PTDF using AC Power Flow

PTDF using AC Power Flow Model

• Power Transfer Distribution Factor (PTDF) relies on voltage and angle sensitivity
• Sensitivity analysis calculates flow change on line l-m
• Change in real power flow on line l-m, (Δ𝑃𝑙𝑚​) is determined through the partial derivatives of Plm with respect to angles (𝛿) and voltage magnitudes (|V|)

PTDF (AC Model)

• Load Flow Solution uses a converged base case
• J represents the Jacobian matrix, used to solve the load flow equations

PTDF (AC Model) for MW Transaction

• The change in power flow, Δ𝑃𝑖​ , equals +𝑃𝑡 for a MW power transaction number t
• The change in power flow, Δ𝑃𝑗, equals −𝑃𝑡
• Δ𝑃𝑘​ and Δ𝑄𝑘 both equal 0, where k ranges from 1 to n, excluding i and j

PTDF (AC Model) Equations

• 𝑑𝑡 represents the PTDF

Available Transfer Capability (ATC) using PTDF

• A new flow equation on a transmission line sums the original MW flow and any changes in flow
• 𝑃𝑙𝑚𝑛𝑒𝑤​ = 𝑃𝑙𝑚0​ + 𝑑𝑙𝑚,𝑖𝑗​ 𝑃𝑖𝑗, where:
  • 𝑃𝑙𝑚𝑛𝑒𝑤 is the new flow on the line
  • 𝑃𝑙𝑚0 is the original flow
  • 𝑑𝑙𝑚,𝑖𝑗 is the PTDF
  • 𝑃𝑖𝑗 is the power transaction from i to j
• The maximum power transferable without overloading line lm is calculated by the formula
• 𝑃𝑙𝑚,𝑖𝑗𝑚𝑎𝑥= (𝑃𝑙𝑚𝑚𝑎𝑥 − 𝑃𝑙𝑚0)/ 𝑑𝑙𝑚,𝑖𝑗, where :
  • 𝑃𝑙𝑚,𝑖𝑗𝑚𝑎𝑥 is the maximum power that can be transferred
  • 𝑃𝑙𝑚𝑚𝑎𝑥 is the limit on the line
  • 𝑃𝑙𝑚0 is the original flow on the line
  • 𝑑𝑙𝑚,𝑖𝑗 is the PTDF

ATC Calculation

• ATC, or Available Transfer Capability, is the lowest of the maximum allowable transactions across all transmission lines
• 𝐴𝑇𝐶𝑖𝑗​ = min(𝑃𝑙𝑚,𝑖𝑗𝑚𝑎𝑥​), where the minimum is taken over all lines (lm)

Congestion Management Mechanisms

• Market Based Methods
  • Explicit auctioning
  • Nodal pricing
  • Zonal pricing
    • Price area management
  • Redispatch
  • Counter trade
• Non-Market Methods
  • Type of contract
  • First come first served
  • Pro-rata method
  • Curtailment

Alternative Classification for Congestion Management

• Allocation methods
• Alleviation methods

Non-Market Methods List

• First come first served basis
• Pro-rata methods
• Type of contract

India's Non-Market Method Approaches

• Type of contract: Utilizes firm/non-firm contracts for capacity allocation
• First come, first served: Employs STOA (Short-Term Open Access) for capacity allocation
• Pro-rata curtailment: Used as an alleviation method
• Market based: E-bidding for transmission capacity is utilized

Focus on Market Based Methods

Classical Derivative Based Optimization Technique

• Unconstrained optimization aims to optimize a function, 𝑓(𝑥)
• At an optimum point 𝑥∗, the derivative of the function is zero: 𝜕𝑓(𝑥∗) / 𝜕𝑥 = 0; this is a necessary condition
• The optimum point can be a maximum, minimum, or saddle point
  • Higher order derivatives determine the nature of the point
• A local or global optima point may exist
• Typically converges to local optima depending on the initial guess

Equality Constrained Optimization

• Minimizes 𝑓(𝑥) subject to the constraint 𝑔(𝑥) = 0
• Mathematical setup: 𝐿(𝑥,𝜆) = 𝑓(𝑥) + 𝜆𝑔(𝑥)
• Optimality condition for unconstrained problem = ∂𝐿(𝑥,𝜆) /∂𝑥= 0 = ∇𝑓(𝑥) + 𝜆∇𝑔(𝑥), ∂𝐿(𝑥,𝜆) /∂𝜆= 0 = 𝑔(𝑥)
• The gradient of f(x) and g(x) must be collinear when optimizing f(x) subject to g(x)

Collinearity in Equality Constrained Optimization

• Solves min 𝑓(𝑥1,𝑥2) = 0.25𝑥12 + 𝑥22, subject to 𝜔(𝑥1,𝑥2) = 5 − 𝑥1 − 𝑥2 = 0 using Lagrange multipliers
• Lagrange function: 𝐿 = 0.25𝑥12 + 𝑥22 + 𝜆(5 − 𝑥1 − 𝑥2)
• This yields 𝜆 = 2, 𝑥1 = 4, 𝑥2 = 1, and 𝑓 = 5

Graphical Interpretation of Equality Constrained Optimization

• Determines the location of the optimal point

Multiple Equality Constrained Optimization

• Aims to minimize a function 𝑓(𝑥1,𝑥2) when subjected to multiple equality constraints
• Subject to: 𝜔1(𝑥1,𝑥2) = 0, 𝜔2(𝑥1,𝑥2) = 0, 𝜔3(𝑥1,𝑥2) = 0
• The solution involves setting the partial derivatives of the Lagrangian function to zero to find the optimum.

