Multi-Machine Transient Stability and Numerical Time-Domain Integration

Unit 5, Lesson 1: Grid-following inverters and synchronization

Course position: Renewables, grid-following and grid-forming inverters. 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 typical grid-following inverter measures network voltage phase and commands a current injection to achieve active/reactive targets. The phase-locked loop can be stressed when the grid voltage is weak, distorted or rapidly changing. Current controls and DC-link limits regulate what actually reaches the network. A correctly behaving current-controlled inverter cannot deliver arbitrary fault current just because a classical generator-equivalent study predicts it.

Governing model, boundaries and assumptions

A stylized PLL phase error e=theta_grid−theta_hat; frequency estimate adjusts with a selected control transfer function

Worked quantitative example

Suppose a grid voltage phase steps by 10° relative to an inverter estimate. A proportional-only toy PLL with angular-frequency correction K_p e and K_p=20 s^-1 gives initial correction ≈20×0.1745=3.49 rad/s, about 0.556 Hz. This is not a real commercial PLL controller and does not predict stable recovery without integral gains, filtering, saturation and delay.

Interpretation and limiting cases

The PLL angle is an estimate with measurement bandwidth and filtering; it is not an independently perfect network angle.

What could make this answer wrong?

In a low-short-circuit-strength area, current injection changes terminal voltage, which alters the input to the PLL. Ignoring this feedback can hide oscillation or loss-of-synchronization risks.

Applied comparison and design reasoning

Place the same inverter behind a strong and a high-impedance grid equivalent. Explain qualitatively how terminal-voltage sensitivity to reactive current changes.

Guided problems

Problem A — first-principles check. Convert a 5° phase mismatch to radians; multiply by a hypothetical proportional coefficient of 10 s^-1. 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 one-step proportional correction cannot establish closed-loop stability. 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. Which recordings are essential to distinguish a PLL tracking problem from a DC-link energy shortage when the delivered active power falls? Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 5, Lesson 2: Grid-forming voltage and frequency references

Course position: Renewables, grid-following and grid-forming inverters. 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 grid-forming inverter may establish an AC voltage waveform with dynamic controls emulating selected generator behavior, droop or virtual oscillator properties. It still has semiconductor current limits and finite DC-side energy. During severe faults, maintaining a clean voltage waveform and limiting current can be conflicting objectives. Grid-forming labels cover multiple proprietary control designs; one simple droop equation cannot describe every vendor implementation.

Governing model, boundaries and assumptions

Illustrative droop law: Delta f = −m_p (P−P_ref), with m_p in Hz/MW for the chosen base

Worked quantitative example

A toy grid-forming controller has f_ref=50 Hz, P_ref=2 MW and m_p=0.04 Hz/MW. At a commanded steady output P=3 MW the droop law gives f=49.96 Hz. At P=0.5 MW it predicts 50.06 Hz. This is an intentional static characteristic, not a transient frequency trace or a guarantee of island stability.

Interpretation and limiting cases

The droop coefficient has both units and a sign. Converting to per-unit requires explicit base frequency and rated power.

What could make this answer wrong?

If current saturation activates, the inverter may depart from the desired voltage-source characteristic. A controller that works during a small step may fail to support a large faulted island.

Applied comparison and design reasoning

Compare a battery-backed grid-forming unit and a PV-only unit near sunset: even identical controls need not offer the same energy sustainability.

Guided problems

Problem A — first-principles check. For fref=50 Hz, Pref=1 MW and slope 0.05 Hz/MW, calculate steady f at P=2 MW. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. State which measurement and current limit would invalidate an unconstrained droop output. 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. Describe an island acceptance test that demonstrates load sharing and recovery without making the unsafe leap from a classroom simulation to site commissioning. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 5, Lesson 3: Voltage support and inverter capability

Course position: Renewables, grid-following and grid-forming inverters. 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

Voltage sensitivity depends on the network resistance-reactance ratio, line loading, transformer taps and reactive resources. Inverter reactive control can improve local voltage when current margin exists, but feeder topology and real-power curtailment may matter as much as Q. Different grid codes and utility agreements specify ranges of capability, ride-through and response; these must be sourced for an actual project.

Governing model, boundaries and assumptions

In a simplified radial circuit, approximate voltage drop Delta V ≈ (R P + X Q)/V at a consistent per-phase base

Worked quantitative example

At V=1 pu and simple R=0.08 pu, X=0.12 pu, P=0.6 pu and Q=0.2 pu consumption, the approximate drop is 0.08×0.6+0.12×0.2=0.072 pu. Compensating 0.15 pu of Q lowers the approximate drop by 0.018 pu, to 0.054 pu. These are first-order small-drop approximations that omit voltage dependence and losses.

