Economic Load Dispatch

Economic Load Dispatch (ELD) Basics

Key Concepts for GATE

Objective: Minimize total generation cost while meeting demand

\[\text{Minimize } C_T = \sum_{i=1}^n C_i(P_{Gi})\]
Cost Function: Typically quadratic

\[C_i(P_{Gi}) = a_i + b_i P_{Gi} + c_i P_{Gi}^2 \text{ Rs/hr}\]
Incremental Cost: Slope of cost curve

\[IC_i = \dfrac{dC_i}{dP_{Gi}} = b_i + 2c_i P_{Gi} \text{ Rs/MWh}\]
Constraints:
- Power balance: \(\sum P_{Gi} = P_D + P_L\)
- Generator limits: \(P_{Gi,min} \leq P_{Gi} \leq P_{Gi,max}\)

ELD Without Transmission Losses

Problem Formulation

Minimize \(C_T = \sum_{i=1}^n (a_i + b_i P_{Gi} + c_i P_{Gi}^2)\)
Subject to: \(\sum_{i=1}^n P_{Gi} = P_D\)

Optimal Condition: All units operate at same incremental cost

\[\dfrac{dC_1}{dP_{G1}} = \dfrac{dC_2}{dP_{G2}} = \cdots = \lambda\]
Solution Steps:
1. Calculate: \(\lambda = \dfrac{\sum b_i + 2\sum c_i P_D}{\sum \dfrac{1}{2c_i}}\)
2. Find: \(P_{Gi} = \dfrac{\lambda - b_i}{2c_i}\)
3. Verify: \(\sum P_{Gi} = P_D\)

ELD Without Losses - Numerical Example

Example

Two generators with costs:

\[\begin{aligned} C_1 &= 0.1 P_{G1}^2 + 40 P_{G1} + 100 \text{ Rs/hr} \\ C_2 &= 0.2 P_{G2}^2 + 30 P_{G2} + 80 \text{ Rs/hr} \\ \text{Demand} &= 200 \text{ MW} \end{aligned}\]

Solution:

\(IC_1 = 0.2 P_{G1} + 40\), \(IC_2 = 0.4 P_{G2} + 30\)
At optimum: \(0.2 P_{G1} + 40 = 0.4 P_{G2} + 30\)
With constraint: \(P_{G1} + P_{G2} = 200\)
Result: \(P_{G1} = 116.67\) MW, \(P_{G2} = 83.33\) MW
\(\lambda = 63.33\) Rs/MWh

ELD With Transmission Losses

Transmission Loss Formula (B-coefficients):

\[P_L = \sum_{i=1}^n \sum_{j=1}^n P_{Gi} B_{ij} P_{Gj}\]
Modified Optimality Condition:

\[\dfrac{dC_i}{dP_{Gi}} \times \dfrac{1}{1 - \dfrac{\partial P_L}{\partial P_{Gi}}} = \lambda\]
Penalty Factor:

\[L_i = \dfrac{1}{1 - \dfrac{\partial P_L}{\partial P_{Gi}}}\]
Loss Sensitivity:

\[\dfrac{\partial P_L}{\partial P_{Gi}} = 2 \sum_{j=1}^n B_{ij} P_{Gj}\]

ELD With Losses - Solution Algorithm

Iterative Solution Method

Assume initial \(\lambda\) (start with loss-free solution)
Calculate penalty factors: \(L_i = \dfrac{1}{1 - \dfrac{\partial P_L}{\partial P_{Gi}}}\)
Find generations: \(\dfrac{dC_i}{dP_{Gi}} = \dfrac{\lambda}{L_i}\)
Calculate total loss: \(P_L = \sum_{i=1}^n \sum_{j=1}^n P_{Gi} B_{ij} P_{Gj}\)
Check power balance: \(\sum P_{Gi} = P_D + P_L\)
If not balanced, adjust \(\lambda\) and repeat

Important GATE Formula

For simplified loss model: \(P_L = B_{00} + \sum B_{0i} P_{Gi} + \sum \sum B_{ij} P_{Gi} P_{Gj}\)

Generator Limits and Lagrange Multipliers

When generator hits limits:
- If \(P_{Gi} < P_{Gi,min}\): Set \(P_{Gi} = P_{Gi,min}\)
- If \(P_{Gi} > P_{Gi,max}\): Set \(P_{Gi} = P_{Gi,max}\)
- Remove from optimization and redistribute load
Lagrangian Method:

\[L = \sum C_i(P_{Gi}) + \lambda(\sum P_{Gi} - P_D - P_L)\]
KKT Conditions:

\[\dfrac{\partial L}{\partial P_{Gi}} = \dfrac{dC_i}{dP_{Gi}} + \lambda(1 - \dfrac{\partial P_L}{\partial P_{Gi}}) = 0\]
Physical Interpretation: \(\lambda\) represents system marginal cost

Compensation Techniques

Series Compensation

Purpose: Improve power transfer capability and stability
Implementation: Series capacitor in transmission line
Effect on Line Reactance:

\[X_{eff} = X_L - X_C\]
Power Transfer Enhancement:

\[P_{max} = \dfrac{V_S V_R}{X_L - X_C} \sin \delta\]
Compensation Degree:

\[K = \dfrac{X_C}{X_L} \times 100\%\]
Typical values: 25-70%

Equivalent Circuit:

