Power System‑II (EX-601) - Unit 1 Short Notes

1.0 LOAD FLOW ANALYSIS

1.1 Significance and Purpose

Load flow (power flow) analysis determines voltage magnitudes and angles at all buses, and real/reactive power flows in lines for specified generation and load conditions.
Purposes:

• Planning: Expansion studies, new line additions.

• Operation: Optimal dispatch, contingency analysis (N-1 security).

• Optimization: Loss minimization, voltage profile improvement.

• Stability studies: Provides initial operating point for transient and dynamic stability analysis.

1.2 Classification of Buses

1.3 Formation of Nodal Admittance Matrix (Y_bus)

• Definition: \(Y_{\text{bus}}\) is a sparse \(n \times n\) matrix where \(Y_{ij} = -y_{ij}\) for \(i \neq j\) (off-diagonal), and \(Y_{ii} = \sum_{k=1}^{n} y_{ik}\) (sum of all admittances connected to bus \(i\)).

• From line data: For line \(i-j\) with series impedance \(z_{ij} = r_{ij} + jx_{ij}\) and shunt admittance \(y_{ij}/2\) at each end:

  \[ y_{ij} = \frac{1}{z_{ij}}, \quad Y_{ij} = -y_{ij}, \quad Y_{ii} \text{ includes } y_{ij} + \frac{y_{ij}}{2} \text{ if shunt considered}. \]

• Systematic formation: Using incidence matrix \(\mathbf{A}\) (size \(n \times b\)) and branch admittance matrix \(\mathbf{y}\) (diagonal \(b \times b\)):

  \[ \boxed{Y_{\text{bus}} = \mathbf{A}^T \mathbf{y} \mathbf{A}} \]

• Example (4-bus system, shunt neglected): Given line impedances (p.u.):

  \(1-2: 0.25+j0.1\), \(1-3: 0.20+j0.8\), \(1-4: 0.30+j1.2\), \(2-3: 0.20+j0.8\), \(3-4: 0.15+j0.6\).

  Compute \(y_{ij} = 1/z_{ij}\):

  \(y_{12} = 3.448 - j1.379\), \(y_{13} = 0.294 - j1.176\), \(y_{14} = 0.196 - j0.784\),

  \(y_{23} = 0.294 - j1.176\), \(y_{34} = 0.392 - j1.569\).

  Then:

  \[ Y_{\text{bus}} = \begin{bmatrix} Y_{11} & Y_{12} & Y_{13} & Y_{14} \\ Y_{21} & Y_{22} & Y_{23} & 0 \\ Y_{31} & Y_{32} & Y_{33} & Y_{34} \\ Y_{41} & 0 & Y_{43} & Y_{44} \end{bmatrix} \]

  with:

  \[ \begin{aligned} Y_{11} &= 3.938 - j3.340, & Y_{22} &= 3.742 - j2.556, \\ Y_{33} &= 0.980 - j3.922, & Y_{44} &= 0.588 - j2.353, \\ Y_{12} &= -3.448 + j1.379, & Y_{13} &= -0.294 + j1.176, \\ Y_{14} &= -0.196 + j0.784, & Y_{23} &= -0.294 + j1.176, \\ Y_{34} &= -0.392 + j1.569. \end{aligned} \]

1.4 Iterative Solution Methods

1.4.1 Gauss-Seidel Method

• Algorithm (for PQ buses only):

  Initialize voltages (flat start: \(1.0\angle 0^\circ\)). For each PQ bus \(i\):

  \[ V_i^{(k+1)} = \frac{1}{Y_{ii}} \left( \frac{P_i - jQ_i}{V_i^{(k)*}} - \sum_{\substack{j=1 \\ j \neq i}}^{n} Y_{ij} V_j^{(k)} \right) \]

  Check convergence: \(\max |V_i^{(k+1)} - V_i^{(k)}| < \varepsilon\).

• Modification for PV buses: After updating \(V_i\), if \(|V_i| \neq |V_i^{\text{spec}}|\), scale \(V_i\) to specified magnitude and compute new \(Q_i\) from:

  \[ Q_i^{(k+1)} = - \left[ V_i^{(k+1)*} \sum_{j=1}^{n} Y_{ij} V_j^{(k)} \right]_{\text{imag}} \]

• Flow chart for PQ buses only:

  1. Initialize \(V_i = 1.0\angle 0^\circ\) for all PQ buses.

  2. For each PQ bus \(i\), compute \(V_i^{\text{new}}\) using equation above.

  3. Check convergence. If not converged, set \(V_i = V_i^{\text{new}}\) and repeat.

• Advantages: Simple, low memory per iteration.

