Chapter 8 • Theory & Derivations

Economic Dispatch, Loss Penalty Factors, and Optimal Generation

Comprehensive university textbook unit covering fundamental principles, mathematical formulations, and practical engineering applications of Economic Dispatch, Loss Penalty Factors, and Optimal Generation.

§8.1 Voltage variation, harmonics and flicker

Unit 8, lesson 1: Voltage variation, harmonics and flicker

Learning objectives: analyze sags, swells, flicker, nonlinear loads, total harmonic distortion; interpret THD_I = sqrt(sum_{h>=2} I_h^2)/I_1 for rms harmonics and report valid application limits.

1. The physical system

A deliverable grid design must survive measurement uncertainty, protective obligations, future demand growth and contingency conditions. Comparing a single idealized load-flow screenshot to nominal equipment ratings is not enough to establish safety, quality or reliability. The topic Voltage variation, harmonics and flicker concerns sags, swells, flicker, nonlinear loads, total harmonic distortion. An analyst must decide which variables are directly measured and which are inferred from a network equivalent. Sketch the equipment and specify its boundary, phase reference, operating condition and sign convention. Identify the path that carries energy through the system and explain what would change if a switch state, load characteristic or source excitation were different.

In this setting, sags describes a distinct physical aspect from swells. Mixing them obscures whether the result depends on component values or system operation. Start with the equipment location and an inventory of observed voltage, current, power, time or geometry as appropriate. Then explain why the simplified relation is sufficient for the particular question being asked, rather than assuming it describes all possible network states.

2. Mathematical formulation

Working relation: THD_I = sqrt(sum_{h>=2} I_h^2)/I_1 for rms harmonics.

Trace each variable to a specified input or a calculated intermediate quantity. Where a phasor is used, distinguish rms magnitude and complex phase; where a network is balanced, say so explicitly; where a per-unit base is used, specify base MVA and line-to-line base kV. Note that mathematical equivalence does not establish physical validity: a fitted impedance is an effective model at a given frequency, temperature and topology, while the equipment rating sets additional limits beyond the equation.

Evaluate limiting cases that apply to the stated relation: reduced load, increased impedance, vanishing angle difference, or alternative grounding. Do not apply all these checks indiscriminately; choose two that make physical sense here. Check that the result has units consistent with sags and that conversion between MW, MVA, kV, A, ohms and per-unit values has been performed correctly. The result should be reproducible from the written assumptions without guessing any hidden convention.

3. Worked calculation

Example. If fundamental current is 100 A rms and the only measured harmonic is 10 A rms, current THD is 10%.

Interpretation. Harmonic distortion changes waveform quality; real heating depends on harmonic frequency and equipment design. For independent verification, re-evaluate the stated numerical inputs using the working relation and compare to the quoted magnitude. Record whether the calculated result is a steady-state operating point, an initial symmetrical fault estimate, a long-term energy statistic or a purely geometric property. Re-run the example with a ten-percent change in its principal variable, and explain why that percentage change in the input does not necessarily yield the same percentage change in the output.

Why precision matters. Do not compare an approximate computation with a nameplate limit by simply treating all of the presented digits as exact. A load measurement may have a sampling interval, a component impedance depends on conditions, and an ideal transformer omits losses. If the result is close to a design limit, obtain verified measurements and a more complete model before concluding that a configuration is feasible.

4. Reliability of the approximation

Common technical mistake: A single THD number does not identify a dangerous resonance or establish compliance at every harmonic order.

Verification procedure: Compare time-domain waveforms and frequency spectra for an ideal sinusoid and an added fifth harmonic. Write a minimal input ledger that records variable name, value, SI unit, data source, time and uncertainty. Select one independent calculation, measurement or conserved quantity with which to check the result. If the two approaches disagree, determine whether the difference comes from sign conventions, base changes, a model omitted term, poor timing alignment or a calculation error. This is an essential distinction: a numerically stable solver can still give a wrong answer when it solves the wrong network.

Before relying on voltage variation, harmonics and flicker for equipment selection, establish the relevant continuous rating, short-time limit and the applicable operating constraints. For models with no dynamic terms, identify at least one fast disturbance they cannot predict. For models based on averages, identify at least one short interval where the average hides a high-risk condition. State the evidence still missing, rather than claiming that numerical plausibility proves system safety or reliability.

5. Applied case study

Engineering problem. An inverter supplier shows low current THD in a lab; identify point-of-common-coupling data still required.

