Energy Storage Dispatch and Round-Trip Efficiency Exercises

These exercises practise dispatch calculations for battery energy storage systems. The focus is not only how much energy is installed, but how energy moves through a real operating schedule: charge energy, discharged energy, state of charge, reserve, efficiency boundary, auxiliary consumption, and validation evidence.

Use the exercises as engineering checks. A dispatch plan is credible only when it preserves state-of-charge limits, respects power limits, states the measurement boundary, accounts for losses, protects contractual reserve, and leaves enough evidence to reconcile the event after operation.

How to Use These Exercises

Before calculating, define four boundaries:

1. the energy basis, such as current DC battery capacity, usable DC energy, inverter AC terminals, transformer secondary, or point of interconnection;
2. the sign convention for charge and discharge power;
3. the reserve that is not available for routine dispatch;
4. the validation data that would prove the dispatch actually happened.

For a discrete dispatch interval, a useful stored-energy update is:

where E_k is stored energy at the selected battery boundary, P_{ch,k} is charging power measured at the AC input boundary, P_{dis,k} is delivered AC discharge power, \eta_{ch} is charge-path efficiency, and \eta_{dis} is discharge-path efficiency.

This expression is a screening model. Real controls may include inverter current limits, cell voltage limits, thermal derating, auxiliary power, state-of-health correction, SOC-estimation uncertainty, communication loss, market dispatch constraints, and warranty restrictions.

Exercise 1: SOC Trajectory During a Simple Dispatch Schedule

A battery has a current usable DC capacity basis of 20\ \text{MWh}. It starts a dispatch period at 60\% state of charge. The controller follows this schedule:

• charge at 4.0\ \text{MW} AC for 2.0\ \text{h};
• hold for 1.0\ \text{h};
• discharge at 5.0\ \text{MW} AC for 1.5\ \text{h}.

The charge-path efficiency is 96\% and the discharge-path efficiency is 95\%. Ignore standby auxiliary load for this exercise. Estimate the final stored energy and final SOC.

Solution

Initial stored energy:

Energy stored during charge:

Stored energy consumed during discharge:

Final stored energy:

Final SOC:

The final SOC is about 58.9\%.

Engineering Comment

The battery charged for more hours than it discharged, but the final SOC is slightly lower than the initial SOC because the discharge power was higher and conversion losses were present. Dispatch review should use an energy balance, not visual intuition from the schedule.

Exercise 2: Round-Trip Efficiency Across a Defined AC Boundary

A storage asset charges with 10.0\ \text{MWh} measured at the AC point of interconnection. The charge path has transformer efficiency 98.5\%, inverter charge efficiency 97.0\%, and battery charge acceptance efficiency 99.0\%.

During discharge, the battery-to-inverter path has efficiency 96.5\% and the transformer again has efficiency 98.5\%. Auxiliary equipment consumes 0.25\ \text{MWh} over the full cycle. Estimate the net AC energy delivered and the cycle round-trip efficiency at the point of interconnection.

Solution

Stored energy after charging:

AC energy delivered before auxiliary deduction:

Net delivered energy:

Round-trip efficiency at the point of interconnection:

The net AC-to-AC round-trip efficiency is about 87.4\%.

Engineering Comment

Round-trip efficiency is meaningful only when the boundary is stated. Cell-only efficiency, inverter AC efficiency, and point-of-interconnection efficiency can differ materially. Auxiliary energy is small here, but it becomes important for low-utilization assets, cold-weather systems, thermal conditioning, and long standby intervals.

Exercise 3: Reserve Breach in a Proposed Peak-Shaving Dispatch

A battery has a current capacity basis of 50\ \text{MWh} and starts at 55\% SOC. The minimum operating SOC is 10\%. The operator must preserve an additional 8.0\ \text{MWh} of protected emergency reserve above the minimum SOC.