• Disadvantages: Slow linear convergence; may diverge for ill-conditioned systems.

• Convergence: Depends on spectral radius of iteration matrix; typically requires more iterations than NR.

1.4.2 Newton-Raphson Method

• Formulation (polar coordinates):

  State variables: \(\delta_i\) for all non-slack buses, \(|V_i|\) for PQ buses.

  Power mismatches:

  \[ \Delta P_i = P_i^{\text{spec}} - P_i^{\text{calc}}, \quad \Delta Q_i = Q_i^{\text{spec}} - Q_i^{\text{calc}} \]

  where

  \[ \begin{aligned} P_i^{\text{calc}} &= |V_i| \sum_{j=1}^{n} |V_j| \left( G_{ij} \cos \delta_{ij} + B_{ij} \sin \delta_{ij} \right), \\ Q_i^{\text{calc}} &= |V_i| \sum_{j=1}^{n} |V_j| \left( G_{ij} \sin \delta_{ij} - B_{ij} \cos \delta_{ij} \right). \end{aligned} \]

• Jacobian matrix:

  \[ J = \begin{bmatrix} \frac{\partial P}{\partial \delta} & \frac{\partial P}{\partial |V|} \\ \frac{\partial Q}{\partial \delta} & \frac{\partial Q}{\partial |V|} \end{bmatrix} \]

  Elements computed analytically; sparse structure exploited.

• Algorithm:

  1. Initialize voltages (flat start).

  2. Compute \(P_i^{\text{calc}}\), \(Q_i^{\text{calc}}\), and mismatches \(\Delta P_i\), \(\Delta Q_i\).

  3. Form Jacobian \(J\).

  4. Solve \(J \Delta X = -\Delta PQ\) for corrections \(\Delta \delta\) (non-slack) and \(\Delta |V|\) (PQ buses).

  5. Update: \(\delta_i^{\text{new}} = \delta_i^{\text{old}} + \Delta \delta_i\); for PQ buses, \(|V_i|^{\text{new}} = |V_i|^{\text{old}} + \Delta |V_i|\); for PV buses, only \(\delta\) updated, \(|V|\) fixed.

  6. Check convergence: \(\max |\Delta| < \varepsilon\).

  7. Repeat if needed.

• Flow chart: Standard NR flow chart with initialization, mismatch calculation, Jacobian formation, solution, update, convergence check.

• Advantages: Quadratic convergence; robust for large systems.

• Disadvantages: High memory (Jacobian storage); complex per iteration (matrix inversion).

• Convergence: Typically 3–5 iterations for well-behaved systems.

1.4.3 Comparison of Gauss-Seidel and Newton-Raphson

1.4.4 Fast Decoupled Load Flow (FDLF)

• Based on assumptions:

  • Voltage magnitudes close to 1.0 p.u.

  • Lines highly reactive (\(R \ll X\)).

• Decouples \(P-\delta\) and \(Q-|V|\) equations:

  \[ \begin{aligned} \Delta P &= B' \Delta \delta, \\ \Delta Q &= B'' \Delta |V|, \end{aligned} \]

  where \(B'\) and \(B''\) are constant approximations of Jacobian submatrices.

• Advantages: Faster per iteration, reduced memory.

• Disadvantages: May require more iterations for systems with high \(R/X\) ratios; less robust.

2.0 POWER SYSTEM STABILITY ANALYSIS

2.1 Fundamental Concepts

• Steady-state stability: Ability to maintain synchronism under small disturbances (e.g., load fluctuations). Analyzed using linearized models around an operating point.

• Transient stability: Ability to maintain synchronism after large disturbances (e.g., faults, line outages). Requires nonlinear time-domain simulation.

• Factors affecting steady-state stability:

  • Higher voltage levels.

  • Lower transmission reactance.

  • Higher generator excitation.

  • Faster excitation systems.

• Factors affecting transient stability:

  • Fault type and location (3-phase worst).

  • Clearing time (shorter better).

