Power System Protection: Overcurrent, Distance, and Differential Relaying

Unit 7, Lesson 1: Radial feeder modeling and voltage profiles

Course position: Distribution automation and microgrid operation. 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 distribution feeder usually has substantial R/X and nonuniform, sometimes phase-unbalanced loads. A source bus voltage may be acceptable while distant customers experience undervoltage at peak demand or overvoltage during midday reverse export. Feeder analysis should include load locations, conductor impedances, phase connections, regulators and capacitor states. A single end-to-end equivalent is useful for screening, not for locating each stressed device.

Governing model, boundaries and assumptions

Approximate feeder segment voltage drop Delta V ≈ (R P + X Q)/V, consistently on chosen base

Worked quantitative example

In a simplified two-segment radial feeder, first segment has R=0.04 pu, X=0.06 pu and downstream flow P=0.7, Q=0.3; approximate drop 0.046 pu. Second segment has R=0.03, X=0.04 and flow P=0.25, Q=0.1; drop ≈0.0115 pu. Total approximate drop is 0.0575 pu for V≈1 pu. A full unbalanced load-flow solution would calculate changing current and voltage at each node.

Interpretation and limiting cases

The upstream segment carries all downstream power plus its local load, not merely the consumption at the immediately adjacent bus.

What could make this answer wrong?

A feeder built from average energy data rather than interval peaks can look comfortable while cable thermal loading exceeds its limit at short peaks.

Applied comparison and design reasoning

Compare a feeder with rooftop PV near the substation and the same PV near its remote end. Explain why hosting capacity depends on location, not only total installed MW.

Guided problems

Problem A — first-principles check. Combine two approximate per-unit drops 0.02 and 0.035 with a nominal source of 1.01 pu. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain why unequal phases require a richer model than a single series branch. 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. Draw nodal power balances at the remote branch and specify how a phase-unbalance study would change the scalar model. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 7, Lesson 2: Volt-VAR coordination and tap changers

Course position: Distribution automation and microgrid operation. 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

On-load tap changers, switched capacitors and inverter VAR controls can all influence customer voltage. Their actions differ in speed, locality and loss consequences. Without coordination, a rapid inverter response may counteract a slower tap changer or cause repeated mechanical operation. The aim is not a single number but a voltage trajectory at all critical locations with acceptable equipment operations and losses.

Governing model, boundaries and assumptions

A first-order sensitivity model writes Delta V ≈ K_Q Delta Q + K_tap Delta tap, with measured or simulated sensitivities

Worked quantitative example

Suppose a feeder end bus reacts to inverter reactive injection with K_Q=0.025 pu voltage per pu injected Q. An injection of 0.40 pu would predict +0.010 pu local voltage. If a regulator tap change simultaneously lowers source-side voltage by 0.006 pu, the combined first-order effect is only +0.004 pu. This is a local linear approximation; both controls may have nonlinear deadbands.

Interpretation and limiting cases

A change that helps the remote end may create high voltage near the source. Evaluate the full voltage envelope, not one sensor.

What could make this answer wrong?

Constant tap operation in a digital simulation ignores device dwell times and mechanical wear. Continuous control optimization cannot be applied unchanged to discrete steps.

Applied comparison and design reasoning

PV inverters oscillate between VAR injection and absorption around a steep Volt-VAR curve. Explain what hysteresis and response delay might change.

Guided problems

Problem A — first-principles check. Calculate net voltage change if one control raises 0.012 pu and another reduces 0.007 pu in a locally linear model. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. State one measure of excessive tap activity in a control study. 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 event log and supervisory data are needed to prove a reduction in tap-change count rather than merely a better voltage snapshot? Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 7, Lesson 3: Microgrid transition and island energy balance

Course position: Distribution automation and microgrid operation. 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 microgrid can operate grid-connected or intentionally islanded only when the network, protection and control systems were designed for that function. In island mode there is no unlimited upstream slack bus; energy availability, frequency, voltage and instantaneous power balance become local responsibilities. Synchronizing back to the utility requires explicit procedures and protection. A basic energy balance can reveal whether a conceptual island is even feasible before detailed controls are modeled.

Governing model, boundaries and assumptions

P_gen + P_battery_discharge − P_battery_charge − P_load − P_losses = dW_kinetic/dt + other stored-power terms

Worked quantitative example

A teaching island has a 2.0 MW load, 0.6 MW PV, 1.1 MW dispatchable generation and a battery that can export at most 0.5 MW. Ignoring losses, the full 0.5 MW battery output exactly balances supply at 2.2 MW, 0.2 MW above load; a controller must adjust generation, charging or other injections. If PV suddenly falls to 0.2 MW, 1.1+0.2+0.5=1.8 MW, leaving 0.2 MW shortfall without additional dispatch or load reduction.

