Exercise set

Power Flow and Grid Stability Exercises

Solved power flow and grid stability exercises for per-unit bases, line loading, voltage drop, reactive power, N-1 capacity, inertia, RoCoF and frequency response.

Branch: Energy Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.0.1 · reviewed

These exercises practise power-flow and grid-stability calculations as network evidence. The goal is not to replace load-flow software or dynamic simulation. The goal is to make first-pass checks explicit: base values, current, line loading, voltage drop, reactive capability, N-1 capacity, short-circuit strength, inertia, RoCoF, frequency response and release margins.

Assume balanced three-phase operation and simplified screening models unless an exercise states otherwise. Real grid studies should also include network topology, transformer taps, voltage-control modes, contingency cases, validated dynamic models, protection settings, grid-code requirements, harmonic limits, commissioning evidence and measurement uncertainty.

How to Use These Exercises

For each problem, state:

the electrical boundary and voltage base;
whether the result is current, MVA, per-unit impedance, voltage deviation, frequency deviation or response headroom;
the operating case, such as normal, N-1, weak grid, export, import, disturbance or commissioning;
the acceptance gate;
which detailed study or field evidence would be needed before release.

The common mistake is treating a MW value as the whole grid answer. Voltage, reactive power, current, fault strength, inertia, converter limits and protection can control the decision even when active power appears acceptable.

Release Evidence Notes

Power-flow and grid-stability calculations should be tied to an explicit electrical boundary. A value at generator terminals is not the same as a value at inverter AC terminals, transformer high-voltage side, collector bus, distribution feeder, transmission bus or point of interconnection. Record the voltage base, MVA base, transformer tap, operating voltage, dispatch state, power-factor or reactive-power mode, online equipment, switching state and applicable contingency before comparing any result with a limit.

Steady-state screens and dynamic screens answer different questions. A feeder-loading calculation can show that conductors and transformers are not overloaded. It does not prove voltage stability, frequency stability, converter control stability, protection selectivity, harmonic compliance or ride-through performance. Similarly, a RoCoF or nadir screen can show that a credible disturbance may be survivable, but it does not replace validated dynamic models, governor data, inverter firmware settings, load-damping assumptions, event-recorder channels or grid-code compliance tests.

Weak-grid decisions require special care because several margins can collapse together. Low short-circuit ratio, high feeder impedance, limited reactive headroom, aggressive voltage control, inverter current priority, transformer tap constraints and protection sensitivity can interact. A single acceptable SCR number should not be used as release evidence if voltage rise, reactive margin, harmonics or protection settings are already close to limits.

For engineering review, separate normal operation, credible N-1 contingencies, minimum-fault cases, maximum-fault cases, export cases, import cases, islanded cases and commissioning tests. Each case may be controlled by a different limit. A case that passes thermal loading may fail voltage rise. A case that passes breaker interrupting duty may fail protection sensitivity at minimum fault. A case that passes RoCoF may still fail reactive headroom during voltage support.

The practical release question is whether current loading, voltage range, reactive capability, short-circuit strength, inertia, frequency response, protection ratings, control settings and field evidence all support the same operating state. If they do not, the result should trigger derating, curtailment, revised settings, reinforcement, additional dynamic study, commissioning retest or refusal to release the operating case.

Engineering Boundary Notes

These exercises use simplified balanced three-phase, steady-state and dynamic screens. They do not replace a validated load-flow case, short-circuit study, protection-coordination study, electromagnetic transient model, harmonic study, grid-code compliance package or commissioning test. A passing screen is a triage result, not final grid approval.

