Power System‑II (EX-601) - Unit 2 Short Notes

UNIT 2: Power System Analysis, Stability, and Control

1. Load Flow Analysis (Power Flow Studies)

Significance and Applications

• Planning: Expansion planning, reinforcement studies.

• Operation: Day-ahead scheduling, contingency analysis, optimal dispatch.

• Optimization: Loss minimization, voltage profile improvement.

• Provides steady-state voltage magnitudes and angles, real/reactive power flows.

Classification of Buses

Justification: Slack bus absorbs transmission losses; PV buses fix voltage; PQ buses have fixed power injection.

Formation of Y_bus (Admittance Matrix)

Step-by-step:

1. Initialize: Create N×N matrix with zeros.

2. Diagonal Elements: $$\displaystyle Y_{ii} = \sum_{k=1}^{N} y_{ik} + y_{shunt,i} $$ (sum of all admittances connected to bus i, including line charging).

3. Off-diagonal Elements: $$\displaystyle Y_{ij} = -y_{ij} $$ for directly connected buses; 0 otherwise.

4. Incorporate:

   • Mutual coupling: Add mutual impedance terms.

   • Transformer taps: Phase-shifting transformer → modify off-diagonals; tap-changing → adjust diagonals.

   • Shunt elements: Add to diagonal.

Sparsity: Y_bus is sparse (mostly zeros). Storage uses compressed sparse row (CSR) or linked-list to save memory.

Solution Methods

Gauss-Seidel Method

• Derivation: From nodal equation $$\displaystyle I = YV $$, iterate: $$\displaystyle V_i^{(k+1)} = \frac{1}{Y_{ii}} \left( I_i - \sum_{j=1, j\neq i}^{N} Y_{ij} V_j^{(k)} \right) $$.

• Algorithm:

  1. Initialize voltages (flat start: 1∠0° p.u.).

  2. For each PQ bus, update V using above equation.

  3. For PV bus: update δ from real power equation, then compute Q; if Q exceeds limit, convert to PQ.

  4. Check convergence: $$\displaystyle |ΔV_i| < ε $$ for all buses.

• Advantages: Simple, low memory.

• Disadvantages: Slow convergence, may diverge for large systems.

• Acceleration factor: α (1.0–1.6) to speed up: $$\displaystyle V_i^{new} = V_i^{old} + αΔV_i $$.

Newton-Raphson Method

• Formulation: Taylor series expansion of power mismatches:

$$ \begin{bmatrix} ΔP \\ ΔQ \end{bmatrix} = \begin{bmatrix} J_{11} & J_{12} \\ J_{21} & J_{22} \end{bmatrix} \begin{bmatrix} Δδ \\ Δ|V| \end{bmatrix} $$

where Jacobian J:

$$ J_{11} = \frac{∂P_i}{∂δ_j},\; J_{12} = \frac{∂P_i}{∂|V_j|},\; J_{21} = \frac{∂Q_i}{∂δ_j},\; J_{22} = \frac{∂Q_i}{∂|V_j|} $$

• Algorithm:

  1. Initialize voltages.

  2. Compute P, Q from $$\displaystyle P_i = |V_i| \sum |V_j| (G_{ij} \cos δ_{ij} + B_{ij} \sin δ_{ij}) $$, similar for Q.

  3. Calculate mismatches ΔP, ΔQ.

  4. Form Jacobian (sparse structure exploited).

  5. Solve for Δδ, Δ|V|.

  6. Update: $$\displaystyle δ_i^{new} = δ_i^{old} + Δδ_i $$, $$\displaystyle |V_i|^{new} = |V_i|^{old} + Δ|V_i| $$.

  7. Repeat until $$\displaystyle |ΔP|, |ΔQ| < ε $$.

• Advantages: Quadratic convergence, robust.

• Disadvantages: Complex Jacobian formation, high memory.

Comparison of Methods

Fast Decoupled Load Flow (FDLF): Exploits strong decoupling between P-δ and Q-V; uses constant Jacobian; very fast but less accurate for high R/X lines.

Flow Charts and Algorithms

• PQ-only system: Iterate all PQ buses until convergence; slack bus updated last.

• PV buses: After δ update, compute Q; if Q > Q_max or < Q_min, set Q to limit and treat as PQ for that iteration.