The proposed peak-shaving event delivers 7.0\ \text{MW} AC for 2.0\ \text{h}. Discharge-path efficiency is 94\%. Check whether the dispatch preserves the protected reserve.

Solution

Initial stored energy:

Minimum stored energy from SOC limit:

Minimum stored energy including protected reserve:

Stored energy consumed by the peak-shaving event:

Final stored energy:

Reserve shortfall:

The dispatch breaches the protected reserve by about 0.39\ \text{MWh}.

Engineering Comment

The event appears feasible if only the minimum SOC limit is checked. It is not feasible under the service contract because the protected reserve is part of the operating constraint. Corrective options include reducing discharge power, shortening the event, increasing starting SOC, relaxing another service, or redefining the reserve policy with explicit priority rules.

Exercise 4: Arbitrage Value After Efficiency and Degradation Cost

A battery charges with 12.0\ \text{MWh} from the grid during a low-price period at 35/\text{MWh}. The expected AC-to-AC round-trip efficiency for the dispatch is 88\%. The evening sale price is 95/\text{MWh}. The operator assigns a degradation cost of 18/\text{MWh} discharged.

Estimate the net value of the cycle before fixed operating costs.

Solution

Delivered discharge energy:

Charging cost:

Discharge revenue:

Degradation cost:

Net value:

The cycle has an estimated net value of about 393 before fixed costs, taxes, fees, availability penalties, and market settlement details.

Engineering Comment

The spread between buy price and sell price is not the profit. The battery sells less energy than it buys because of round-trip losses, and the cycle consumes part of the asset life. If degradation cost is ignored, dispatch logic may look profitable while accelerating loss of future capability.

Exercise 5: Solar Clipping Capture with Charge Limit and SOC Headroom

A solar plant would otherwise curtail energy because of an export limit. The clipped power available for charging is:

Interval	Duration	Clipped power
1	0.5\ \text{h}	3.0\ \text{MW}
2	0.5\ \text{h}	5.0\ \text{MW}
3	0.5\ \text{h}	4.0\ \text{MW}
4	0.5\ \text{h}	2.0\ \text{MW}

The battery AC charge limit is 4.0\ \text{MW}. Available DC energy headroom is 5.0\ \text{MWh}. Charge efficiency from AC input to stored DC energy is 96\%, and discharge efficiency from stored DC energy to AC output is 94\%.

Estimate clipped energy, captured AC charge energy, stored energy, remaining curtailment, and later AC discharge energy.

Solution

Total clipped energy before battery limits:

Charge-power-limited AC energy that could be accepted:

AC energy needed to fill the available DC headroom:

The battery becomes energy-headroom limited before it can accept all charge-power-limited clipped energy. Captured AC charge energy is therefore 5.21\ \text{MWh}.

Stored energy:

Remaining curtailment:

Later AC discharge energy:

Engineering Comment

Two different limits appear. The battery cannot accept all clipped energy because the second interval exceeds the charge power limit, and then the available SOC headroom becomes the active constraint. A larger inverter would not solve the final part of the problem unless additional energy headroom were also available.

Exercise 6: Recharge Time After an Emergency Discharge

A battery delivers 18.0\ \text{MWh} AC during an emergency support event. Discharge-path efficiency was 94\%. The operator wants to restore the consumed stored energy during an off-peak recharge window.

Available AC charging power is 5.0\ \text{MW}, charge-path efficiency is 96\%, and average auxiliary load during recharge is 0.20\ \text{MW}. Estimate the minimum recharge duration and the total AC energy drawn at the site meter during recharge.

Solution

Stored energy consumed during the emergency event:

Stored-energy restoration rate:

Minimum recharge duration:

AC charge energy:

Auxiliary energy during recharge:

Total site-meter recharge energy:

Engineering Comment

Restoring the battery is part of the service. If the off-peak window is shorter than about 4.0\ \text{h}, the battery will not be ready for the next event unless charge power is increased, the required reserve is reduced, or another resource covers the gap. Recharge energy at the site meter is larger than the stored energy because charge losses and auxiliary load are real energy flows.