  • System strength (lower \(X/R\) ratio improves stability).

  • Generator inertia (higher \(H\) slows rotor acceleration).

  • Post-fault network strength.

2.2 Swing Equation and Rotor Dynamics

• Derivation (single machine-infinite bus):

  Rotor dynamics: \(M \frac{d^2 \delta}{dt^2} = T_m - T_e\), where \(M = \frac{2H}{\omega_s}\) (inertia constant \(H\) in seconds, \(\omega_s = 2\pi f_s\) rad/s).

  In power terms (\(P = T \omega\)):

  \[ \boxed{M \frac{d^2 \delta}{dt^2} = P_m - P_e} \]

  where \(P_e = \frac{E V}{X} \sin \delta\) for simple model.

• Physical significance: Describes rotor acceleration/deceleration due to imbalance between mechanical input \(P_m\) and electrical output \(P_e\). Stability depends on whether \(\delta\) remains bounded.

• Linearized swing equation (small disturbances):

  Let \(\delta = \delta_0 + \Delta\delta\), \(P_e \approx P_{e0} + K \Delta\delta\), where \(K = \left. \frac{dP_e}{d\delta} \right|_{\delta_0}\).

  \[ M \frac{d^2 \Delta\delta}{dt^2} = \Delta P_m - K \Delta\delta \]

  For constant \(\Delta P_m\), this is a second-order system.

• Steady-state stability limit: From linearized equation, maximum \(\Delta P_m\) without instability occurs when \(K \Delta\delta_{\max} = \Delta P_m\). But for nonlinear \(P_e(\delta)\), the limit is at \(\frac{dP_e}{d\delta} = 0\), i.e., \(\delta = 90^\circ\) for simple model, so \(P_{\max} = \frac{EV}{X}\).

2.3 Equal Area Criterion (EAC)

• Statement: For a single machine-infinite bus system, the system is stable if the accelerating area \(A_1\) (when \(P_m > P_e\)) equals the decelerating area \(A_2\) (when \(P_e > P_m\)) during a disturbance.

• Derivation: Multiply swing equation by \(d\delta/dt\) and integrate:

  \[ \int_{\delta_0}^{\delta} (P_m - P_e) d\delta = \frac{1}{2} M \left( \frac{d\delta}{dt} \right)^2 \Big|_{\delta_0}^{\delta} \]

  At \(\delta = \delta_{\text{cr}}\), \(\frac{d\delta}{dt} = 0\), so \(\int_{\delta_0}^{\delta_{\text{cr}}} (P_m - P_e) d\delta = 0 \Rightarrow A_1 = A_2\).

• Graphical interpretation:

  DiagramCANVAS: Plot P_e vs δ curve. Draw horizontal line at P_m. Mark pre-fault curve (P_e1), during-fault curve (P_e2, lower), post-fault curve (P_e3). Accelerating area A1 between P_m and P_e2 from δ0 to δ_cr. Decelerating area A2 between P_e3 and P_m from δ_cr to δ_max where P_e3 = P_m again. Shade areas.

• Critical clearing angle \(\delta_{\text{cr}}\): Angle at which \(A_1 = A_2\). Found by solving:

  \[ \int_{\delta_0}^{\delta_{\text{cr}}} (P_m - P_e^{\text{fault}}) d\delta = \int_{\delta_{\text{cr}}}^{\delta_{\max}} (P_e^{\text{post}} - P_m) d\delta \]

  where \(\delta_{\max}\) satisfies \(P_e^{\text{post}}(\delta_{\max}) = P_m\).

• Critical clearing time \(t_{\text{cr}}\): Time corresponding to \(\delta_{\text{cr}}\) from integration of swing equation during fault.

• Application to faults with changed reactance:

  • Pre-fault: \(X = X_1\), \(P_{e1} = \frac{EV}{X_1} \sin \delta\).

  • During fault: \(X = X_f\), \(P_{e2} = \frac{EV}{X_f} \sin \delta\).

  • Post-fault: \(X = X_2\), \(P_{e3} = \frac{EV}{X_2} \sin \delta\).

  Compute areas accordingly.

2.4 Methods for Improving Transient Stability

• Fast valving: Rapidly reduce steam input to turbine during fault.

• Dynamic braking: Insert resistors in generator terminals to absorb power.