Interpretation and limiting cases

On resistive distribution networks, active power may strongly influence voltage. “Only reactive power controls voltage” is not universally valid.

What could make this answer wrong?

Treating a voltage-control point as identical to a customer service point can hide poor voltage at remote nodes or create a reverse-power overvoltage.

Applied comparison and design reasoning

Consider midday PV export versus evening import on the same feeder. Identify which measurement points and control conflicts should enter a Volt-VAR review.

Guided problems

Problem A — first-principles check. For R=0.02, X=0.06, P=0.8 and Q=0.3 pu at V≈1, calculate the first-order voltage drop. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain whether active power sensitivity can be neglected on a high-R/X feeder. 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 whether the current margin needed for the toy reactive injection is available from the inverter apparent-power capability at its existing active output. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 5, Lesson 4: Ride-through and protection coordination

Course position: Renewables, grid-following and grid-forming inverters. 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

During faults or abnormal frequency, renewable generators may be required to stay connected for specified operating envelopes, yet they must also protect hardware and personnel. The correct envelope is project-specific and evolves with applicable standards and contracts. System protection becomes complex when conventional overcurrent relays see limited converter fault current and controlled negative-sequence behavior. Proper validation combines electromagnetic and phasor-domain studies with field tests.

Governing model, boundaries and assumptions

For an illustrative inverter current limit I_max, |I_d+j I_q| ≤ I_max in a chosen synchronized reference frame

Worked quantitative example

Suppose current magnitude must stay at or below 1.2 pu. If the controller requests reactive current I_q=1.0 pu, the available active-axis current is at most sqrt(1.2²−1.0²)=0.663 pu. A controller requesting both I_q=1.0 and I_d=1.0 would request magnitude sqrt(2)=1.414 pu, exceeding the assumed limit.

Interpretation and limiting cases

The d-axis and q-axis meanings depend on controller alignment conventions. In the classroom, define the axes before associating d with active and q with reactive current.

What could make this answer wrong?

A ride-through requirement does not imply an obligation or physical capacity to sustain maximum real-power output during a voltage depression.

Applied comparison and design reasoning

A feeder was coordinated assuming fault current 6 pu from local generators. Replace some generators with current-limited converters and explain which protection studies must be repeated.

Guided problems

Problem A — first-principles check. At Imax=1.1 pu and requested Iq=1.0 pu, calculate permissible |Id|. 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 controller may curtail active output when required to prioritize reactive current. 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. Why can an inverter pass a steady reactive-capability test yet still fail a fast ride-through performance test? Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 5 project: Converter control and P-Q envelope screening

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

Assume a 5 MVA converter AC interface at a specified voltage. At P=1,2,3,4 and 5 MW compute maximum reactive magnitude from the circular apparent-power boundary. Include a current-priority fault scenario requesting Iq=1.0 pu and Id=0.8 pu with a total current ceiling of 1.1 pu. Compute the requested vector norm and demonstrate how allocating Iq priority changes possible Id. Sketch a small reactive setpoint sweep and model a feeder bus sensitivity DeltaV=0.03 DeltaQ.

Deliverables and acceptance checks

Create a capability table, plot, current circle, and decision memo explaining why a grid-forming label does not remove the converter current limit. List two questions needing actual grid-code and OEM data.

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 5 assessment: Renewables, grid-following and grid-forming inverters

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

Question group 1: Grid-following inverters and synchronization

Q1.1 — calculation. Convert a 5° phase mismatch to radians; multiply by a hypothetical proportional coefficient of 10 s^-1. 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 one-step proportional correction cannot establish closed-loop stability. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q1.3 — applied critique. Which recordings are essential to distinguish a PLL tracking problem from a DC-link energy shortage when the delivered active power falls?

Question group 2: Grid-forming voltage and frequency references

Q2.1 — calculation. For fref=50 Hz, Pref=1 MW and slope 0.05 Hz/MW, calculate steady f at P=2 MW. 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 which measurement and current limit would invalidate an unconstrained droop output. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q2.3 — applied critique. Describe an island acceptance test that demonstrates load sharing and recovery without making the unsafe leap from a classroom simulation to site commissioning.

Question group 3: Voltage support and inverter capability

Q3.1 — calculation. For R=0.02, X=0.06, P=0.8 and Q=0.3 pu at V≈1, calculate the first-order voltage drop. 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 whether active power sensitivity can be neglected on a high-R/X feeder. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q3.3 — applied critique. Calculate whether the current margin needed for the toy reactive injection is available from the inverter apparent-power capability at its existing active output.