Exercise 7: Standby Loss and Short Frequency-Response Event

A battery is held ready for a fast frequency-response service for 18\ \text{h}. Thermal management and control auxiliaries consume 35\ \text{kW} on average during standby. A frequency event then requires 10.0\ \text{MW} AC for 6\ \text{min}. Discharge-path efficiency for the event is 95\%.

Estimate standby auxiliary energy, AC event energy, stored energy consumed by the event, and the total energy that must be considered in the operating review.

Solution

Standby auxiliary energy:

Event duration:

AC event energy:

Stored energy consumed by the event:

Energy relevant to the operating review:

Engineering Comment

For short-duration grid services, standby energy can be comparable to event energy over long readiness periods. The asset may be technically valuable even with low utilization, but the efficiency, emissions, and economic review should include the energy consumed while staying available.

Exercise 8: Correcting a Round-Trip Efficiency Acceptance Test

During an acceptance test, the site meter records 24.0\ \text{MWh} of AC charging energy and 20.7\ \text{MWh} of AC discharge energy. The test should compare equal initial and final stored energy, but the final stored energy is estimated to be 0.40\ \text{MWh} higher than the initial stored energy. If the discharge-path efficiency is 95\%, estimate the corrected round-trip efficiency.

The contract acceptance threshold is 88.0\%.

Solution

Because the final stored energy is higher than the initial stored energy, the discharge did not return all energy that could be associated with the charged cycle. Equivalent additional AC discharge energy is:

Corrected AC discharge energy:

Corrected round-trip efficiency:

The corrected round-trip efficiency is about 87.8\%.

Engineering Comment

The corrected result is close to the 88.0\% threshold but still slightly below it. A responsible acceptance decision should check metering accuracy, SOC-estimation uncertainty, auxiliary-energy treatment, temperature, test duration, and contractual rounding rules. The engineering conclusion is not just pass or fail; it is whether the evidence is clean enough to support the contractual decision.

Exercise 9: Dispatch Evidence Reconciliation

A storage controller reports that a battery delivered a 4.0\ \text{MW} demand-response dispatch for 90\ \text{min}. The revenue meter shows 5.85\ \text{MWh} delivered during the event. The historian shows average inverter active power of 3.95\ \text{MW}, one 4\ \text{min} communication dropout, no inverter trip, and a stored-energy decrease of 6.30\ \text{MWh}. The expected discharge-path efficiency for the event is 94\%.

Check whether the event record is internally consistent enough for engineering review.

Solution

Commanded AC energy:

Metered delivered energy:

Historian energy from average inverter power:

Expected stored-energy decrease from the meter:

Measured stored-energy decrease:

Difference:

The meter, historian, and stored-energy change are broadly consistent. The revenue meter is 0.15\ \text{MWh} below the commanded energy, or:

Engineering Comment

The dispatch should not be rejected automatically from these numbers. The energy balance is close, and the historian agrees with the revenue meter. However, the communication dropout must be explained because service qualification may require continuous telemetry. The event record should include whether the dropout affected only communications or also control, metering, and compliance evidence.

Dispatch Review Checklist

For a storage dispatch schedule, a good engineering review should confirm:

1. the SOC trajectory stays within operating limits at every time step;
2. protected reserve remains available after routine dispatch;
3. charge and discharge efficiency use the correct boundary;
4. auxiliary energy is included when it affects the service or settlement;
5. power limits, ramp limits, inverter apparent-power limits, and thermal limits are not exceeded;
6. the recharge window restores the service before the next required event;
7. the economic result includes losses and degradation, not only price spread;
8. metered energy, controller logs, SOC change, alarms, and historian data reconcile after operation.

Dispatch is an engineering problem before it is a market schedule. The schedule is acceptable only if the asset can deliver the service, preserve its constraints, recover for the next obligation, and prove what happened.