Common Release Mistakes

comparing per-unit quantities from different MVA or voltage bases;
using MW loading while ignoring MVA, voltage, reactive power and current limits;
treating one normal case as proof of N-1 or weak-grid acceptability;
checking maximum fault duty but not minimum-fault protection sensitivity;
quoting RoCoF or inertia without governor, inverter and load-response evidence;
accepting SCR while reactive headroom, harmonics or voltage control are already weak.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Per-unit and loading	1, 2, 3, 4	base current, impedance base, line current and transformer loading	Establish common bases before comparing equipment.
Voltage and reactive power	5, 6, 7, 8	voltage drop/rise, apparent-power circle and capacitor/inverter support	Decide whether voltage control has enough headroom.
Contingency and strength	9, 10, 11, 12	N-1 capacity, short-circuit level, SCR and breaker margin	Check whether the network is strong and rated enough.
Dynamic stability	13, 14, 15, 16, 17	inertia, RoCoF, droop, primary response and reserve	Decide whether the system can survive credible disturbances.
Release evidence	18	combined voltage, current, SCR, RoCoF and reactive margins	Accept, derate or hold the operating case.

Validation Package Checklist

electrical boundary, MVA base, voltage base and operating state are stated;
topology, transformer taps, dispatch and contingency assumptions are recorded;
current, voltage, MVA, reactive and thermal limits are checked together;
short-circuit strength, maximum fault duty and minimum-fault sensitivity are separated;
inertia, RoCoF, frequency response and reserve assumptions are documented;
protection, harmonic, ride-through and commissioning evidence are identified;
final release decision states accept, derate, curtail, reinforce, retest or hold.

Exercise 1: Per-Unit Current Base

A study uses a three-phase base of:

S_{base}=100\ \text{MVA}

and:

V_{base}=132\ \text{kV}

Find the line-current base.

Solution

For a three-phase system:

\displaystyle I_{base}=\frac{S_{base}}{\sqrt{3}V_{base}}

Using MVA and kV gives kA:

\displaystyle I_{base}=\frac{100}{\sqrt{3}\times 132}

I_{base}=0.437\ \text{kA}

So:

I_{base}=437\ \text{A}

Engineering Comment

Current base must match the selected voltage base. A per-unit current calculated on the wrong transformer side can make a conductor, CT or breaker appear incorrectly loaded.

Plausibility Check

A $100\ \text{MVA}$ circuit at high voltage should have current in the hundreds of amperes, not thousands. The result is plausible.

Exercise 2: Impedance Base and Per-Unit Line Reactance

On the same $100\ \text{MVA}$ , $132\ \text{kV}$ base, a line has reactance:

X=38\ \Omega

Find $Z_{base}$ and $X_{pu}$ .

Solution

Impedance base:

\displaystyle Z_{base}=\frac{V_{base}^2}{S_{base}}

Using kV and MVA gives ohms:

\displaystyle Z_{base}=\frac{132^2}{100}=174.24\ \Omega

Per-unit reactance:

\displaystyle X_{pu}=\frac{38}{174.24}=0.218

Engineering Comment

Per-unit conversion makes the network easier to compare across voltage levels, but the base must be documented. A per-unit value without base data is not reviewable.

Plausibility Check

The actual reactance is about one fifth of the base impedance, so $0.218\ pu$ is credible.

Exercise 3: Three-Phase Line Current from MVA

A feeder carries:

S=42\ \text{MVA}

at:

V_{LL}=33\ \text{kV}

Estimate line current.

Solution

Three-phase apparent power:

S=\sqrt{3}V_{LL}I

Therefore:

\displaystyle I=\frac{S}{\sqrt{3}V_{LL}}

\displaystyle I=\frac{42}{\sqrt{3}\times 33}=0.735\ \text{kA}

So:

I=735\ \text{A}

Engineering Comment

Apparent power controls current loading, not MW alone. This current must be compared with conductor ampacity, transformer rating, switchgear rating and protection settings.

Plausibility Check

Tens of MVA at $33\ \text{kV}$ usually produce hundreds of amperes. The result fits that scale.

Exercise 4: Transformer Loading and Margin

A transformer is rated $75\ \text{MVA}$ . A contingency case loads it to $68\ \text{MVA}$ . Find percent loading and MVA margin.