• Excitation control: High initial response excitation (e.g., PSS) to increase \(P_e\) during fault.

• HVDC links: Fast power flow control.

• Series capacitors: Reduce line reactance, increase \(P_{\max}\).

• Load shedding: Reduce \(P_m\) during emergency.

• Single-pole switching: For line faults, open only faulty pole to maintain coupling.

3.0 LOAD FREQUENCY CONTROL (LFC) AND GOVERNOR MODELING

3.1 Importance of Strict Frequency Control

• Frequency variation affects:

  • Induction motor speeds (industrial processes).

  • Synchronous motor speeds (constant-speed drives).

  • Consumer devices (clocks, electronics).

  • System stability (frequency instability can cause cascading outages).

• Standards: Typically \(\pm 0.5\ \text{Hz}\) for interconnected systems.

3.2 Modeling of Turbine Speed Governing System

• Components:

  • Governor: Senses speed/frequency, adjusts valve position.

  • Speed changer (set point): Manual/automatic adjustment of desired speed.

  • Turbine: Converts steam/water flow to mechanical power.

  • Valve motor: Time delay due to inertia.

• Transfer function block diagram:

  DiagramCANVAS: Block diagram: Speed changer (gain 1/R) → Governor (1/(1+sT_g)) → Valve motor (1/(1+sT_v)) → Turbine (1/(1+sT_t)) → ΔP_m. Input: ΔP_ref - Δf (with feedback from Δf).

  Standard model:

  \[ \Delta P_m(s) = \frac{1}{1+sT_g} \cdot \frac{1}{1+sT_t} \cdot \frac{1}{R} \left( \Delta P_{\text{ref}}(s) - \Delta f(s) \right) \]

  where \(R\) is droop (Hz per unit power).

3.3 Inertia Constant (H) and Rotor Dynamics

• Definition:

  \[ \boxed{H = \frac{\frac{1}{2} I \omega_s^2}{S_{\text{base}}} \ \text{(seconds)}} \]

  where \(I\) = moment of inertia (kg·m²), \(\omega_s = 2\pi f_s\) rad/s, \(S_{\text{base}}\) = generator MVA rating.

• Physical meaning: Time (in seconds) for which generator can supply full load without input power. Higher \(H\) means more stored kinetic energy, slower frequency changes.

• Equivalent inertia constant for multiple machines:

  \[ H_{\text{eq}} = \frac{\sum_{i=1}^{n} H_i S_i}{\sum_{i=1}^{n} S_i} \]

• Kinetic energy stored:

  \[ \text{KE} = 2 H S_{\text{base}} \ \text{(MW·s or MJ)} \]

• Rotor acceleration: From swing equation in per unit:

  \[ M \frac{d^2 \delta}{dt^2} = P_m - P_e, \quad M = \frac{2H}{\omega_s} \]

  Since \(\frac{d\delta}{dt} = \Delta \omega = 2\pi \Delta f\),

  \[ \frac{d(\Delta f)}{dt} = \frac{f_s}{2H} (P_m - P_e) \]

  For constant power imbalance \(\Delta P\) during acceleration phase (before governor response):

  \[ \boxed{\Delta f = \frac{f_0}{2H} \Delta P \cdot t} \]

  where \(\Delta P\) in per unit on system base, \(t\) in seconds.

3.4 Load Frequency Control in Interconnected Systems

• Automatic Load Frequency Control (ALFC) / Automatic Generation Control (AGC):

  • Primary control (Governor droop):

    • Droop \(R = \frac{\Delta f}{\Delta P}\) (Hz per MW or per unit).

    • For a load change \(\Delta P_D\), frequency changes by \(\Delta f = -\frac{\Delta P_D}{1/R + \sum 1/R_i}\) (for parallel generators).

    • Generators share load proportional to \(1/R\) (inverse droop).

    • Frequency regulation: \(R\) determines steady-state frequency deviation.

  • Secondary control (Area Control Error - ACE):

    • ACE = \(\Delta P_{\text{tie}} + B \Delta f\), where \(\Delta P_{\text{tie}}\) = tie-line power deviation from schedule, \(B\) = frequency bias constant (often \(B = 1/R\) or based on area load).

    • ACE integrated to adjust speed changer set points (\(\Delta P_{\text{ref}}\)) to drive ACE to zero, restoring frequency and tie-line power to schedule.

