Power System Analysis

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This section covers the computation of series and shunt parameters of transmission line equivalent circuits, the concepts of GMD and GMR, proximity and skin effects, corona phenomena, and per unit system representation.

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Series & Shunt Parameters A transmission line's series parameters are resistance (R, ohmic loss) and inductance (L, from the magnetic flux linking the conductors); its shunt parameter is capacitance (C, from the electric field between conductors and ground/each other), with shunt conductance (leakage) usually neglected.

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Line inductance per phase depends on conductor geometry, expressed via the Geometric Mean Radius (GMR) — an equivalent radius accounting for internal flux linkage of a (possibly bundled/stranded) conductor — and the Geometric Mean Distance (GMD) — the equivalent spacing between phase conductors for unsymmetrically-spaced or transposed lines:

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Line capacitance per phase similarly depends on conductor radius and GMD:

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C ∝ 1/ln(GMD/r), where r is the actual conductor radius (capacitance calculation uses actual radius, not GMR, since it depends on the external electric field, not internal flux).

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Transposition of conductors (cyclically interchanging phase positions along the line) equalizes the mutual inductance/capacitance among phases despite unsymmetrical spacing, using GMD as the equivalent symmetrical spacing.

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Proximity Effect & Skin Effect Effect Description Skin effect At AC, current density is higher near the conductor surface and lower at the core, effectively reducing the usable cross-section and increasing effective AC resistance compared to DC resistance; increases with frequency and conductor size Proximity effect Current distribution within a conductor is further distorted by the magnetic field of a nearby current- carrying conductor, also increasing effective AC resistance; more significant when conductors are closely spaced Corona Corona is a partial electrical discharge (ionization of air) occurring around a conductor when the surface voltage gradient exceeds the critical (corona inception) voltage, a threshold depending on conductor radius, spacing, and atmospheric conditions (air density, humidity).

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Corona power loss increases with system voltage above the inception threshold, and is influenced by conductor surface condition (smoother, larger-diameter or bundled conductors reduce corona), weather (worse in rain/fog/high humidity), and altitude (lower air density reduces breakdown strength, increasing corona).

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Bundled conductors (two or more sub-conductors per phase) are commonly used on EHV lines specifically to increase the effective conductor diameter/reduce surface voltage gradient, reducing corona loss and radio interference.

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Per Unit System Representation The per unit (p.u.) system expresses voltages, currents, powers, and impedances as fractions of chosen base quantities (typically base MVA and base kV), simplifying calculations across a network with multiple voltage levels via transformers.

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Per unit impedance = actual impedance / base impedance, where base impedance = (base kV)² / base MVA; a consistent base MVA is used system-wide, while base kV is chosen appropriate to each voltage level (transformed by turns ratio across transformers).

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A single-line diagram represents a three-phase balanced power system using one line (since all three phases are identical under balanced conditions), from which impedance and reactance diagrams are derived for load flow, fault, and stability studies — the impedance diagram includes resistance and reactance, while the reactance diagram (used for fault calculations) typically neglects resistance and static loads for simplicity.

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This section covers lumped and distributed parameter modeling of transmission lines, the ABCD parameter representation, efficiency and regulation calculations, the Ferranti effect, and surge impedance loading.

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Distributed Parameter Modeling Line Length Model Short line (up to ~80 km) Lumped series impedance only (R + jX); shunt capacitance neglected Medium line (~80–250 km) Lumped series impedance with shunt capacitance concentrated (nominal-π or nominal-T) at one or two points Long line (>~250 km) Parameters are actually distributed continuously along the line; distributed-parameter (long-line) model uses hyperbolic functions of propagation constant × length, more accurate than any lumped approximation at this length ABCD Parameters A transmission line (of any length/model) can be represented as a two-port network relating sending-end quantities to receiving-end quantities via ABCD (generalized circuit) constants:

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VS = A·VR + B·IR, and IS = C·VR + D·IR.

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For a symmetrical (physically identical from either end) transmission line, A = D, and for a passive, reciprocal network, AD − BC = 1 — these identities are used to check the consistency of computed ABCD parameters.

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The constants have physical meaning:

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A and D are dimensionless (voltage/current ratios), B has units of impedance (ohms), and C has units of admittance (siemens/mhos).

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Efficiency & Regulation Calculations Using ABCD constants, voltage regulation can be found from the no-load receiving voltage (= VS/A when load is removed) compared to the full-load receiving voltage, and transmission efficiency from the ratio of receiving-end to sending-end real power, consistent with the basic definitions covered for short/medium lines.

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These calculations allow direct comparison of line performance across different lengths and loading conditions using a common (ABCD-parameter-based) framework.

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Ferranti Effect & Surge Impedance Loading The Ferranti effect is the phenomenon where the receiving-end voltage of a long transmission line exceeds the sending-end voltage under light-load or no-load conditions, caused by the line's shunt capacitive charging current flowing through the line's series inductive reactance — more pronounced on long, lightly-loaded, high-voltage lines.