Solution

Percent loading:

\displaystyle L_{\%}=100\frac{68}{75}

L_{\%}=90.7\%

MVA margin:

M_S=75-68=7\ \text{MVA}

Engineering Comment

A $7\ \text{MVA}$ margin may be acceptable or weak depending on ambient temperature, cooling state, overload rules, harmonic loading and forecast growth. A thermal rating is not only a nameplate number.

Plausibility Check

$68$ is a little over nine tenths of $75$ , so $90.7\%$ loading is plausible.

Exercise 5: Simplified Voltage Drop

A feeder has:

R_{pu}=0.035,\quad X_{pu}=0.110

The load is:

P_{pu}=0.70,\quad Q_{pu}=0.30

Estimate voltage drop:

\Delta V_{pu}\approx R_{pu}P_{pu}+X_{pu}Q_{pu}

Solution

Compute:

\Delta V_{pu}=0.035(0.70)+0.110(0.30)

\Delta V_{pu}=0.0245+0.0330=0.0575

So the approximate voltage drop is:

5.75\%

Engineering Comment

Reactive power dominates this voltage drop because the feeder is more reactive than resistive. Voltage support may be more effective than active-power curtailment in this case.

Plausibility Check

Both terms are positive for load, and the result is a few percent, which is typical for a stressed feeder screen.

Exercise 6: Voltage Rise from Export

A distributed generator exports:

P_{pu}=0.45

with:

Q_{pu}=0

on a feeder with:

R_{pu}=0.060,\quad X_{pu}=0.080

Estimate voltage rise from active export using:

\Delta V_{pu}\approx R_{pu}P_{pu}

Solution

Compute:

\Delta V_{pu}=0.060(0.45)=0.027

Voltage rise is:

2.7\%

Engineering Comment

Voltage rise can limit distributed generation even when thermal current is acceptable. The sign convention should be explicit: export tends to raise voltage at the local bus on a resistive feeder.

Plausibility Check

The export is less than half per unit, and the resistance is modest, so a few percent rise is plausible.

Exercise 7: Reactive Power Headroom of an Inverter

An inverter has apparent-power rating:

S_{max}=50\ \text{MVA}

It is exporting:

P=42\ \text{MW}

Find maximum reactive power magnitude:

Q_{max}=\sqrt{S_{max}^2-P^2}

Solution

Compute:

Q_{max}=\sqrt{50^2-42^2}

Q_{max}=\sqrt{2500-1764}=\sqrt{736}=27.1\ \text{MVAr}

Engineering Comment

Reactive support consumes inverter current headroom. If the plant must provide more reactive power, it may need active-power curtailment or a larger converter.

Plausibility Check

At $42\ \text{MW}$ on a $50\ \text{MVA}$ circle, there is still meaningful but not unlimited reactive margin. $27\ \text{MVAr}$ is credible.

Exercise 8: Active-Power Curtailment for Reactive Support

The same inverter must provide:

Q=35\ \text{MVAr}

while staying within:

S_{max}=50\ \text{MVA}

Find the maximum active power.

Solution

Use:

P_{max}=\sqrt{S_{max}^2-Q^2}

P_{max}=\sqrt{50^2-35^2}

P_{max}=\sqrt{2500-1225}=\sqrt{1275}=35.7\ \text{MW}

Engineering Comment

If the plant was scheduled for $42\ \text{MW}$ , it cannot provide $35\ \text{MVAr}$ at the same time without exceeding MVA rating. Voltage-support obligations should be included in export forecasts.

Plausibility Check

Providing large reactive power leaves less active-power headroom. The active limit dropping below $40\ \text{MW}$ is expected.

Exercise 9: N-1 Firm Capacity

A substation has three transformers rated:

40,\quad 40,\quad 25\ \text{MVA}

Estimate firm N-1 capacity by subtracting the largest unit from total installed capacity.

Solution

Total capacity:

S_{total}=40+40+25=105\ \text{MVA}

Largest unit:

S_{largest}=40\ \text{MVA}

N-1 firm capacity:

S_{firm,N-1}=105-40=65\ \text{MVA}

Engineering Comment

This is a capacity screen, not a full contingency study. Bus arrangement, parallel loading, impedance mismatch, protection, cooling and physical separation can reduce usable firm capacity.