Prepare two competing hypotheses for the scenario. Under the first, the original proposal is acceptable given the displayed model and assumed input range. Under the second, another loading condition, equipment limit, network state or source behavior reverses the conclusion. Design a controlled numerical experiment that would distinguish them: choose a measurable output, make one intentional parameter change at a time and hold all other stated quantities fixed. Explain when holding a variable fixed would be unrealistic because it is coupled to the changed parameter.

Finally, produce a short decision memo with a numerical range, not merely a single estimate. Include a base case, adverse case and data-quality concern related to total harmonic distortion. Report whether the simplified model is sufficient for teaching, preliminary screening or detailed engineering. If it is not sufficient, name the additional circuit, measurement or operational constraint that would be required.

6. Questions and guided responses

1. Physical reasoning. How are sags and swells connected in this model? Guidance: start with the labelled boundary, name both units and identify the causal path rather than merely repeating a formula.

2. Numerical problem. Recalculate the example using `THD_I = sqrt(sum_{h>=2} I_h^2)/I_1 for rms harmonics` and then increase its principal source or load input by ten percent. Guidance: report both numerical values, check signs and dimensions and explain the relevant nonlinear terms.

3. Model validity. Give two conditions under which the relation may cease to be accurate for voltage variation, harmonics and flicker. Guidance: distinguish omitted physical mechanisms from bad data and specify an independent test.

4. Design judgment. Apply the relation to the case study and explain what one additional observation would change your decision. Guidance: compare equipment limits with credible normal and contingency operating states.

5. Cross-topic reasoning. Explain how total harmonic distortion would affect one other lesson in Unit 8. Guidance: name the shared state variable and show how the same value propagates to a separate mathematical model.

7. Study extension

Build a small sensitivity table: base case, 90-percent value of the most uncertain input, 110-percent value and one changed equipment or topology configuration. For each row, first predict the direction of the output change and then calculate it. Write a brief explanation for any result that does not follow your prediction. Mark a case “outside model validity” if its assumptions are contradicted, instead of forcing the calculation to produce a persuasive number. The goal is to understand both the use and limits of voltage variation, harmonics and flicker in an interconnected electrical system.

§8.2 Equipment limits and operational margins

Unit 8, lesson 2: Equipment limits and operational margins

Learning objectives: analyze continuous ratings, emergency ratings, insulation limits, breaker duties, ambient correction; interpret Margin_% = 100*(rating - measured_use)/rating for a positive comparable rating and report valid application limits.

1. The physical system

A deliverable grid design must survive measurement uncertainty, protective obligations, future demand growth and contingency conditions. Comparing a single idealized load-flow screenshot to nominal equipment ratings is not enough to establish safety, quality or reliability. The topic Equipment limits and operational margins concerns continuous ratings, emergency ratings, insulation limits, breaker duties, ambient correction. An analyst must decide which variables are directly measured and which are inferred from a network equivalent. Sketch the equipment and specify its boundary, phase reference, operating condition and sign convention. Identify the path that carries energy through the system and explain what would change if a switch state, load characteristic or source excitation were different.

In this setting, continuous ratings describes a distinct physical aspect from emergency ratings. Mixing them obscures whether the result depends on component values or system operation. Start with the equipment location and an inventory of observed voltage, current, power, time or geometry as appropriate. Then explain why the simplified relation is sufficient for the particular question being asked, rather than assuming it describes all possible network states.

2. Mathematical formulation

Working relation: Margin_% = 100*(rating - measured_use)/rating for a positive comparable rating.

Evaluate limiting cases that apply to the stated relation: reduced load, increased impedance, vanishing angle difference, or alternative grounding. Do not apply all these checks indiscriminately; choose two that make physical sense here. Check that the result has units consistent with continuous ratings and that conversion between MW, MVA, kV, A, ohms and per-unit values has been performed correctly. The result should be reproducible from the written assumptions without guessing any hidden convention.

3. Worked calculation

Example. An 80 MVA transformer carrying 68 MVA has 15 percent unused nameplate apparent-power margin.

Interpretation. Electrical MVA loading is only one limit; thermal hotspot, cooling state and duration matter. For independent verification, re-evaluate the stated numerical inputs using the working relation and compare to the quoted magnitude. Record whether the calculated result is a steady-state operating point, an initial symmetrical fault estimate, a long-term energy statistic or a purely geometric property. Re-run the example with a ten-percent change in its principal variable, and explain why that percentage change in the input does not necessarily yield the same percentage change in the output.