  • Tie-line modeling: For two areas, tie-line power deviation:

    \[ \Delta P_{\text{tie}}(s) = T_{12} \left( \Delta \delta_1(s) - \Delta \delta_2(s) \right) \]

    where \(T_{12} = \frac{V_1 V_2}{X_{12}} \cos \delta_{12}^0\). Since \(\Delta \delta_i(s) = \frac{2\pi}{s} \Delta f_i(s)\),

    \[ \Delta P_{\text{tie}}(s) = \frac{2\pi T_{12}}{s} (\Delta f_1 - \Delta f_2) = \frac{K_{\text{tie}}}{s} (\Delta f_1 - \Delta f_2) \]

    with \(K_{\text{tie}} = 2\pi T_{12}\).

3.5 Numerical Problem Example

Problem: A 100 MVA, 50 Hz turbo alternator with \(H = 4.5\ \text{kW·s/kVA}\) (i.e., 4.5 s) operates at no load. A load of 25 MW is suddenly applied. Steam valves open after 0.6 s due to governor delay. Find frequency before steam flow increases.
Solution:

\(\Delta P = 25/100 = 0.25\ \text{p.u.}\), \(t = 0.6\ \text{s}\), \(f_0 = 50\ \text{Hz}\), \(H = 4.5\ \text{s}\).

Acceleration phase:

\[ \Delta f = \frac{f_0}{2H} \Delta P \cdot t = \frac{50}{2 \times 4.5} \times 0.25 \times 0.6 = \frac{50}{9} \times 0.15 = 0.833\ \text{Hz} \]

Frequency drops to \(50 - 0.833 = 49.167\ \text{Hz}\).

4.0 VOLTAGE CONTROL AND REACTIVE POWER MANAGEMENT

4.1 Relationship between Reactive Power Flow and Voltage Profile

• From transmission line equations (ignoring resistance for long lines):

  \[ |V_s| - |V_r| \approx \frac{Q X}{|V|} \]

  where \(Q\) is reactive power flow at receiving end.

  • \(Q > 0\) (inductive, absorbing reactive) → voltage drop.

  • \(Q < 0\) (capacitive, generating reactive) → voltage rise.

• Methods for protection and absorption of reactive power:

  • Protection (voltage support): Shunt capacitors, series capacitors, synchronous condensers (over-excited), SVC/STATCOM (generating mode).

  • Absorption (voltage reduction): Shunt reactors, synchronous condensers (under-excited), SVC/STATCOM (absorbing mode).

4.2 Generation and Absorption of Reactive Power by Components

4.3 Voltage Control Methods

4.3.1 Excitation Systems

• Need and function: Maintain terminal voltage, control reactive power output, improve transient stability.

• Types:

  • DC exciters with amplidyne: DC generator driven by prime mover, output amplified by amplidyne (high-gain DC amplifier). Largely obsolete.

  • AC static exciters:

    • AC generator (brushless) with rotating rectifier, or static thyristor rectifier from stator terminals.

    • Fast response, common in modern plants.

  • Brushless excitation system:

    DiagramCANVAS: Functional block diagram: AC exciter (small AC generator) on same shaft → rotating rectifier (thyristor/diode) → DC output to main generator rotor field. Excitation control via stationary regulator sending firing pulses to rotating thyristors via transformer/telemetry.
    • Advantages: No brushes/slip rings, low maintenance, fast response.

  • Limiting features:

    • V/Hz limit: Prevents core overflux at low frequency/high voltage.

    • Over-excitation limit: Thermal limit on field winding.

    • Under-excitation limit: Stability limit to prevent loss of synchronism.

• Automatic Voltage Regulator (AVR) for turbo generators:

  DiagramCANVAS: Block diagram: Voltage sensor (PT) → comparator (with reference \(V_{\text{ref}}\)) → error amplifier → exciter (static/rotating) → generator field. Optional: Power system stabilizer (PSS) input from speed/power.

  Components: Sensing, excitation control, power stage, limiters.

4.3.2 Shunt and Series Compensation

• Shunt capacitors/Reactors:

  • Capacitors: Generate reactive power, boost voltage at buses.

  • Reactors: Absorb reactive power, lower voltage.

  • Switched based on voltage levels (e.g., on/off at substations).

• Series capacitors:

  • Connected in series with line to compensate reactance.