Plausibility Check

Losing one of the two $40\ \text{MVA}$ units leaves the other $40\ \text{MVA}$ unit and the $25\ \text{MVA}$ unit, totaling $65\ \text{MVA}$ .

Exercise 10: Short-Circuit Ratio

A renewable plant has rating:

S_{plant}=120\ \text{MVA}

The grid short-circuit level at the point of interconnection is:

S_{sc}=520\ \text{MVA}

Find short-circuit ratio:

\displaystyle SCR=\frac{S_{sc}}{S_{plant}}

Solution

Compute:

\displaystyle SCR=\frac{520}{120}=4.33

Engineering Comment

An SCR of $4.33$ is not extremely weak, but it still requires inverter-model validation, control-interaction review and grid-code evidence. SCR is a warning metric, not a proof of stable operation.

Plausibility Check

The grid fault level is a little over four times the plant rating, so the ratio near $4.3$ is expected.

Exercise 11: Fault Current from Short-Circuit MVA

A bus has:

S_{sc}=750\ \text{MVA}

at:

V_{LL}=66\ \text{kV}

Estimate symmetrical three-phase fault current.

Solution

Use:

\displaystyle I_{sc}=\frac{S_{sc}}{\sqrt{3}V_{LL}}

\displaystyle I_{sc}=\frac{750}{\sqrt{3}\times 66}=6.56\ \text{kA}

Engineering Comment

This current is a screening value. Breaker duty also depends on X/R ratio, DC offset, making duty, clearing time, future reinforcement and contribution from local machines or converters.

Plausibility Check

Hundreds of MVA at tens of kV should produce several kiloamperes. $6.56\ \text{kA}$ is plausible.

Exercise 12: Breaker Interrupting Margin

A breaker is rated:

I_{rating}=25\ \text{kA}

The studied maximum symmetrical fault current is:

I_{fault}=21.5\ \text{kA}

Find interrupting margin.

Solution

Margin:

M=I_{rating}-I_{fault}

M=25-21.5=3.5\ \text{kA}

Percent margin relative to rating:

\displaystyle \frac{3.5}{25}\times 100=14\%

Engineering Comment

The positive margin is useful but not final. If the grid is reinforced, if motors are added, or if X/R produces high peak duty, the breaker may need reevaluation.

Plausibility Check

$21.5\ \text{kA}$ is below $25\ \text{kA}$ , so the margin should be positive but modest.

Exercise 13: Stored Kinetic Energy from Inertia Constant

A synchronous unit has:

H=4.5\ \text{s}

on a:

S_{rated}=200\ \text{MVA}

base. Estimate stored kinetic energy:

E_k=HS_{rated}

Solution

Compute:

E_k=4.5 \times 200=900\ \text{MW s}

This is also:

900\ \text{MJ}

because $1\ \text{MW s}=1\ \text{MJ}$ .

Engineering Comment

Inertia supports frequency during the first moments after an imbalance. It does not replace primary response, reserve, protection or governor action.

Plausibility Check

Large rotating machines commonly store hundreds to thousands of megajoules, so $900\ \text{MJ}$ is credible.

Exercise 14: Initial RoCoF After a Generation Loss

A system has aggregate inertia energy:

E_k=18{,}000\ \text{MW s}

At nominal frequency:

f_0=50\ \text{Hz}

a generation trip creates imbalance:

\Delta P=450\ \text{MW}

Estimate initial RoCoF magnitude using:

\displaystyle \left|\frac{df}{dt}\right|\approx \frac{f_0\Delta P}{2E_k}

Solution

Compute:

\displaystyle \left|\frac{df}{dt}\right|=\frac{50 \times 450}{2 \times 18{,}000}

\displaystyle \left|\frac{df}{dt}\right|=\frac{22{,}500}{36{,}000}=0.625\ \text{Hz/s}

Engineering Comment

RoCoF is sensitive to both imbalance size and inertia. Low-inertia systems need faster response, grid-forming support, smaller credible contingencies or adjusted protection settings.