4. Reliability of the approximation

Common technical mistake: Treating a short-time emergency rating as an all-day continuous rating can damage equipment.

Verification procedure: Create a dashboard combining transformer loading, bus voltage, branch current and breaker capacity information. Write a minimal input ledger that records variable name, value, SI unit, data source, time and uncertainty. Select one independent calculation, measurement or conserved quantity with which to check the result. If the two approaches disagree, determine whether the difference comes from sign conventions, base changes, a model omitted term, poor timing alignment or a calculation error. This is an essential distinction: a numerically stable solver can still give a wrong answer when it solves the wrong network.

Before relying on equipment limits and operational margins for equipment selection, establish the relevant continuous rating, short-time limit and the applicable operating constraints. For models with no dynamic terms, identify at least one fast disturbance they cannot predict. For models based on averages, identify at least one short interval where the average hides a high-risk condition. State the evidence still missing, rather than claiming that numerical plausibility proves system safety or reliability.

5. Applied case study

Engineering problem. A facility expects demand growth: decide when to uprate a transformer versus redistribute feeder load.

Finally, produce a short decision memo with a numerical range, not merely a single estimate. Include a base case, adverse case and data-quality concern related to ambient correction. Report whether the simplified model is sufficient for teaching, preliminary screening or detailed engineering. If it is not sufficient, name the additional circuit, measurement or operational constraint that would be required.

6. Questions and guided responses

1. Physical reasoning. How are continuous ratings and emergency ratings connected in this model? Guidance: start with the labelled boundary, name both units and identify the causal path rather than merely repeating a formula.

2. Numerical problem. Recalculate the example using `Margin_% = 100(rating - measured_use)/rating for a positive comparable rating` and then increase its principal source or load input by ten percent. Guidance:* report both numerical values, check signs and dimensions and explain the relevant nonlinear terms.

3. Model validity. Give two conditions under which the relation may cease to be accurate for equipment limits and operational margins. Guidance: distinguish omitted physical mechanisms from bad data and specify an independent test.

4. Design judgment. Apply the relation to the case study and explain what one additional observation would change your decision. Guidance: compare equipment limits with credible normal and contingency operating states.

5. Cross-topic reasoning. Explain how ambient correction would affect one other lesson in Unit 8. Guidance: name the shared state variable and show how the same value propagates to a separate mathematical model.

7. Study extension

Build a small sensitivity table: base case, 90-percent value of the most uncertain input, 110-percent value and one changed equipment or topology configuration. For each row, first predict the direction of the output change and then calculate it. Write a brief explanation for any result that does not follow your prediction. Mark a case “outside model validity” if its assumptions are contradicted, instead of forcing the calculation to produce a persuasive number. The goal is to understand both the use and limits of equipment limits and operational margins in an interconnected electrical system.

§8.3 SCADA, state estimation and metering

Unit 8, lesson 3: SCADA, state estimation and metering

Learning objectives: analyze telemetry, time synchronization, bad-data detection, observability, phasor measurement; interpret Weighted least squares seeks x minimizing (z-h(x))^T W (z-h(x)) and report valid application limits.

1. The physical system

A deliverable grid design must survive measurement uncertainty, protective obligations, future demand growth and contingency conditions. Comparing a single idealized load-flow screenshot to nominal equipment ratings is not enough to establish safety, quality or reliability. The topic SCADA, state estimation and metering concerns telemetry, time synchronization, bad-data detection, observability, phasor measurement. An analyst must decide which variables are directly measured and which are inferred from a network equivalent. Sketch the equipment and specify its boundary, phase reference, operating condition and sign convention. Identify the path that carries energy through the system and explain what would change if a switch state, load characteristic or source excitation were different.

In this setting, telemetry describes a distinct physical aspect from time synchronization. Mixing them obscures whether the result depends on component values or system operation. Start with the equipment location and an inventory of observed voltage, current, power, time or geometry as appropriate. Then explain why the simplified relation is sufficient for the particular question being asked, rather than assuming it describes all possible network states.

2. Mathematical formulation

Working relation: Weighted least squares seeks x minimizing (z-h(x))^T W (z-h(x)).

Evaluate limiting cases that apply to the stated relation: reduced load, increased impedance, vanishing angle difference, or alternative grounding. Do not apply all these checks indiscriminately; choose two that make physical sense here. Check that the result has units consistent with telemetry and that conversion between MW, MVA, kV, A, ohms and per-unit values has been performed correctly. The result should be reproducible from the written assumptions without guessing any hidden convention.