Interpretation and limiting cases

Energy capacity matters after immediate power balance. Even if battery power covers a shortfall, discharge duration must be checked from state of charge.

What could make this answer wrong?

A microgrid feasibility chart that assumes solar output is constant through the night cannot support a resilience claim.

Applied comparison and design reasoning

Compare a hospital microgrid with a campus classroom microgrid: critical-load prioritization and verification requirements differ even for identical kW ratings.

Guided problems

Problem A — first-principles check. Given 1.4 MW load, 0.4 MW PV, 0.7 MW dispatchable and 0.2 MW battery output, find residual deficit. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Calculate one-hour deficit energy if it remains constant and identify which load-priority information is missing. 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. List the information needed to determine which noncritical loads can be deferred under a cloudy three-hour outage; distinguish engineering needs from permissions and life-safety rules. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 7, Lesson 4: Reliability indices and feeder restoration

Course position: Distribution automation and microgrid operation. 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

Distribution reliability uses customer interruptions and restoration times, not just generator reserve margins. SAIFI estimates average sustained interruption frequency and SAIDI estimates average duration for the customers considered, under a defined reporting standard and event inclusion rule. Their ratios are useful but can hide highly uneven customer experiences. Restoration plans should be evaluated for feasible topology, conductor ratings and safe backfeed paths.

Governing model, boundaries and assumptions

SAIFI=sum(interrupted customers for each included event)/total customers; SAIDI=sum(customer interruption hours)/total customers

Worked quantitative example

A utility study area has 1000 customers. One 2-hour event interrupts 200 customers and one 0.5-hour event interrupts 400 customers, with event sets possibly overlapping. SAIFI=(200+400)/1000=0.60 interruptions/customer and SAIDI=(200×2+400×0.5)/1000=0.60 h/customer. In this arithmetic the same customer can be counted in multiple events.

Interpretation and limiting cases

Indices depend on reporting definitions such as which short interruptions or major events are included. The scenario is not benchmark data from a real distribution system.

What could make this answer wrong?

Average SAIDI alone masks a small vulnerable customer group that repeatedly suffers very long outages. Averages should be paired with distributions and critical-site flags.

Applied comparison and design reasoning

A tie switch could restore 300 customers but would overload the adjacent feeder during peak load. Explain why a restoration study must simulate the post-switch state.

Guided problems

Problem A — first-principles check. For 500 customers, one event affects 100 for 2 h and another 200 for 0.5 h. Calculate SAIFI and SAIDI. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain which highly affected customers might be hidden by the averages. 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. Construct an event table with start, restoration, affected customers and event category, then independently recompute the indices and one high-quantile customer impact. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 7 project: Microgrid island adequacy by hour

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

Create a notional critical-load profile [1.2,1.4,1.6,1.3] MW, solar profile [0.5,0.3,0.0,0.0] MW, and a generator able to deliver 1.0 MW continuously. Battery initial usable energy above minimum is 2 MWh, max discharge power is 0.6 MW and discharge efficiency 0.9. Use four 1-hour steps to calculate hourly power gaps, battery feasible discharge, unmet critical load, and the next energy state. You may curtail surplus energy but should not charge the battery without explicitly including charging efficiency.

Deliverables and acceptance checks

Report any uncovered MW and MWh, state-of-charge constraint violations and a revised critical-load schedule that reduces unserved energy. Do not assert that the physical island is safe; include the missing grounding, relay and transfer-control studies.

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 7 assessment: Distribution automation and microgrid operation

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

Question group 1: Radial feeder modeling and voltage profiles

Q1.1 — calculation. Combine two approximate per-unit drops 0.02 and 0.035 with a nominal source of 1.01 pu. 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 unequal phases require a richer model than a single series branch. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q1.3 — applied critique. Draw nodal power balances at the remote branch and specify how a phase-unbalance study would change the scalar model.

Question group 2: Volt-VAR coordination and tap changers

Q2.1 — calculation. Calculate net voltage change if one control raises 0.012 pu and another reduces 0.007 pu in a locally linear model. 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. State one measure of excessive tap activity in a control study. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q2.3 — applied critique. What event log and supervisory data are needed to prove a reduction in tap-change count rather than merely a better voltage snapshot?

