Chapter 4 • Theory & Derivations

Per-Unit System Formulations and Network Representation

Comprehensive university textbook unit covering fundamental principles, mathematical formulations, and practical engineering applications of Per-Unit System Formulations and Network Representation.

§4.1 Bus types and nodal equations

Unit 4, lesson 1: Bus types and nodal equations

Learning objectives: analyze slack reference, PQ load bus, PV generator bus, injection convention; interpret S_i = V_i conjugate(sum_j(Y_ijV_j)) and report valid application limits.

1. The physical system

A power-flow solution describes one algebraic operating point at a specified load and generation dispatch. It requires a reference/slack bus to anchor angle, PV-bus voltage targets, PQ-bus power demands and a suitable model of generator reactive limits. Numerical convergence is not the same as physical feasibility. The topic Bus types and nodal equations concerns slack reference, PQ load bus, PV generator bus, injection convention. 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, slack reference describes a distinct physical aspect from PQ load bus. 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: S_i = V_i conjugate(sum_j(Y_ijV_j)).

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 slack reference 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. A PQ bus with 20 MW load and 8 MVAr load is represented by negative net complex-power injection if injection is positive for generation.

Interpretation. A slack bus is a numerical reference whose computed power balances losses and the specified injections. 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 PV bus cannot maintain its set voltage if the generator reaches a Q capability limit; algorithms must switch bus treatment.

Verification procedure: Write the known and unknown quantities for a three-bus case consisting of slack, PV and PQ buses. 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 bus types and nodal equations 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 changes a generator real-power schedule without revising a slack-power limit; identify the model-risk.

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 injection convention. 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 slack reference and PQ load bus 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 `S_i = V_i * conjugate(sum_j(Y_ijV_j))` 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 bus types and nodal equations. 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 injection convention would affect one other lesson in Unit 4. 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 bus types and nodal equations in an interconnected electrical system.

§4.2 Gauss-Seidel iteration

Unit 4, lesson 2: Gauss-Seidel iteration

Learning objectives: analyze complex bus-voltage iteration, relaxation, initialization, mismatch tolerance; interpret V_i_new = (conj(S_i)/conj(V_i_old) - sum_(j neq i) Y_ij V_j) / Y_ii for PQ and report valid application limits.

1. The physical system

A power-flow solution describes one algebraic operating point at a specified load and generation dispatch. It requires a reference/slack bus to anchor angle, PV-bus voltage targets, PQ-bus power demands and a suitable model of generator reactive limits. Numerical convergence is not the same as physical feasibility. The topic Gauss-Seidel iteration concerns complex bus-voltage iteration, relaxation, initialization, mismatch tolerance. 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, complex bus-voltage iteration describes a distinct physical aspect from relaxation. 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: V_i_new = (conj(S_i)/conj(V_i_old) - sum_(j neq i) Y_ij V_j) / Y_ii for PQ.

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 complex bus-voltage iteration 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. In a two-bus model with slack 1+j0 pu and a 0.1+j0.2 pu series line, the no-load receiving voltage is exactly 1+j0 pu.

Interpretation. Sequential updates use the latest available voltage estimates and should be evaluated with power mismatches. 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 small voltage change between iterations does not automatically mean the power-balance residual is small.

Verification procedure: Perform three iterations on a simple PQ-bus case and print mismatch as well as voltage magnitude. 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 gauss-seidel iteration 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 poorly conditioned network is repeatedly relaxed until results appear stable; explain when a different solver is necessary.

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 mismatch tolerance. 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 complex bus-voltage iteration and relaxation 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 `V_i_new = (conj(S_i)/conj(V_i_old) - sum_(j neq i) Y_ij V_j) / Y_ii for PQ` 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 gauss-seidel iteration. 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 mismatch tolerance would affect one other lesson in Unit 4. 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 gauss-seidel iteration in an interconnected electrical system.

§4.3 Newton-Raphson power flow

Unit 4, lesson 3: Newton-Raphson power flow

Learning objectives: analyze power mismatch, voltage angles, Jacobian, linearized correction, conditioning; interpret J(x_k) * delta_x = mismatch(x_k); x_(k+1)=x_k+delta_x and report valid application limits.

1. The physical system

A power-flow solution describes one algebraic operating point at a specified load and generation dispatch. It requires a reference/slack bus to anchor angle, PV-bus voltage targets, PQ-bus power demands and a suitable model of generator reactive limits. Numerical convergence is not the same as physical feasibility. The topic Newton-Raphson power flow concerns power mismatch, voltage angles, Jacobian, linearized correction, conditioning. 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, power mismatch describes a distinct physical aspect from voltage angles. 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: J(x_k) * delta_x = mismatch(x_k); x_(k+1)=x_k+delta_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 power mismatch 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. A linear correction with mismatch 0.02 pu and local sensitivity 0.4 pu/rad gives an angle correction of about 0.05 rad in a one-variable illustration.

Interpretation. The Jacobian couples active and reactive mismatches to bus-angle and voltage-magnitude changes. 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 singular or ill-conditioned Jacobian may signal low-voltage behavior or bad modeling; blindly accepting a step is unsafe.

Verification procedure: Compare convergence traces on identical network data from flat-start and a deliberately perturbed starting point. 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 newton-raphson power flow 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. The program reports convergence but a Q-limited PV bus is still treated as fixed-voltage; identify the violated physical condition.

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 conditioning. 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 power mismatch and voltage angles 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 `J(x_k) * delta_x = mismatch(x_k); x_(k+1)=x_k+delta_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 newton-raphson power flow. 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 conditioning would affect one other lesson in Unit 4. 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 newton-raphson power flow in an interconnected electrical system.

§4.4 Interpreting a solved power flow

Unit 4, lesson 4: Interpreting a solved power flow

Learning objectives: analyze branch P and Q, real losses, transformer loading, voltage bands, feasibility; interpret Total generation - total demand = calculated system real losses (steady state) and report valid application limits.

1. The physical system

A power-flow solution describes one algebraic operating point at a specified load and generation dispatch. It requires a reference/slack bus to anchor angle, PV-bus voltage targets, PQ-bus power demands and a suitable model of generator reactive limits. Numerical convergence is not the same as physical feasibility. The topic Interpreting a solved power flow concerns branch P and Q, real losses, transformer loading, voltage bands, feasibility. 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, branch P and Q describes a distinct physical aspect from real losses. 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: Total generation - total demand = calculated system real losses (steady state).

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 branch P and Q 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. A system with 60 MW total generation and 57 MW total demand has 3 MW losses if storage is unchanged and accounting boundaries match.

Interpretation. The slack result, reactive limits and branch ratings should be inspected before any recommendation is made. 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 physically impossible negative loss from passive lines typically indicates a sign, unit or data problem unless power flow direction was mislabeled.

Verification procedure: Generate an exception report listing undervoltage, overvoltage, overloaded branches and generators at reactive limits. 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 interpreting a solved power flow 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 case study shows all voltage bands satisfied but one transformer exceeds its continuous thermal limit; rank the corrective options.

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 feasibility. 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 branch P and Q and real losses 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 `Total generation - total demand = calculated system real losses (steady state)` 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 interpreting a solved power flow. 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 feasibility would affect one other lesson in Unit 4. 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 interpreting a solved power flow 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 4.1: Per-Unit System Formulations and Network Representation Numerical Problem 1

Determine the operating parameters, terminal responses, and stability criteria for an engineered power system or electrical machine operating under Per-Unit System Formulations and Network Representation conditions.