3. Worked calculation

Example. Two independent sensors observing one constant 100 V state return 99 V and 101 V with equal uncertainty; their least-squares estimate is 100 V.

Interpretation. State estimation reconciles redundant noisy readings with the network model, rather than treating each measurement as exact. For independent verification, re-evaluate the stated numerical inputs using the working relation and compare to the quoted magnitude. Record whether the calculated result is a steady-state operating point, an initial symmetrical fault estimate, a long-term energy statistic or a purely geometric property. Re-run the example with a ten-percent change in its principal variable, and explain why that percentage change in the input does not necessarily yield the same percentage change in the output.

4. Reliability of the approximation

Common technical mistake: A network may be unobservable if too few independent measurements are available or a breaker status is wrong.

Verification procedure: Create a small noisy-data example and compare unweighted average with variance-weighted average. Write a minimal input ledger that records variable name, value, SI unit, data source, time and uncertainty. Select one independent calculation, measurement or conserved quantity with which to check the result. If the two approaches disagree, determine whether the difference comes from sign conventions, base changes, a model omitted term, poor timing alignment or a calculation error. This is an essential distinction: a numerically stable solver can still give a wrong answer when it solves the wrong network.

Before relying on scada, state estimation and metering for equipment selection, establish the relevant continuous rating, short-time limit and the applicable operating constraints. For models with no dynamic terms, identify at least one fast disturbance they cannot predict. For models based on averages, identify at least one short interval where the average hides a high-risk condition. State the evidence still missing, rather than claiming that numerical plausibility proves system safety or reliability.

5. Applied case study

Engineering problem. An operator sees a power-flow mismatch after a switch operates; separate stale telemetry from model-topology problems.

Finally, produce a short decision memo with a numerical range, not merely a single estimate. Include a base case, adverse case and data-quality concern related to phasor measurement. Report whether the simplified model is sufficient for teaching, preliminary screening or detailed engineering. If it is not sufficient, name the additional circuit, measurement or operational constraint that would be required.

6. Questions and guided responses

1. Physical reasoning. How are telemetry and time synchronization connected in this model? Guidance: start with the labelled boundary, name both units and identify the causal path rather than merely repeating a formula.

2. Numerical problem. Recalculate the example using `Weighted least squares seeks x minimizing (z-h(x))^T W (z-h(x))` and then increase its principal source or load input by ten percent. Guidance: report both numerical values, check signs and dimensions and explain the relevant nonlinear terms.

3. Model validity. Give two conditions under which the relation may cease to be accurate for scada, state estimation and metering. Guidance: distinguish omitted physical mechanisms from bad data and specify an independent test.

4. Design judgment. Apply the relation to the case study and explain what one additional observation would change your decision. Guidance: compare equipment limits with credible normal and contingency operating states.

5. Cross-topic reasoning. Explain how phasor measurement would affect one other lesson in Unit 8. Guidance: name the shared state variable and show how the same value propagates to a separate mathematical model.

7. Study extension

Build a small sensitivity table: base case, 90-percent value of the most uncertain input, 110-percent value and one changed equipment or topology configuration. For each row, first predict the direction of the output change and then calculate it. Write a brief explanation for any result that does not follow your prediction. Mark a case “outside model validity” if its assumptions are contradicted, instead of forcing the calculation to produce a persuasive number. The goal is to understand both the use and limits of scada, state estimation and metering in an interconnected electrical system.

§8.4 Reliability planning and system studies

Unit 8, lesson 4: Reliability planning and system studies

Learning objectives: analyze N-1 screening, outage frequency, interrupted energy, risk ranking, scenarios; interpret EENS = sum_s probability(s) * unserved_energy(s) over defined scenarios and report valid application limits.

1. The physical system

A deliverable grid design must survive measurement uncertainty, protective obligations, future demand growth and contingency conditions. Comparing a single idealized load-flow screenshot to nominal equipment ratings is not enough to establish safety, quality or reliability. The topic Reliability planning and system studies concerns N-1 screening, outage frequency, interrupted energy, risk ranking, scenarios. An analyst must decide which variables are directly measured and which are inferred from a network equivalent. Sketch the equipment and specify its boundary, phase reference, operating condition and sign convention. Identify the path that carries energy through the system and explain what would change if a switch state, load characteristic or source excitation were different.

In this setting, N-1 screening describes a distinct physical aspect from outage frequency. Mixing them obscures whether the result depends on component values or system operation. Start with the equipment location and an inventory of observed voltage, current, power, time or geometry as appropriate. Then explain why the simplified relation is sufficient for the particular question being asked, rather than assuming it describes all possible network states.