Question group 3: Microgrid transition and island energy balance

Q3.1 — calculation. Given 1.4 MW load, 0.4 MW PV, 0.7 MW dispatchable and 0.2 MW battery output, find residual deficit. 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. Calculate one-hour deficit energy if it remains constant and identify which load-priority information is missing. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q3.3 — applied critique. List the information needed to determine which noncritical loads can be deferred under a cloudy three-hour outage; distinguish engineering needs from permissions and life-safety rules.

Question group 4: Reliability indices and feeder restoration

Q4.1 — calculation. For 500 customers, one event affects 100 for 2 h and another 200 for 0.5 h. Calculate SAIFI and SAIDI. 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 which highly affected customers might be hidden by the averages. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q4.3 — applied critique. Construct an event table with start, restoration, affected customers and event category, then independently recompute the indices and one high-quantile customer impact.

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 7: Distribution automation and microgrid operation

Distribution networks combine radial topology, unbalanced single-phase loads, embedded generation and customer-facing reliability constraints. Automated reconfiguration, islanded microgrids and volt/VAR controls can improve operations when they are designed and tested within equipment ratings and protection requirements. They can also create new failure modes: unexpected backfeeds, loss of grounding reference, unsafe islanding, control competition and inadequate fault detection. Classroom network examples focus on system reasoning; all actual switching and island formation belong to approved utility procedures.

Deep reading — interacting limitations

A radial feeder’s voltage and thermal problems are spatial. PV located near a remote customer can create different voltage impacts than equal PV near the substation. Distribution phases are often unequal, and a scalar balanced network can miss neutral and phase-loading problems. Voltage regulators, capacitor banks and inverter controllers may respond on different time scales, creating oscillations or excessive switching unless coordinated. An islanding design must maintain a grounding and protection philosophy; mere availability of enough MW does not imply a safe standalone supply. Reliability indices help aggregate service but hide distributional inequity. Event metadata and critical-facility consequences should accompany averaged SAIDI and SAIFI results.

Case synthesis: an acceptable substation voltage is not an acceptable feeder

Consider a radial distribution circuit beginning at 1.01 pu, with a heavily loaded upstream segment and a lightly loaded remote branch. In a basic approximate model, assume the first segment contributes a 0.040 pu drop and the second a 0.025 pu drop. The calculated remote bus voltage is 0.945 pu, although the substation measures 1.01 pu. The particular voltage may or may not comply with the applicable supply requirements; those depend on jurisdiction, point of measurement, time basis and voltage level. The mathematical lesson is that the substation meter cannot certify every customer’s service quality.

Now add local solar generation close to the remote bus. It reduces power drawn over the upstream network and may improve low daytime voltage, but sufficiently large midday export may push the same remote point above its appropriate limit. On a high-R/X circuit, active-power reversal can affect voltage substantially even with unchanged reactive controls. A simple balanced model can miss one phase carrying markedly more single-phase rooftop generation than another. Tap changers, line regulators, switched capacitors and inverter VAR controls operate with different sensors and delays; independent control objectives can create repeated tap operations or cycling.

A proposed microgrid boundary introduces a distinct problem. If the upstream grid is intentionally disconnected, a source must form a viable local voltage/frequency reference and maintain sufficient instant power and energy. A battery rated at 2 MW with only 0.5 MWh available cannot support 2 MW for a full hour. Dispatchable generation may reduce the deficit but still needs fuel supply, protection, grounding arrangements and a safe re-synchronization philosophy. Classroom balancing cannot substitute for approved islanding and commissioning. Priority of life-safety critical loads may also change the optimization objective relative to an ordinary commercial feeder.

Reliability statistics require similarly explicit boundaries. SAIDI and SAIFI are customer-weighted averages over a defined reporting population and event set. Two feeders can share the same average while one repeatedly interrupts a small group of vulnerable customers. A tie-switch restoration plan may reduce outage hours but overload another feeder or create a prohibited backfeed path. Therefore a valid planning proposal identifies network topology, loading under restored state, protection zones, affected critical customers, and event-data confidence before proposing automated operation.

Unit 7, Lesson 1: Radial feeder modeling and voltage profiles

Course position: Distribution automation and microgrid operation. 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 distribution feeder usually has substantial R/X and nonuniform, sometimes phase-unbalanced loads. A source bus voltage may be acceptable while distant customers experience undervoltage at peak demand or overvoltage during midday reverse export. Feeder analysis should include load locations, conductor impedances, phase connections, regulators and capacitor states. A single end-to-end equivalent is useful for screening, not for locating each stressed device.