• Convergence criteria: Max $$\displaystyle |ΔV| < 0.001 $$ p.u. or $$\displaystyle |ΔP|, |ΔQ| < 0.001 $$ MW/Mvar.

2. Power System Stability

Stability Classification

• Steady-state (Small-signal): Stability to small disturbances (e.g., load fluctuations). Analyzed via linearized models.

• Transient: Stability to large disturbances (faults, line trips). Non-linear analysis, time-domain simulation or EAC.

• Dynamic (Long-term): Stability over seconds to minutes; involves slow controls (governor, AVR, tap changers).

Swing Equation

Derivation:

Rotor dynamics: $$\displaystyle J \frac{d^2θ_m}{dt^2} = T_m - T_e $$, where θ_m = mechanical angle.

Convert to electrical angle δ = (P/2) θ_m (for P poles):

$$ M \frac{d^2δ}{dt^2} = P_m - P_e $$

where $$\displaystyle M = \frac{2H}{ω_s} $$ (inertia constant H in MJ/MVA, ω_s=2πf rad/s).

Physical Significance: Relates accelerating power (P_m - P_e) to rate of change of rotor angle δ. If P_m > P_e, δ increases (accelerating); else decelerating.

Linearized for small-signal:

$$ M \frac{d^2(Δδ)}{dt^2} + D \frac{d(Δδ)}{dt} + ω_s^2 K_{δ} Δδ = 0 $$

where D = damping coefficient, K_δ = synchronizing coefficient.

Steady-State Stability

• Power-angle characteristic (single machine-infinite bus):

$$ P_e = \frac{E V}{X} \sin δ $$

Maximum at δ = 90°: $$\displaystyle P_{max} = \frac{EV}{X} $$.

• Stability limit: Maximum P_e transferable without losing synchronism.

• Improvement methods:

  • Reduce transfer reactance X (parallel lines, series compensation).

  • Increase excitation (higher E).

  • Use high-speed excitation systems.

Transient Stability

Equal Area Criterion (EAC)

• Derivation: From swing equation: $$\displaystyle M \frac{d^2δ}{dt^2} = P_m - P_e(δ) $$. Multiply by dδ/dt and integrate:

$$ \int_{δ_0}^{δ_{cr}} (P_m - P_e) dδ = 0 $$

Accelerating area (A_acc) = Decelerating area (A_dec).

• Application:

  1. Pre-fault: P_m = constant, P_e = P_e0(δ).

  2. Fault: P_e = P_ef(δ) (lower due to higher X).

  3. Post-fault: P_e = P_ep(δ).

  4. Find δ_cr where A_acc = A_dec.

  5. Critical clearing time t_cr from $$\displaystyle \frac{dδ}{dt} = ω - ω_s $$ integration.

• Step-by-step:

  • Plot P_m vs δ, P_e0, P_ef, P_ep.

  • Determine δ₀ (initial), δ_max (post-fault intersection).

  • Calculate areas: $$\displaystyle A_{acc} = \int_{δ_0}^{δ_c} (P_m - P_{ef}) dδ $$, $$\displaystyle A_{dec} = \int_{δ_c}^{δ_{max}} (P_{ep} - P_m) dδ $$.

  • Solve for δ_c (δ_cr).

• Critical clearing angle δ_cr: Angle at fault clearing where system just stable.

• Critical clearing time t_cr: Maximum fault duration without instability.

Factors Affecting Transient Stability

1. Fault type/location: More severe (3-phase) and closer faults reduce P_ef more.

2. Clearing time: Longer t_c → larger A_acc → less stable.

3. System strength: Higher X (weak system) → lower P_max → less stable.

4. Inertia constant H: Higher H → slower acceleration → more time to clear fault.

5. Initial loading: Higher P_m → smaller δ₀ margin to δ_max.

6. Automatic controls: Fast valving, braking resistors, FACTS improve stability.

Methods to Improve Transient Stability

• Fast valving: Quickly reduce steam input during fault.

• Braking resistors: Insert resistors to reduce electrical power temporarily.

• Single-pole reclosing: For single-line-to-ground faults, only faulty pole opened.

• FACTS: SVC/STATCOM (dynamic V support), TCSC (reduce X).

• Energy storage: Fast injection of power.

Multi-Machine Stability (Brief)

• Challenges: Many machines, complex interactions; EAC not directly applicable.