2. Mathematical formulation

Working relation: EENS = sum_s probability(s) * unserved_energy(s) over defined scenarios.

Evaluate limiting cases that apply to the stated relation: reduced load, increased impedance, vanishing angle difference, or alternative grounding. Do not apply all these checks indiscriminately; choose two that make physical sense here. Check that the result has units consistent with N-1 screening and that conversion between MW, MVA, kV, A, ohms and per-unit values has been performed correctly. The result should be reproducible from the written assumptions without guessing any hidden convention.

3. Worked calculation

Example. If a 2-hour, 4 MW interruption occurs with probability 0.05 in a planning period and no other interruptions occur, its expected unserved energy contribution is 0.4 MWh per period.

Interpretation. A contingency that never occurs in a studied sample is not thereby proven impossible. For independent verification, re-evaluate the stated numerical inputs using the working relation and compare to the quoted magnitude. Record whether the calculated result is a steady-state operating point, an initial symmetrical fault estimate, a long-term energy statistic or a purely geometric property. Re-run the example with a ten-percent change in its principal variable, and explain why that percentage change in the input does not necessarily yield the same percentage change in the output.

4. Reliability of the approximation

Common technical mistake: System-average reliability indices can conceal severe impacts on a small critical-customer group.

Verification procedure: List credible contingencies, calculate affected customers and restoration time, and identify evidence gaps. Write a minimal input ledger that records variable name, value, SI unit, data source, time and uncertainty. Select one independent calculation, measurement or conserved quantity with which to check the result. If the two approaches disagree, determine whether the difference comes from sign conventions, base changes, a model omitted term, poor timing alignment or a calculation error. This is an essential distinction: a numerically stable solver can still give a wrong answer when it solves the wrong network.

Before relying on reliability planning and system studies for equipment selection, establish the relevant continuous rating, short-time limit and the applicable operating constraints. For models with no dynamic terms, identify at least one fast disturbance they cannot predict. For models based on averages, identify at least one short interval where the average hides a high-risk condition. State the evidence still missing, rather than claiming that numerical plausibility proves system safety or reliability.

5. Applied case study

Engineering problem. A cheap design satisfies normal peak load but fails under loss of a single transformer; assess whether redundancy or restoration planning is more appropriate.

Finally, produce a short decision memo with a numerical range, not merely a single estimate. Include a base case, adverse case and data-quality concern related to scenarios. Report whether the simplified model is sufficient for teaching, preliminary screening or detailed engineering. If it is not sufficient, name the additional circuit, measurement or operational constraint that would be required.

6. Questions and guided responses

1. Physical reasoning. How are N-1 screening and outage frequency connected in this model? Guidance: start with the labelled boundary, name both units and identify the causal path rather than merely repeating a formula.

2. Numerical problem. Recalculate the example using `EENS = sum_s probability(s) * unserved_energy(s) over defined scenarios` and then increase its principal source or load input by ten percent. Guidance: report both numerical values, check signs and dimensions and explain the relevant nonlinear terms.

3. Model validity. Give two conditions under which the relation may cease to be accurate for reliability planning and system studies. Guidance: distinguish omitted physical mechanisms from bad data and specify an independent test.

4. Design judgment. Apply the relation to the case study and explain what one additional observation would change your decision. Guidance: compare equipment limits with credible normal and contingency operating states.

5. Cross-topic reasoning. Explain how scenarios would affect one other lesson in Unit 8. Guidance: name the shared state variable and show how the same value propagates to a separate mathematical model.

7. Study extension

Build a small sensitivity table: base case, 90-percent value of the most uncertain input, 110-percent value and one changed equipment or topology configuration. For each row, first predict the direction of the output change and then calculate it. Write a brief explanation for any result that does not follow your prediction. Mark a case “outside model validity” if its assumptions are contradicted, instead of forcing the calculation to produce a persuasive number. The goal is to understand both the use and limits of reliability planning and system studies in an interconnected electrical system.

THREE WORKED QUESTIONS

Step-by-Step Power Systems I Solutions

Three original questions connect calculations, definitions and interpretation. Open each solution to follow the reasoning.

Solved Problem Example 8.1: Economic Dispatch, Loss Penalty Factors, and Optimal Generation Numerical Problem 1

Determine the operating parameters, terminal responses, and stability criteria for an engineered power system or electrical machine operating under Economic Dispatch, Loss Penalty Factors, and Optimal Generation conditions.