Question group 4: Ride-through and protection coordination

Q4.1 — calculation. At Imax=1.1 pu and requested Iq=1.0 pu, calculate permissible |Id|. 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 a controller may curtail active output when required to prioritize reactive current. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q4.3 — applied critique. Why can an inverter pass a steady reactive-capability test yet still fail a fast ride-through performance test?

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 5: Renewables, grid-following and grid-forming inverters

High shares of wind, solar PV and batteries change grid behavior because power electronic converters often limit fault currents, follow control references and respond through algorithms rather than direct synchronous-machine electromechanics. A grid-following inverter uses a reference, often a phase-locked loop, to synchronize to an existing voltage waveform. A grid-forming controller establishes a controlled voltage/frequency reference within its own physical limits. Real projects require manufacturer models, protection coordination, commissioning studies and grid-code compliance; simple labels do not guarantee behavior under faulted conditions.

Deep reading — interacting limitations

Inverter controls are layered: an outer active/reactive target or frequency/voltage reference produces commands, inner current loops enforce limits, measurement filters condition grid quantities and protection overrides normal behavior during stress. Grid-following and grid-forming describe different reference roles, not blanket guarantees of grid support. The converter’s silicon and thermal limits remain regardless of marketing terminology. The distinction between a phase-locked current source and a voltage-reference-forming controller is most consequential when the external grid voltage is weak, absent or distorted. Modeling quality depends on use case: a quasi-static feeder study may suffice for daily Volt-VAR behavior, while a fast control interaction requires suitable electromagnetic representations. Report vendor-model availability as a limitation whenever it is missing.

Case synthesis: control architecture is not a substitute for current capacity

Suppose a feeder hosts a 5 MVA solar inverter producing 4 MW. Within a simple circular apparent-power envelope, its remaining reactive capability is sqrt(5²−4²)=3 MVAr. This is a useful steady-state calculation. If the inverter tries to provide 4 MVAr simultaneously, apparent demand becomes sqrt(4²+4²)=5.657 MVA and exceeds the assumed envelope. Active curtailment, reactive reduction or another permitted action must follow. The arithmetic alone does not reveal how a commercial controller prioritizes quantities under abnormal voltage, nor does it establish a grid-code obligation.

A grid-following inverter synchronizes to a measured network waveform, frequently through a PLL. In a weak connection its injected current affects local voltage and therefore changes the very signal used as a phase reference. This coupling can create dynamic interactions not represented by a static P–Q circle. A grid-forming inverter instead aims to establish a controlled voltage reference, but during a fault its semiconductor current limit still binds. The word grid-forming does not guarantee that a storage-poor plant can sustain an islanded load or that it can tolerate unlimited faults. Different manufacturers implement distinct current-limiting and recovery strategies.

To make the comparison quantitative at a basic level, represent the feeder with approximate R=0.05 pu, X=0.08 pu, net P=0.6 pu and Q demand=0.3 pu at 1 pu source voltage. A first-order drop estimate is 0.05×0.6+0.08×0.3=0.054 pu. Reactive injection of 0.2 pu could reduce the local approximation by 0.016 pu, but the effect varies with voltage, power-flow direction and actual feeder impedance. Midday PV reverse power may raise voltage and alter which control action is beneficial. A transformer tap changer may change voltage on a slower timeline; competing fast inverter VAR actions can influence its mechanical operation count.

A credible integration study should distinguish the hours of normal operation, voltage excursions, unbalanced faults, fast electromagnetic interactions and restoration after disturbances. It needs the actual interconnection agreement and accepted dynamic model; it cannot infer current limits, ride-through envelopes or protection behavior from a generic diagram. Record controller version, control mode, voltage/current limits, sensor assumptions, source strength and validation scenario before reporting inverter hosting capability.

Unit 5, Lesson 1: Grid-following inverters and synchronization

Course position: Renewables, grid-following and grid-forming inverters. 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 typical grid-following inverter measures network voltage phase and commands a current injection to achieve active/reactive targets. The phase-locked loop can be stressed when the grid voltage is weak, distorted or rapidly changing. Current controls and DC-link limits regulate what actually reaches the network. A correctly behaving current-controlled inverter cannot deliver arbitrary fault current just because a classical generator-equivalent study predicts it.