• Solution: Numerical integration (e.g., modified Euler, Runge-Kutta) of swing equations for each machine.

• Need: For detailed stability studies, especially with controls.

3. Frequency Control and Load Frequency Control (LFC)

Necessity of Strict Frequency Control

• Motor speeds: AC motors synchronous speed ∝ f.

• Industrial processes: Sensitive equipment (e.g., paper mills, aluminum smelting).

• Grid synchronization: All generators must run at same f; deviations cause instability.

• Interconnected systems: Frequency oscillations → cascading outages → blackouts.

Turbine Governing System Modeling

• Governor characteristics:

  • Droop (speed regulation): $$\displaystyle R = \frac{Δf_{no-load}}{ΔP_{full-load}} $$ (typically 4–5%).

  • No-load speed: Speed at zero load.

• Time constants:

  • Delay time (T_D): Relay dead-band, valve closure time.

  • Transient droop compensation: Temporary reduction in droop for fast response.

• Block diagram:

  
  Δf → [Speed relay (T_D)] → [Governor (R)] → [Servo (T_G)] → [Valve] → ΔP_m
  
  

Inertia Constant (H) and Equivalent Inertia

• Definition: $$\displaystyle H = \frac{\text{Kinetic energy at } ω_s}{\text{MVA rating}} $$ (MJ/MVA or kW-sec/kVA).

• Kinetic energy: $$\displaystyle KE = \frac{1}{2} J ω_s^2 = H \cdot S_{base} $$.

• Equivalent H for multiple machines on common base MVA_base:

$$ H_{eq} = \frac{\sum (H_i \cdot S_{i})}{S_{base}} $$

where S_i is machine MVA rating.

Load Frequency Control (LFC) / ALFC

Primary Control (Speed Control)

• Governor responds to Δf: ΔP_m = -Δf/R.

• Steady-state frequency error: Non-zero because ΔP_m limited by R.

• Free governor action: Primary control without coordination → frequency deviation remains.

Secondary Control (Area Control)

• Area Control Error (ACE):

$$ ACE = ΔP_{tie} + B Δf $$

where B = frequency bias factor (MW/Hz).

• Tie-line power model (two-area):

$$ ΔP_{tie} = \frac{P_{tie,0}}{δ_0} Δδ = T \cdot Δδ $$

T = tie-line synchronizing coefficient.

• Bias factor B: Should equal area's load-frequency characteristic (D + 1/R) for zero steady-state error.

• Integral control: ACE integrated → control signal → set-point adjustment → eliminates steady-state error.

Block Diagram of ALFC

• Single-area:

  
  ΔP_load → [Δf] → [1/R] → ΔP_gov → [1/(T_G s + 1)] → ΔP_turbine → [ΔP_m]
  
  

  with feedback from Δf.

• Two-area:

  Each area has ACE computed from ΔP_tie and Δf; integral of ACE adjusts governor set-point.

Numerical Problems

1. Frequency deviation with delay:

$$ Δf = \frac{ΔP_m}{D + 1/R} \left(1 - e^{-t/T_D}\right) \text{ (approx)} $$

before valve opens.

2. ACE computation: Given ΔP_tie (positive if import), Δf, B → ACE.

3. Equivalent H: Use formula above.

4. Voltage Control and Reactive Power Management

Reactive Power and Voltage Relationship

• Phasor diagram: Voltage drop across line: $$\displaystyle ΔV ≈ \frac{P R + Q X}{V} $$.

  For typical lines (R << X): $$\displaystyle ΔV ≈ \frac{Q X}{V} $$ → voltage drop primarily due to reactive power.

• Impact: High Q flow → low V; capacitive Q → V rise; inductive Q → V drop.

• Voltage stability: System's ability to maintain V; depends on reactive power support.

Generation and Absorption of Reactive Power

Excitation Systems

Need for Excitation Control

• Voltage regulation at generator terminals.

• Reactive power support.

• Stability enhancement (transient, dynamic).

Types of Excitation Systems

1. DC excitation: DC generator → slip-rings → field. Slow, maintenance.

2. AC static excitation: AC exciter → thyristor rectifier → field. Fast, reliable.

3. Brushless excitation: AC excitor → rotating rectifier → field. No brushes, medium speed.

Functional Block Diagrams

Common components:

• Voltage regulator: Compares V_t with reference → error signal.

• Exciter: Amplifies signal (DC/AC/static).