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Surge Impedance Loading (SIL) is the power delivered by a line when it is terminated in a resistive load equal to its surge (characteristic) impedance ZC = √(L/C); at SIL, the line's reactive power generation (from shunt capacitance) exactly balances its reactive power absorption (in series inductance), so the line operates at unity power factor with flat voltage profile along its length.

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Loading a line below SIL causes voltage rise toward the receiving end (capacitive/Ferranti-like effect dominates); loading above SIL causes voltage drop toward the receiving end (inductive effect dominates) — SIL is thus a key reference for assessing a long line's loading condition and reactive power behavior.

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This section covers the method of symmetrical components for analyzing unbalanced faults, the effect of system grounding, and the analysis of the major unbalanced and balanced fault types.

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Symmetrical Components The method of symmetrical components resolves any unbalanced set of three-phase phasors into three balanced sets: positive sequence (same phase rotation as the normal system, a-b-c), negative sequence (opposite phase rotation, a-c-b), and zero sequence (all three phasors identical in magnitude and phase, no rotation) — the actual unbalanced phasors are the sum of these three sequence components.

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Each sequence network is analyzed independently using its own sequence impedance (Z1 positive, Z2 negative, Z0 zero sequence impedance), since a balanced/symmetrical power system has no coupling between sequence networks — this decouples the otherwise complex unbalanced fault problem into three simpler balanced problems.

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Zero sequence current can only flow in a circuit with a return path (e.g. through a grounded neutral or earth return); it does not flow in a system with no path to ground, which is central to why grounding configuration affects fault behavior.

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Ungrounded Systems In a solidly or effectively grounded system, zero sequence current can flow freely during a ground fault, typically resulting in high ground fault current but limited transient overvoltage on healthy phases.

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In an ungrounded (or high-impedance grounded) system, zero sequence current during a single line-to- ground fault is limited (often to a small capacitive current), but the voltage on the healthy phases can rise significantly (toward line-to-line voltage), and locating the fault or clearing it promptly becomes more important to avoid a second fault developing into a more severe condition.

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Fault Types Fault Type Description Single line-to- ground (L-G) One phase conductor faults to ground/earth; the most common fault type on overhead systems; involves all three sequence networks connected in series Line-to-line (L- L) Two phase conductors fault to each other, no ground involvement; involves positive and negative sequence networks connected in parallel/opposition, no zero sequence current flows Double line-to- ground (L-L-G) Two phase conductors fault to each other and to ground; involves all three sequence networks connected in parallel Three-phase (L-L-L or L-L-L- G) All three phases fault together (with or without ground); the only balanced fault type, so it can be analyzed using only the positive sequence network (per-phase equivalent) — generally produces the highest fault current on most systems

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This section covers the formation of the Y-bus (bus admittance matrix) representation of a power system network, and the two classical iterative methods used to solve the load flow (power flow) problem:

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Gauss- Seidel and Newton-Raphson.

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Y-Bus of a Power System Network The bus admittance matrix (Y-bus) represents a power network's topology and line/transformer admittances in a compact matrix form, where the diagonal element Yii is the sum of all admittances connected to bus i (including line charging and any shunt elements), and the off-diagonal element Yij is the negative of the admittance directly connecting buses i and j (zero if no direct connection exists).

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Y-bus is typically sparse for realistic power networks (most buses connect to only a few neighbors), which is exploited by efficient sparse-matrix computational techniques in large-scale load flow programs.

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Each bus in a load flow study is classified as a slack (swing) bus (reference bus, voltage magnitude and angle specified, absorbs the system's power balance/losses), a PV (generator) bus (voltage magnitude and real power specified, reactive power and angle to be solved), or a PQ (load) bus (real and reactive power specified, voltage magnitude and angle to be solved).

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Gauss-Seidel Method The Gauss-Seidel method iteratively updates each bus voltage using the most recently computed values of other bus voltages (within the same iteration), based on the power balance equation at each bus, continuing until voltage changes between iterations fall below a specified tolerance.

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Gauss-Seidel is simple to program and requires relatively low memory per iteration, but generally exhibits slow (linear) convergence, requiring a comparatively large number of iterations, and convergence can be sensitive to the choice of slack bus and system conditioning — making it less suited to very large modern networks.

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Newton-Raphson Method The Newton-Raphson method linearizes the nonlinear power flow equations at each iteration using the Jacobian matrix (partial derivatives of real/reactive power with respect to voltage magnitude and angle), solving for voltage corrections that drive the power mismatch toward zero.

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Newton-Raphson exhibits fast (quadratic) convergence near the solution, typically requiring far fewer iterations than Gauss-Seidel for large systems, though each iteration is more computationally intensive (requiring Jacobian formation and a linear system solution) — overall, it is the preferred method for large- scale practical power system load flow studies.

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A widely-used variant, the Fast Decoupled Load Flow, exploits the typical power system property that real power is primarily affected by voltage angle while reactive power is primarily affected by voltage magnitude, simplifying/approximating the Jacobian for faster computation with only a modest accuracy trade-off.