Governing model, boundaries and assumptions

Approximate feeder segment voltage drop Delta V ≈ (R P + X Q)/V, consistently on chosen base

Worked quantitative example

In a simplified two-segment radial feeder, first segment has R=0.04 pu, X=0.06 pu and downstream flow P=0.7, Q=0.3; approximate drop 0.046 pu. Second segment has R=0.03, X=0.04 and flow P=0.25, Q=0.1; drop ≈0.0115 pu. Total approximate drop is 0.0575 pu for V≈1 pu. A full unbalanced load-flow solution would calculate changing current and voltage at each node.

Interpretation and limiting cases

The upstream segment carries all downstream power plus its local load, not merely the consumption at the immediately adjacent bus.

What could make this answer wrong?

A feeder built from average energy data rather than interval peaks can look comfortable while cable thermal loading exceeds its limit at short peaks.

Applied comparison and design reasoning

Compare a feeder with rooftop PV near the substation and the same PV near its remote end. Explain why hosting capacity depends on location, not only total installed MW.

Guided problems

Problem A — first-principles check. Combine two approximate per-unit drops 0.02 and 0.035 with a nominal source of 1.01 pu. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain why unequal phases require a richer model than a single series branch. 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. Draw nodal power balances at the remote branch and specify how a phase-unbalance study would change the scalar model. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 7, Lesson 2: Volt-VAR coordination and tap changers

Course position: Distribution automation and microgrid operation. 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

On-load tap changers, switched capacitors and inverter VAR controls can all influence customer voltage. Their actions differ in speed, locality and loss consequences. Without coordination, a rapid inverter response may counteract a slower tap changer or cause repeated mechanical operation. The aim is not a single number but a voltage trajectory at all critical locations with acceptable equipment operations and losses.

Governing model, boundaries and assumptions

A first-order sensitivity model writes Delta V ≈ K_Q Delta Q + K_tap Delta tap, with measured or simulated sensitivities

Worked quantitative example

Suppose a feeder end bus reacts to inverter reactive injection with K_Q=0.025 pu voltage per pu injected Q. An injection of 0.40 pu would predict +0.010 pu local voltage. If a regulator tap change simultaneously lowers source-side voltage by 0.006 pu, the combined first-order effect is only +0.004 pu. This is a local linear approximation; both controls may have nonlinear deadbands.

Interpretation and limiting cases

A change that helps the remote end may create high voltage near the source. Evaluate the full voltage envelope, not one sensor.

What could make this answer wrong?

Constant tap operation in a digital simulation ignores device dwell times and mechanical wear. Continuous control optimization cannot be applied unchanged to discrete steps.

Applied comparison and design reasoning

PV inverters oscillate between VAR injection and absorption around a steep Volt-VAR curve. Explain what hysteresis and response delay might change.

Guided problems

Problem A — first-principles check. Calculate net voltage change if one control raises 0.012 pu and another reduces 0.007 pu in a locally linear model. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. State one measure of excessive tap activity in a control study. 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 event log and supervisory data are needed to prove a reduction in tap-change count rather than merely a better voltage snapshot? Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 7, Lesson 3: Microgrid transition and island energy balance

Course position: Distribution automation and microgrid operation. 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 microgrid can operate grid-connected or intentionally islanded only when the network, protection and control systems were designed for that function. In island mode there is no unlimited upstream slack bus; energy availability, frequency, voltage and instantaneous power balance become local responsibilities. Synchronizing back to the utility requires explicit procedures and protection. A basic energy balance can reveal whether a conceptual island is even feasible before detailed controls are modeled.

Governing model, boundaries and assumptions

P_gen + P_battery_discharge − P_battery_charge − P_load − P_losses = dW_kinetic/dt + other stored-power terms

Worked quantitative example

A teaching island has a 2.0 MW load, 0.6 MW PV, 1.1 MW dispatchable generation and a battery that can export at most 0.5 MW. Ignoring losses, the full 0.5 MW battery output exactly balances supply at 2.2 MW, 0.2 MW above load; a controller must adjust generation, charging or other injections. If PV suddenly falls to 0.2 MW, 1.1+0.2+0.5=1.8 MW, leaving 0.2 MW shortfall without additional dispatch or load reduction.

Interpretation and limiting cases

Energy capacity matters after immediate power balance. Even if battery power covers a shortfall, discharge duration must be checked from state of charge.

What could make this answer wrong?

A microgrid feasibility chart that assumes solar output is constant through the night cannot support a resilience claim.

Applied comparison and design reasoning

Compare a hospital microgrid with a campus classroom microgrid: critical-load prioritization and verification requirements differ even for identical kW ratings.