Governing model, boundaries and assumptions

A stylized PLL phase error e=theta_grid−theta_hat; frequency estimate adjusts with a selected control transfer function

Worked quantitative example

Suppose a grid voltage phase steps by 10° relative to an inverter estimate. A proportional-only toy PLL with angular-frequency correction K_p e and K_p=20 s^-1 gives initial correction ≈20×0.1745=3.49 rad/s, about 0.556 Hz. This is not a real commercial PLL controller and does not predict stable recovery without integral gains, filtering, saturation and delay.

Interpretation and limiting cases

The PLL angle is an estimate with measurement bandwidth and filtering; it is not an independently perfect network angle.

What could make this answer wrong?

In a low-short-circuit-strength area, current injection changes terminal voltage, which alters the input to the PLL. Ignoring this feedback can hide oscillation or loss-of-synchronization risks.

Applied comparison and design reasoning

Place the same inverter behind a strong and a high-impedance grid equivalent. Explain qualitatively how terminal-voltage sensitivity to reactive current changes.

Guided problems

Problem A — first-principles check. Convert a 5° phase mismatch to radians; multiply by a hypothetical proportional coefficient of 10 s^-1. 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 one-step proportional correction cannot establish closed-loop stability. 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. Which recordings are essential to distinguish a PLL tracking problem from a DC-link energy shortage when the delivered active power falls? Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 5, Lesson 2: Grid-forming voltage and frequency references

Course position: Renewables, grid-following and grid-forming inverters. 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 grid-forming inverter may establish an AC voltage waveform with dynamic controls emulating selected generator behavior, droop or virtual oscillator properties. It still has semiconductor current limits and finite DC-side energy. During severe faults, maintaining a clean voltage waveform and limiting current can be conflicting objectives. Grid-forming labels cover multiple proprietary control designs; one simple droop equation cannot describe every vendor implementation.

Governing model, boundaries and assumptions

Illustrative droop law: Delta f = −m_p (P−P_ref), with m_p in Hz/MW for the chosen base

Worked quantitative example

A toy grid-forming controller has f_ref=50 Hz, P_ref=2 MW and m_p=0.04 Hz/MW. At a commanded steady output P=3 MW the droop law gives f=49.96 Hz. At P=0.5 MW it predicts 50.06 Hz. This is an intentional static characteristic, not a transient frequency trace or a guarantee of island stability.

Interpretation and limiting cases

The droop coefficient has both units and a sign. Converting to per-unit requires explicit base frequency and rated power.

What could make this answer wrong?

If current saturation activates, the inverter may depart from the desired voltage-source characteristic. A controller that works during a small step may fail to support a large faulted island.

Applied comparison and design reasoning

Compare a battery-backed grid-forming unit and a PV-only unit near sunset: even identical controls need not offer the same energy sustainability.

Guided problems

Problem A — first-principles check. For fref=50 Hz, Pref=1 MW and slope 0.05 Hz/MW, calculate steady f at P=2 MW. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. State which measurement and current limit would invalidate an unconstrained droop output. 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. Describe an island acceptance test that demonstrates load sharing and recovery without making the unsafe leap from a classroom simulation to site commissioning. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 5, Lesson 3: Voltage support and inverter capability

Course position: Renewables, grid-following and grid-forming inverters. 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

Voltage sensitivity depends on the network resistance-reactance ratio, line loading, transformer taps and reactive resources. Inverter reactive control can improve local voltage when current margin exists, but feeder topology and real-power curtailment may matter as much as Q. Different grid codes and utility agreements specify ranges of capability, ride-through and response; these must be sourced for an actual project.

Governing model, boundaries and assumptions

In a simplified radial circuit, approximate voltage drop Delta V ≈ (R P + X Q)/V at a consistent per-phase base

Worked quantitative example

At V=1 pu and simple R=0.08 pu, X=0.12 pu, P=0.6 pu and Q=0.2 pu consumption, the approximate drop is 0.08×0.6+0.12×0.2=0.072 pu. Compensating 0.15 pu of Q lowers the approximate drop by 0.018 pu, to 0.054 pu. These are first-order small-drop approximations that omit voltage dependence and losses.

Interpretation and limiting cases

On resistive distribution networks, active power may strongly influence voltage. “Only reactive power controls voltage” is not universally valid.

What could make this answer wrong?

Treating a voltage-control point as identical to a customer service point can hide poor voltage at remote nodes or create a reverse-power overvoltage.

Applied comparison and design reasoning

Consider midday PV export versus evening import on the same feeder. Identify which measurement points and control conflicts should enter a Volt-VAR review.

Guided problems

Problem A — first-principles check. For R=0.02, X=0.06, P=0.8 and Q=0.3 pu at V≈1, calculate the first-order voltage drop. Show the relationship used, a fully labeled unit conversion if needed, the computed result and an interpretive sentence.