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This section covers steady state and transient stability limits, the swing equation, the equal area criterion, stability enhancement techniques, and voltage stability.

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Steady State & Transient Stability Stability Type Description Steady state stability The power system's ability to maintain synchronism under small, slow (quasi-static) disturbances/load changes; the steady state stability limit is the maximum power that can be transferred without loss of synchronism under such gradual changes Transient stability The power system's ability to maintain synchronism following a large, sudden disturbance (e.g. a fault, sudden loss of a line or generator); assessed over the first swing (typically ~1 second) after the disturbance, and depends on fault location, clearing time, and system/generator characteristics Swing Equation The swing equation describes the dynamic behavior of a synchronous generator's rotor angle (δ) relative to a synchronously rotating reference, following a disturbance: it relates the rotor's moment of inertia (via the inertia constant H) and angular acceleration to the difference between mechanical input power and electrical output power (the accelerating power).

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When mechanical input power exceeds electrical output power, the rotor accelerates (δ increases); when electrical output exceeds mechanical input, the rotor decelerates — stable operation requires the rotor angle to settle to a new stable equilibrium (or oscillate back) rather than continuously increasing (which would indicate loss of synchronism).

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Equal Area Criterion The equal area criterion is a graphical method for assessing transient stability (for a single-machine system, or a machine against an infinite bus) without formally solving the swing equation, based on the power-angle (P-δ) curve.

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It states that following a disturbance, the system remains transiently stable if the accelerating area (energy gained by the rotor while mechanical power exceeds electrical power) can be exactly balanced by an available decelerating area (energy that can be absorbed as electrical power exceeds mechanical power) before the rotor angle reaches its critical limit — if the decelerating area available is insufficient to equal the accelerating area, the system loses synchronism.

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The equal area criterion is commonly used to determine the critical clearing angle/time — the maximum fault duration before which the fault must be cleared for the system to remain stable.

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Stability Enhancement & Voltage Stability Stability enhancement techniques include: fast fault clearing (reduced circuit breaker/relay operating time, directly reducing the accelerating area during a fault), high-speed reclosing (restoring the line quickly after a transient fault), reducing transfer reactance (e.g. via series compensation or additional parallel lines), increasing generator inertia or using fast-responding excitation systems (AVR) and power system stabilizers (PSS), and installing FACTS devices for dynamic reactive support.

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Voltage stability is a system's ability to maintain acceptable voltages at all buses under normal conditions and after a disturbance; voltage collapse is a progressive, uncontrollable decline in system voltage (often associated with heavily loaded systems unable to meet reactive power demand), distinct from (though sometimes related to) rotor-angle (synchronism) instability.

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This final section of the chapter covers real and reactive power flow through a transmission line, and the principles of series and shunt compensation used for voltage control and reactive power (VAR) management.

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Real & Reactive Power Flow Through a Transmission Line For a transmission line connecting two buses with voltage magnitudes VS, VR and angle difference δ, real power flow is approximately P ≈ (VSVR/X) sinδ, and reactive power flow is approximately Q ≈ (VSVR/X) (1−cosδ)/… more precisely Q depends on the voltage magnitude difference, for a predominantly reactive (inductive) line where resistance is neglected.

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This leads to the key practical relationships: real power flow is primarily controlled by the angle difference δ between the two bus voltages (larger angle difference → more real power transfer, up to a maximum at δ=90°), while reactive power flow is primarily controlled by the voltage magnitude difference between the two buses (reactive power flows from the higher-voltage bus toward the lower- voltage bus).

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This angle-real-power / magnitude-reactive-power decoupling is the physical basis for both the Fast Decoupled Load Flow method (Section 9.4) and for the general practice of using reactive power injection (VAR compensation) as the primary tool for local voltage control.

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Series Compensation Series compensation (typically series capacitors inserted in the line) reduces the effective series reactance X of the line, which directly increases real power transfer capability for a given angle δ and voltage levels (since P ∝ 1/X), and also helps reduce voltage drop along the line.

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Series compensation is particularly valuable for long transmission lines, where the naturally high series reactance limits power transfer capability well below thermal limits; however, series capacitors introduce risks such as sub-synchronous resonance (SSR) with nearby generator shafts, requiring careful protection/design.

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Shunt Compensation Shunt compensation (capacitor banks, reactors, synchronous condensers, or power-electronic devices such as SVC/STATCOM connected in parallel at a bus) primarily controls local bus voltage by injecting or absorbing reactive power.

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Shunt capacitors supply reactive power, raising voltage and supporting power factor (used where voltage tends to sag under heavy inductive loading); shunt reactors absorb reactive power, lowering voltage (used at lightly-loaded line receiving ends to counter Ferranti-effect voltage rise).

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Dynamic shunt devices (SVC, STATCOM) can rapidly and continuously vary their reactive power output/absorption in response to changing system conditions, offering superior voltage regulation and stability support compared to fixed or mechanically-switched shunt compensation.