  • Increase power transfer capability: \(P_{\max} \propto 1/X_{\text{eq}}\).

  • Issues: Subsynchronous resonance (SSR), need protection (e.g., bypass switch).

4.3.3 Tap-changing Transformers

• On-load tap changers (OLTC): Adjust turns ratio under load to control voltage at a bus. Used for voltage regulation along feeders.

• Off-circuit tap changers: Manual adjustment when de-energized.

4.3.4 SVC and STATCOM

• SVC (Static VAR Compensator): Combination of thyristor-controlled reactor (TCR) and thyristor-switched capacitor (TSC). Provides continuous reactive power control.

• STATCOM (Static Synchronous Compensator): Voltage source converter (VSC) based. Better dynamic response, wider operating range (can generate/absorb), and supports low voltage better.

4.4 Modern Voltage Control Strategies

• Coordinated control: Multiple devices (AVR, OLTC, SVC, STATCOM) coordinated via centralized or distributed control.

• Wide-area measurement systems (WAMS): Use PMU data for global voltage stability assessment and control.

• Adaptive and predictive control: Based on real-time system conditions and forecasts.

5.0 ECONOMIC DISPATCH AND ENERGY PRICING

5.1 Economic Operation of Power Systems

• Objective: Minimize total fuel cost while meeting load demand and operating constraints (generation limits, line flows).

• Economic dispatch: Determine optimal power output of each committed generating unit.

5.2 Economic Dispatch Problem

• Neglecting transmission losses:

  • Incremental fuel cost: \(\lambda_i = \frac{dF_i}{dP_i}\) ($/MWh).

  • Equal incremental cost criterion: \(\lambda_i = \lambda\) for all generators at optimum.

  • Solution: Find \(\lambda\) such that \(\sum P_i = P_D\).

• Including transmission losses:

  • Losses approximated by B-coefficients (loss formula):

    \[ P_{\text{loss}} = \sum_{i=1}^{n} \sum_{j=1}^{n} B_{ij} P_i P_j \]

    where \(B_{ij}\) are symmetric constants (\(B_{ii} > 0\), \(B_{ij} \geq 0\)).

  • Lagrangian method:

    \[ \mathcal{L} = \sum_{i=1}^{n} F_i(P_i) + \lambda \left( P_D + P_{\text{loss}} - \sum_{i=1}^{n} P_i \right) \]

    Conditions:

    \[ \frac{\partial \mathcal{L}}{\partial P_i} = \frac{dF_i}{dP_i} + \lambda \left( \frac{\partial P_{\text{loss}}}{\partial P_i} - 1 \right) = 0 \]

    \[ \Rightarrow \frac{dF_i}{dP_i} = \lambda \left(1 - \frac{\partial P_{\text{loss}}}{\partial P_i} \right) \]

  • Penalty factor for plant \(i\):

    \[ \boxed{L_i = \frac{1}{1 - \frac{\partial P_{\text{loss}}}{\partial P_i}}} \]

    Then condition becomes: \(\frac{dF_i}{dP_i} \cdot L_i = \lambda\). Plants with high loss contribution (large \(\partial P_{\text{loss}}/\partial P_i\)) have higher \(L_i\), meaning their effective incremental cost is higher.

5.3 Pricing of Energy and Transmission Service

• In deregulated markets, locational marginal pricing (LMP) is common:

  \[ \text{LMP}_i = \text{Marginal generation cost} + \text{Loss component} + \text{Congestion component} \]

  • Marginal generation cost: Incremental cost of the marginal generator at that location.

  • Loss component: Cost of incremental losses.

  • Congestion component: Arises when transmission constraints cause price differences across locations.

• Transmission service pricing:

  • Congestion rent: Revenue from LMP differences.

  • Grid usage charges: Postage stamp, path-based, or nodal charges.

  • Ancillary services: Separate markets for frequency control, voltage support, etc.

6.0 MODERN POWER SYSTEM ISSUES: DEREGULATION & RESTRUCTURING

6.1 Need for Interconnected Power Systems

• Necessity:

  • Reliability: Backup during outages, reserve sharing.

  • Economy: Peak load sharing, use of distant cheap generation (e.g., hydro-thermal complementation).

  • Reduced spinning reserve: Shared reserves lower overall requirement.