• Power system stabilizer (PSS): Inputs Δω or ΔP → supplementary damping.

• Limiting features:

  • V/Hz limit: Over-fluxing of core.

  • Over-excitation limit: Field thermal limit.

  • Under-excitation limit: Stator core saturation, stability.

  • Stator current limit: Armature heating.

Automatic Voltage Regulator (AVR)

• Block diagram for turbo-generator:

  
  V_ref → [Error amp] → [AVR] → [Exciter] → [Generator field] → V_t
  
  

  with feedback from V_t (often through PT).

• Role: Maintains terminal voltage within limits; supports system voltage during disturbances.

Methods of Voltage Control

1. Reactive Power Compensation:

   • Shunt capacitors: Fixed/switched → supply Q locally → raise V.

   • Series capacitors: Compensate line X → increase P_max, improve stability.

   • Shunt reactors: Absorb excess Q (light load) → prevent overvoltage.

   • SVC/STATCOM: Dynamic V control via thyristors/IGBTs; fast response.

2. Transformer Tap Changing:

   • OLTC: Adjusts turns ratio → controls secondary voltage.

3. Generator Voltage Control: Via AVR set-point.

4. Coordinated Volt/VAR Optimization: Centralized control of capacitors, OLTCs, generator AVRs for optimal V profile and loss reduction.

5. Economic Dispatch and Optimal Power Flow

Economic Operation of Power Systems

• Objective: Minimize total fuel cost $$\displaystyle F = \sum_{i=1}^{n} F_i(P_i) $$ subject to $$\displaystyle \sum P_i = P_D + P_{loss} $$.

• Incremental fuel cost (λ): $$\displaystyle \frac{dF_i}{dP_i} $$ ($/MWh). At optimum, λ equal for all generators.

Economic Dispatch Without Losses

• Criterion: $$\displaystyle λ_i = λ $$ for all i (no losses).

• Solution: $$\displaystyle P_i = \frac{λ - a_i}{2b_i} $$ for quadratic cost $$\displaystyle F_i = a_i P_i + b_i P_i^2 $$.

Economic Dispatch With Losses

Loss Coefficients (B Coefficients)

• Loss formula: $$\displaystyle P_{loss} = \sum_{i} \sum_{j} B_{ij} P_i P_j + \sum_{i} B_{i0} P_i + B_{00} $$.

• Derivation: From $$\displaystyle I = YV $$, assume constant |V|, neglect phase angles → quadratic in P.

• B matrix symmetric; B_i0, B₀₀ from network data.

Penalty Factor

• Definition: $$\displaystyle L_i = \frac{1}{1 - \frac{∂P_{loss}}{∂P_i}} $$.

• Interpretation: Effective incremental cost for generator i including its share of losses.

• Optimality: $$\displaystyle λ_i \cdot L_i = λ $$ (common λ).

Lagrangian Multiplier Method

• Lagrangian: $$\displaystyle \mathcal{L} = \sum F_i(P_i) + λ (P_D + P_{loss} - \sum P_i) $$.

• Coordination equations:

$$ \frac{dF_i}{dP_i} - λ \left(1 - \frac{∂P_{loss}}{∂P_i}\right) = 0 \Rightarrow \frac{dF_i}{dP_i} = λ L_i $$

• λ-iteration method:

  1. Guess λ.

  2. Compute P_i from $$\displaystyle \frac{dF_i}{dP_i} = λ L_i $$ (using B coefficients to get L_i).

  3. Calculate P_loss from B formula.

  4. Check power balance: $$\displaystyle \sum P_i \stackrel{?}{=} P_D + P_{loss} $$.

  5. Adjust λ (increase if excess, decrease if deficit).

  6. Iterate until balance satisfied.

Numerical Problems

• Calculate B coefficients from line admittances.

• Compute penalty factors.

• λ-iteration for optimal P_i.

Pricing of Energy and Transmission Services

• Marginal cost: λ from economic dispatch → cost of next MWh.

• Locational Marginal Pricing (LMP): λ + congestion cost + loss cost at each bus.

• Transmission pricing methods:

  • Postage stamp: Uniform charge per MWh.

  • Contract path: Cost based on designated path.

  • MW-mile: Charge proportional to flow × distance.

  • LMP-based: Reflects real-time congestion/losses.

• Unbundling: Separate charges for generation, transmission, distribution.