Plausibility Check

The contingency is significant but the inertia is not tiny. A RoCoF below $1\ \text{Hz/s}$ is plausible.

Exercise 15: Droop Frequency Response

A resource has droop response:

K=120\ \text{MW/Hz}

for frequency deviations outside a deadband. Frequency falls from $50.00\ \text{Hz}$ to $49.82\ \text{Hz}$ . Estimate active-power increase.

Solution

Frequency deviation:

\Delta f=50.00-49.82=0.18\ \text{Hz}

Response:

\Delta P=K\Delta f

\Delta P=120(0.18)=21.6\ \text{MW}

Engineering Comment

This response is useful only if the resource has headroom, controls enabled, ramp capability and measurement quality to deliver it. Droop gain alone is not service evidence.

Plausibility Check

A $0.18\ \text{Hz}$ deviation is less than a quarter hertz. At $120\ \text{MW/Hz}$ , a response in the low tens of MW is plausible.

Exercise 16: Primary Reserve Sufficiency

A system operator requires primary response of:

180\ \text{MW}

within the response window. Available providers can deliver:

60,\quad 45,\quad 35,\quad 25\ \text{MW}

Check sufficiency and deficit.

Solution

Available response:

P_{avail}=60+45+35+25=165\ \text{MW}

Deficit:

180-165=15\ \text{MW}

The response is insufficient by $15\ \text{MW}$ .

Engineering Comment

The system may need additional reserve, lower contingency size, faster storage response, demand response or operating restriction. Counting unavailable or already-saturated resources would create a false stability margin.

Plausibility Check

The four resources nearly meet the requirement, so a small deficit is consistent.

Exercise 17: Frequency Nadir Screening

A simplified event model estimates frequency nadir as:

f_{nadir}=f_0-\Delta f

with:

f_0=50.0\ \text{Hz},\quad \Delta f=0.72\ \text{Hz}

The under-frequency load-shedding threshold is $49.2\ \text{Hz}$ . Check the margin.

Solution

Nadir:

f_{nadir}=50.0-0.72=49.28\ \text{Hz}

Margin to threshold:

M=49.28-49.2=0.08\ \text{Hz}

The simplified screen passes with $0.08\ \text{Hz}$ margin.

Engineering Comment

The margin is narrow. Detailed dynamic simulation, governor response, load damping, relay settings and measurement uncertainty should be checked before approving the contingency.

Plausibility Check

The frequency drop is less than $1\ \text{Hz}$ , and the nadir remains slightly above the trip threshold, so the small positive margin is credible.

Exercise 18: Grid Stability Release Gate

A proposed operating case has the following evidence:

Check	Result	Gate
Feeder loading	$94\%$	$\le 95\%$
Voltage rise	$3.8\%$	$\le 3.0\%$
SCR	$3.4$	$\ge 3.0$
RoCoF	$0.72\ \text{Hz/s}$	$\le 0.8\ \text{Hz/s}$
Reactive headroom	$9\ \text{MVAr}$	$\ge 12\ \text{MVAr}$

Decide whether the case can be released as-is.

Solution

Feeder loading:

94\%\le 95\%

passes.

Voltage rise:

3.8\%>3.0\%

fails.

SCR:

3.4\ge 3.0

passes.

RoCoF:

0.72\le 0.8

passes.

Reactive headroom:

9<12

fails.

The operating case should not be released as-is. It needs voltage mitigation, reactive support, active-power curtailment, network reinforcement, control change or a restricted export limit.

Engineering Comment

Three checks pass, but the failed checks are directly tied to voltage control. Releasing from feeder loading and RoCoF alone would ignore the grid condition most likely to violate the interconnection requirement.

Plausibility Check

The case is close to limits in multiple places. A hold or derate decision is consistent with the failed voltage-rise and reactive-headroom gates.

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