• Problems:

  • Complex control and operation: Need for coordination among many control areas.

  • Stability issues: Synchronizing many machines, interarea oscillations.

  • Protection coordination: More complex fault currents.

  • Communication and coordination: Essential for AGC and emergency control.

6.2 Deregulation and Restructuring

• Concepts: Unbundling of generation, transmission, distribution into independent entities; introduction of competition in generation and retail.

• Objectives: Reduce costs, improve efficiency, encourage innovation, customer choice.

• Models:

  • POOL: Single buyer (system operator) purchases all generation and sells to retailers/consumers.

  • Bilateral: Direct contracts between generators and consumers/retailers.

  • Hybrid: Combination of pool and bilateral markets.

• Effects and challenges:

  • Positive: Lower prices, innovation, customer choice.

  • Challenges:

    • Transmission congestion: Need for congestion management (e.g., LMP).

    • Market power: Abuse by dominant players; mitigation via market design.

    • Investment incentives: Transmission and generation investment in competitive environment.

    • Ancillary services: Separate procurement and pricing.

    • Reliability assurance: Independent system operator (ISO) responsibilities.

6.3 Distributed Generation (DG)

• Definition: Small generators (renewable: solar, wind; or conventional: diesel, gas) connected at distribution level or near load centers.

• Impact:

  • Positive: Reduced transmission losses, improved voltage profile (if properly placed), renewable integration.

  • Challenges:

    • Protection coordination: Bi-directional flows complicate relay settings.

    • Voltage regulation: Reverse power flows may cause overvoltage.

    • Islanding: Unintentional islands during outages.

    • Power quality: Harmonics, flicker.

  • Requires smart grid technologies (advanced inverters, communication).

6.4 Transmission Open Access (TOA)

• Definition: Non-discriminatory access to transmission network for all generators and consumers.

• Role of system operator: Independent entity to:

  • Operate grid securely.

  • Facilitate markets (dispatch, congestion management).

  • Procure ancillary services.

  • Ensure reliability (N-1 security).

• Issues:

  • Tariff design: Fair transmission charges (postage stamp, path-based, nodal).

  • Congestion management: Methods like re-dispatch, load shedding, LMP.

  • Coordination with distribution networks: Especially with DG.

Key Formulas Summary (Boxed)

• Y_bus formation: \(Y_{\text{bus}} = \mathbf{A}^T \mathbf{y} \mathbf{A}\)

• Swing equation: \(M \frac{d^2 \delta}{dt^2} = P_m - P_e\), \(M = \frac{2H}{\omega_s}\)

• Linearized swing: \(M \frac{d^2 \Delta\delta}{dt^2} = \Delta P_m - K \Delta\delta\)

• Frequency deviation (acceleration): \(\Delta f = \frac{f_0}{2H} \Delta P \cdot t\)

• Tie-line power deviation: \(\Delta P_{\text{tie}}(s) = \frac{K_{\text{tie}}}{s} (\Delta f_1 - \Delta f_2)\)

• Penalty factor: \(L_i = \frac{1}{1 - \frac{\partial P_{\text{loss}}}{\partial P_i}}\)

• Loss formula: \(P_{\text{loss}} = \sum_i \sum_j B_{ij} P_i P_j\)

[!TIP] Exam Tips and Common Pitfalls

• Y_bus formation: Always include shunt admittance if given; off-diagonals are negative.

• Gauss-Seidel: For PV buses, after updating voltage, scale to specified |V| and recalculate Q.

• Newton-Raphson: Remember the structure of Jacobian; for PV buses, rows/columns for |V| are omitted.

• Swing equation: Use consistent units. \(H\) in seconds, \(f_0\) in Hz, \(\Delta P\) in per unit.

• EAC: Accelerating area is above \(P_m\) and below \(P_e^{\text{fault}}\); decelerating area is below \(P_m\) and above \(P_e^{\text{post}}\).

• LFC: Primary control reduces frequency deviation but does not restore nominal frequency; secondary control (ACE) does.

• Voltage control: Reactive power flow primarily affects voltage magnitude; active power affects angle.

• Economic dispatch: With losses, plants with high loss contribution (large \(\partial P_{\text{loss}}/\partial P_i\)) have higher penalty factors.

• Deregulation: Understand the difference between pool (single buyer) and bilateral (direct contracts) models.