6. Modern Power System Concepts

Interconnected Power Systems

Necessity and Advantages

• Economies of scale: Share large generators.

• Reliability: Mutual support during contingencies.

• Load diversity: Reduced peak demand.

• Reserve sharing: Lower spinning reserve requirement.

Problems

• Stability: Angle, frequency, voltage stability more complex.

• Control coordination: LFC, AVR must work in unison.

• Cascading outages: One area's problem propagates.

• Operational complexity: Scheduling, protection, market operations.

Deregulation and Restructuring

Definitions

• Deregulation: Removal of regulatory controls; introduce competition.

• Restructuring: Reorganization into separate entities: generation, transmission, distribution, retail.

Effects

• Market-based pricing: Bidding, spot prices.

• ISO role: Independent operator for grid access, congestion management.

• Challenges:

  • Market power: Large players manipulating prices.

  • Congestion management: Need for transmission rights.

  • Investment incentives: Insufficient transmission investment.

How to Overcome

• Proper market design: Uniform pricing, bid caps, monitoring.

• Regulatory frameworks: FERC/CEA rules, open access.

• Financial transmission rights (FTRs): Hedge against congestion revenue.

Distributed Generation (DG)

Definition and Examples

• Small-scale generation (≤10 MW) near load: solar PV, wind turbines, microturbines, fuel cells.

Impact on Power Systems

• Positive:

  • Reduced transmission losses.

  • Improved reliability (backup).

  • Deferred T&D upgrades.

• Challenges:

  • Voltage regulation: Reverse power flow → overvoltage.

  • Protection coordination: Fault currents from DG.

  • Stability: Islanding detection, inertia reduction.

• Integration issues:

  • Interconnection standards (IEEE 1547).

  • Grid support functions (voltage ride-through, frequency response).

7. Integrated Problem-Solving and Applications

Combined Frequency-Voltage Control

• In interconnected areas, LFC and AVR interact: frequency changes affect reactive power via load characteristics; voltage changes affect real power via governor droop.

• Coordination: Use of power system stabilizers (PSS) to damp oscillations; coordinated tuning of AVR and governor.

Coordination between Economic Dispatch and Security Constraints

• Economic dispatch must respect:

  • Thermal limits: Line flows < ratings.

  • Stability limits: P < P_max (steady-state), δ < δ_cr (transient).

  • Voltage limits: |V| within 0.95–1.05 p.u.

• Security-Constrained OPF (SCOPF): Includes contingency constraints in optimization.

Numerical Problems Integrating Multiple Concepts

1. Load flow → Economic dispatch:

   • Perform load flow to get Y_bus, losses.

   • Compute B coefficients.

   • Run λ-iteration for optimal dispatch.

2. Stability with controls:

   • Swing equation with governor/AVR models.

   • Simulate fault → clearing → post-fault response.

   • Use EAC with modified P_e due to controls.

3. Frequency/voltage response:

   • Load change → Δf from LFC model → ΔV from load V sensitivity.

   • Include capacitor switching, OLTC actions.

Case Studies (Brief)

• 2003 Northeast blackout: Frequency instability, lack of situational awareness.

• Economic dispatch in PJM: LMP-based markets, FTR auctions.

• High DG penetration in Germany: Voltage rise issues, smart inverter requirements.

Priority Topics from Past Exams (Very High Frequency)

1. Load flow methods: Gauss-Seidel (algorithm, PV bus handling), Newton-Raphson (Jacobian, flowchart).

2. Y_bus formation: Step-by-step with line data.

3. Swing equation: Derivation, physical significance.

4. Equal Area Criterion: Derivation, application to find δ_cr.

5. Frequency control: Governor modeling, H constant, ACE, tie-line modeling.

6. Voltage control: Reactive power sources, excitation systems, AVR.

7. Economic dispatch: Lagrangian method, penalty factors, loss coefficients.

[!TIP] Exam Focus:

• Y_bus: Always draw matrix, fill diagonal/off-diagonal.

• EAC: Sketch P-δ curves, label areas, show δ_cr calculation.

• LFC: Write ACE formula, explain B factor, draw block diagram.

• Economic dispatch: Derive coordination equation, show λ-iteration steps.

• Common Pitfall: Confusing PV and PQ buses; forgetting slack bus power balance; misapplying EAC for multi-machine.