\(V_S\)

\(\downarrow\)

\(X_L\)

\(\downarrow\)

\(-jX_C\)

\(\downarrow\)

\(V_R\)

Series Compensation - Benefits and Drawbacks

Benefits

Increases power transfer capability
Improves transient stability
Reduces voltage drop across line
Better voltage regulation
Reduces line losses

Drawbacks

Sub-synchronous resonance (SSR) problems
High installation and maintenance cost
Protection complexity
Fault current may increase

GATE Important

Maximum compensation limited to 70% due to SSR concerns

Shunt Compensation

Types:
- Shunt capacitors (lagging PF loads)
- Shunt reactors (leading PF, long lines)
Applications:
- Voltage regulation
- Power factor correction
- Reactive power compensation
Location: Load centers, substations
Voltage Effect:

\[\Delta V = \dfrac{Q_C X}{V}\]

Shunt Capacitor:

\(I_L\)

\(\rightarrow\)

Load

\(\uparrow\)

\(I_C\)

\(\uparrow\)

\(C\)

Key Formula

Reactive power from capacitor: \(Q_C = V^2 \omega C\)

Shunt Compensation - Detailed Analysis

Capacitive Compensation:
- Provides leading reactive power
- Improves lagging power factor
- Reduces line current: \(I_{new} = \sqrt{P^2 + (Q_L - Q_C)^2}/V\)
Inductive Compensation:
- Absorbs leading reactive power
- Used for long transmission lines
- Prevents voltage rise during light load
Optimal Location:
- Capacitors: At load centers (2/3 distance from source)
- Reactors: At line terminals

Power Factor Correction

Objective: Improve power factor from \(\cos \theta_1\) to \(\cos \theta_2\)
Capacitor Size Calculation:

\[Q_C = P (\tan \theta_1 - \tan \theta_2)\]
where:
- \(\theta_1 = \cos^{-1}(\text{PF}_1)\) (original)
- \(\theta_2 = \cos^{-1}(\text{PF}_2)\) (desired)
Alternative Form:

\[Q_C = P \left(\dfrac{\sin \theta_1}{\cos \theta_1} - \dfrac{\sin \theta_2}{\cos \theta_2}\right)\]
Capacitor Rating:

\[C = \dfrac{Q_C}{\omega V^2} \text{ Farads}\]

Power Factor Correction - Benefits

Economic Benefits

Reduced line losses: \(P_{loss} = I^2 R\) (lower current)
Improved voltage regulation
Increased system capacity
Reduced penalty charges from utility

Technical Benefits

Reduced transformer and cable loading
Better voltage stability
Improved motor starting capability
Reduced reactive power flow

Example

For 100 kW load at 0.8 lagging PF, to improve to 0.95 PF: \(Q_C = 100(\tan 36.87° - \tan 18.19°) = 100(0.75 - 0.33) = 42\) kVAR

Static VAR Compensators (SVC)

Components:
- Thyristor Switched Capacitor (TSC)
- Thyristor Controlled Reactor (TCR)
- Fixed Capacitor (FC)
Operating Principle:

\[Q_{SVC} = Q_C - Q_L(\alpha)\]
where \(\alpha\) is firing angle
Advantages:
- Continuous reactive power control
- Fast response (2-3 cycles)
- Voltage regulation capability
- Harmonic filtering
V-I Characteristic: Drooping characteristic for stable operation

Comparison of Compensation Methods

Compensation Methods Comparison
Feature	Series	Shunt	SVC
Primary purpose	Power transfer	Voltage control	Dynamic control
Response time	Instantaneous	Instantaneous	2-3 cycles
Control	Fixed	Fixed/Switched	Continuous
Cost	High	Moderate	High
Stability impact	Transient	Voltage	Both
Installation	Line	Substation	Substation

Key Formulas Summary for GATE

Economic Dispatch without losses:

\[\dfrac{dC_i}{dP_{Gi}} = \lambda\]
Economic Dispatch with losses:

\[\dfrac{dC_i}{dP_{Gi}} = \lambda (1 - \dfrac{\partial P_L}{\partial P_{Gi}})\]
Series Compensation:

\[\% \text{Compensation} = \dfrac{X_C}{X_L} \times 100,\quad P_{max} = \dfrac{V_S V_R}{X_L - X_C}\]
Power Factor Correction:

\[Q_C = P (\tan \cos^{-1} \text{PF}_1 - \tan \cos^{-1} \text{PF}_2)\]
Shunt Compensation:

\[Q_C = V^2 \omega C,\quad \Delta V = \dfrac{Q_C X}{V}\]

GATE Problem-Solving Tips

Economic Load Dispatch

Always check if generator limits are violated
For losses, use iterative method systematically
Remember: \(\lambda\) has units of Rs/MWh

Compensation

Series compensation: Focus on power transfer improvement
Shunt compensation: Focus on voltage and PF improvement
Power factor correction: Use trigonometric identities carefully

Common Mistakes to Avoid

Forgetting to include losses in power balance
Wrong signs in reactive power calculations
Mixing up series and shunt compensation effects