Problem B — alternate assumption. Explain whether active power sensitivity can be neglected on a high-R/X feeder. 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 whether the current margin needed for the toy reactive injection is available from the inverter apparent-power capability at its existing active output. Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 5, Lesson 4: Ride-through and protection coordination

Course position: Renewables, grid-following and grid-forming inverters. 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

During faults or abnormal frequency, renewable generators may be required to stay connected for specified operating envelopes, yet they must also protect hardware and personnel. The correct envelope is project-specific and evolves with applicable standards and contracts. System protection becomes complex when conventional overcurrent relays see limited converter fault current and controlled negative-sequence behavior. Proper validation combines electromagnetic and phasor-domain studies with field tests.

Governing model, boundaries and assumptions

For an illustrative inverter current limit I_max, |I_d+j I_q| ≤ I_max in a chosen synchronized reference frame

Worked quantitative example

Suppose current magnitude must stay at or below 1.2 pu. If the controller requests reactive current I_q=1.0 pu, the available active-axis current is at most sqrt(1.2²−1.0²)=0.663 pu. A controller requesting both I_q=1.0 and I_d=1.0 would request magnitude sqrt(2)=1.414 pu, exceeding the assumed limit.

Interpretation and limiting cases

The d-axis and q-axis meanings depend on controller alignment conventions. In the classroom, define the axes before associating d with active and q with reactive current.

What could make this answer wrong?

A ride-through requirement does not imply an obligation or physical capacity to sustain maximum real-power output during a voltage depression.

Applied comparison and design reasoning

A feeder was coordinated assuming fault current 6 pu from local generators. Replace some generators with current-limited converters and explain which protection studies must be repeated.

Guided problems

Problem A — first-principles check. At Imax=1.1 pu and requested Iq=1.0 pu, calculate permissible |Id|. 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 controller may curtail active output when required to prioritize reactive current. 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. Why can an inverter pass a steady reactive-capability test yet still fail a fast ride-through performance test? Write a short technical response containing one quantitative or logical check, one serious counterargument, and one criterion that would falsify your initial conclusion.

Unit 5 project: Converter control and P-Q envelope screening

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

Assume a 5 MVA converter AC interface at a specified voltage. At P=1,2,3,4 and 5 MW compute maximum reactive magnitude from the circular apparent-power boundary. Include a current-priority fault scenario requesting Iq=1.0 pu and Id=0.8 pu with a total current ceiling of 1.1 pu. Compute the requested vector norm and demonstrate how allocating Iq priority changes possible Id. Sketch a small reactive setpoint sweep and model a feeder bus sensitivity DeltaV=0.03 DeltaQ.

Deliverables and acceptance checks

Create a capability table, plot, current circle, and decision memo explaining why a grid-forming label does not remove the converter current limit. List two questions needing actual grid-code and OEM data.

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 5 assessment: Renewables, grid-following and grid-forming inverters

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

Question group 1: Grid-following inverters and synchronization

Q1.1 — calculation. Convert a 5° phase mismatch to radians; multiply by a hypothetical proportional coefficient of 10 s^-1. 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 one-step proportional correction cannot establish closed-loop stability. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q1.3 — applied critique. Which recordings are essential to distinguish a PLL tracking problem from a DC-link energy shortage when the delivered active power falls?

Question group 2: Grid-forming voltage and frequency references

Q2.1 — calculation. For fref=50 Hz, Pref=1 MW and slope 0.05 Hz/MW, calculate steady f at P=2 MW. 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 which measurement and current limit would invalidate an unconstrained droop output. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q2.3 — applied critique. Describe an island acceptance test that demonstrates load sharing and recovery without making the unsafe leap from a classroom simulation to site commissioning.

Question group 3: Voltage support and inverter capability

Q3.1 — calculation. For R=0.02, X=0.06, P=0.8 and Q=0.3 pu at V≈1, calculate the first-order voltage drop. 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 whether active power sensitivity can be neglected on a high-R/X feeder. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q3.3 — applied critique. Calculate whether the current margin needed for the toy reactive injection is available from the inverter apparent-power capability at its existing active output.

Question group 4: Ride-through and protection coordination

Q4.1 — calculation. At Imax=1.1 pu and requested Iq=1.0 pu, calculate permissible |Id|. 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 a controller may curtail active output when required to prioritize reactive current. Explain which term changes, then compare the result with the original simplified physical interpretation.

Q4.3 — applied critique. Why can an inverter pass a steady reactive-capability test yet still fail a fast ride-through performance test?

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.