Guided problems

Problem A — first-principles check. Given 1.4 MW load, 0.4 MW PV, 0.7 MW dispatchable and 0.2 MW battery output, find residual deficit. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Calculate one-hour deficit energy if it remains constant and identify which load-priority information is missing. 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. List the information needed to determine which noncritical loads can be deferred under a cloudy three-hour outage; distinguish engineering needs from permissions and life-safety rules. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 7, Lesson 4: Reliability indices and feeder restoration

Course position: Distribution automation and microgrid operation. 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

Distribution reliability uses customer interruptions and restoration times, not just generator reserve margins. SAIFI estimates average sustained interruption frequency and SAIDI estimates average duration for the customers considered, under a defined reporting standard and event inclusion rule. Their ratios are useful but can hide highly uneven customer experiences. Restoration plans should be evaluated for feasible topology, conductor ratings and safe backfeed paths.

Governing model, boundaries and assumptions

SAIFI=sum(interrupted customers for each included event)/total customers; SAIDI=sum(customer interruption hours)/total customers

Worked quantitative example

A utility study area has 1000 customers. One 2-hour event interrupts 200 customers and one 0.5-hour event interrupts 400 customers, with event sets possibly overlapping. SAIFI=(200+400)/1000=0.60 interruptions/customer and SAIDI=(200×2+400×0.5)/1000=0.60 h/customer. In this arithmetic the same customer can be counted in multiple events.

Interpretation and limiting cases

Indices depend on reporting definitions such as which short interruptions or major events are included. The scenario is not benchmark data from a real distribution system.

What could make this answer wrong?

Average SAIDI alone masks a small vulnerable customer group that repeatedly suffers very long outages. Averages should be paired with distributions and critical-site flags.

Applied comparison and design reasoning

A tie switch could restore 300 customers but would overload the adjacent feeder during peak load. Explain why a restoration study must simulate the post-switch state.

Guided problems

Problem A — first-principles check. For 500 customers, one event affects 100 for 2 h and another 200 for 0.5 h. Calculate SAIFI and SAIDI. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain which highly affected customers might be hidden by the averages. 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. Construct an event table with start, restoration, affected customers and event category, then independently recompute the indices and one high-quantile customer impact. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 7 project: Microgrid island adequacy by hour

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

Create a notional critical-load profile [1.2,1.4,1.6,1.3] MW, solar profile [0.5,0.3,0.0,0.0] MW, and a generator able to deliver 1.0 MW continuously. Battery initial usable energy above minimum is 2 MWh, max discharge power is 0.6 MW and discharge efficiency 0.9. Use four 1-hour steps to calculate hourly power gaps, battery feasible discharge, unmet critical load, and the next energy state. You may curtail surplus energy but should not charge the battery without explicitly including charging efficiency.

Deliverables and acceptance checks

Report any uncovered MW and MWh, state-of-charge constraint violations and a revised critical-load schedule that reduces unserved energy. Do not assert that the physical island is safe; include the missing grounding, relay and transfer-control studies.

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 7 assessment: Distribution automation and microgrid operation

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

Question group 1: Radial feeder modeling and voltage profiles

Q1.1 — calculation. Combine two approximate per-unit drops 0.02 and 0.035 with a nominal source of 1.01 pu. 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 unequal phases require a richer model than a single series branch. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q1.3 — applied critique. Draw nodal power balances at the remote branch and specify how a phase-unbalance study would change the scalar model.

Question group 2: Volt-VAR coordination and tap changers

Q2.1 — calculation. Calculate net voltage change if one control raises 0.012 pu and another reduces 0.007 pu in a locally linear model. 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. State one measure of excessive tap activity in a control study. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q2.3 — applied critique. What event log and supervisory data are needed to prove a reduction in tap-change count rather than merely a better voltage snapshot?

Question group 3: Microgrid transition and island energy balance

Q3.1 — calculation. Given 1.4 MW load, 0.4 MW PV, 0.7 MW dispatchable and 0.2 MW battery output, find residual deficit. 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. Calculate one-hour deficit energy if it remains constant and identify which load-priority information is missing. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q3.3 — applied critique. List the information needed to determine which noncritical loads can be deferred under a cloudy three-hour outage; distinguish engineering needs from permissions and life-safety rules.

Question group 4: Reliability indices and feeder restoration

Q4.1 — calculation. For 500 customers, one event affects 100 for 2 h and another 200 for 0.5 h. Calculate SAIFI and SAIDI. 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 which highly affected customers might be hidden by the averages. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q4.3 — applied critique. Construct an event table with start, restoration, affected customers and event category, then independently recompute the indices and one high-quantile customer impact.

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.