% round trip and $45 cycle cost.

Question group 3: Demand response and rebound effects

Q3.1 — calculation. Reduce load by 0.5 MW for two hours, then raise by 0.2 MW for three hours; find net energy change. 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. Explain why event-period peak reduction can still have value when net energy saving is small. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q3.3 — applied critique. How could a randomized control group or suitable comparison days help establish whether a claimed 1 MW response actually occurred?

Question group 4: Storage participation in ancillary services

Q4.1 — calculation. A store can deliver 6 MW and has 1.5 MWh above the minimum with ηd=0.9. Find max continuous full-power duration. 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. Explain why one committed reserve product can conflict with a scheduled arbitrage discharge. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q4.3 — applied critique. Design a minimum data table for a 24-hour schedule showing charge/discharge, reserve up, reserve down, energy state and infeasibility flags.

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.