Optimization with Equality and Inequality Constraints

• Minimize 𝑓(𝑥) subject to equality constraint 𝑔(𝑥)=0, and inequality constraint ℎ(𝑥)≤0
• Inequality constraints can be binding or non-binding
• Solved with Karush-Kuhn-Tucker (KKT) optimality conditions
• KKT Optimality Condition:
  • Lagrangian Function: 𝐿(𝑥,𝜆,𝜇) = 𝑓(𝑥) + 𝜆𝑔(𝑥) + 𝜇ℎ(𝑥)
  • 𝜕𝐿(𝑥,𝜆,𝜇)/𝜕𝑥=0= ∇𝑓(𝑥) + 𝜆∇𝑔(𝑥) + 𝜇∇ℎ(𝑥)
  • 𝜕𝐿(𝑥,𝜆,𝜇)/𝜕𝜆=0= 𝑔(𝑥)
  • Complementary Slackness condition: 0 = 𝜇ℎ(𝑥), 𝜇≥0
  • Binding constraints: ℎ(𝑥)=0 and 𝜇≠0
  • Non-binding constraints: ℎ(𝑥)<0 and 𝜇=0

Equality and Inequality Constraints Applied

• Shows example optimization

Application of Lagrangian

• Demonstrates the solution process using Lagrangian multipliers under both equality and inequality constraints
• μ = 0 represents the equality constraint
• μ > 0 represents the inequality constraints

Optimal Point Location

Nodal Pricing

• An Optimal Power Flow (OPF) analysis identifies a power transfer approach that respects network constraints
• OPF Objectives
  • Minimize the total cost of production
  • Maximize total social welfare
  • Minimize total system loss
  • Minimize the redispatch cost
  • Minimize total adjustment
  • Minimize load curtailment

OPF Variable Types

• Fixed parameters
• Control variables
• State variables
• Potential control variables
  • Real power output of generator
  • Generator bus voltage
  • Transformer tap position
  • Switched capacitor setting
  • Static VAR compensator setting
  • Load shedding

Economic Load Dispatch (ELD) Formulation

• Mathematical optimization to minimize the cost function subject to generation limits
• The problem converts to DCOPF by adding power flow equations as equality constraints and line flow limits as inequality constraints

DC OPF Assumptions in Power Systems

• Voltage magnitudes remain constant
• Only angles vary
• Angle variation is small
• Lines are lossless

DC OPF Formulation in Power Systems

• Includes the objective function to minimize total generation cost subject to constraints

AC OPF Formulation Details

• Details real and reactive power balance

AC OPF Formulation Components

• Minimize F(PG, QG, V, δ), which can be solved if:
  • G(PG, QG, V, δ) = 0
  • H(PG, QG, V, δ) ≤ 0

Centralized Market Characteristics

• Centralized optimization uses system resources
• Models the full transmission network
• Employs Optimal Power Flow (OPF)
• Seeks objectives such as
  • Cost minimization (inelastic loads)
  • Social welfare maximization (elastic loads)

Centralized Market Attributes

• Locational Marginal Price (LMP) clearing is a by-product of market
• LMP aids market settlement
  • Participation requires payment at LMP
• The system has possible LMP variations due to losses and congestion,
• Otherwise LMP remains constant in a lossless system

LMP Defined

• The marginal cost provides energy at a specific location
• The next MW of load is served at a particular location-lowest production cost, while abiding loading
• Supplying next increment cost
• The network price is related to location and includes network congestion losses effects
• A byproduct of clearing markets mechanism
• Optimal Power Flow (OPF) is used for clearing markets

Locational Marginal Price (LMP) Calculation Details

• LMP is the rate the real power at the bus
• A bus LMP is the derivative
• 𝐿𝑀𝑃𝑘 =∂𝐿/ ∂𝑃𝑘𝐷=λ𝑘

LMP Case 1

• A simple two-generator, one-load system illustrates LMP principles

LMP Case 2

• Shows adjustment to LMP

LMP Case 3

• Shows adjustment to LMP

Line Shadow Price (LSP) Defined

• The shadow price of a line represents the decrease in the cost with respect to the increase
• 𝐿𝑆𝑃𝑖=∂𝐿 /∂𝑓𝑖𝑚𝑎𝑥 −
• 𝐿𝑆𝑃𝑖​ = − 𝜕𝐿 /𝜕𝑓𝑖𝑚𝑎𝑥​ = 𝜇𝑖​

LSP Case 1

• Details on power flow system

LSP Case 2

• Details on power flow system

LSP Case 3

• Details on power flow system

LMP Analogy

• Real world connection, LMP analogous to air fare pricing
• Light traffic -> Predictable
• Heavy traffic like Diwali/New Year -> Exorbitant

Use of LMPs in Practice

• Payment
  • Generators at a rate-LMP
  • Loads at a rate equal to LMP
• Bilateral contracts require congestion costs paid based on LMP system difference
  • Negotiation of prices amongst groups
  • Calculated with charge = (LMP sink – LMP source) x MW
  • modeled as schedule and schedules in OPF

Ways to Calculate LMP

• DC Model
  • No reactive power flow effects
  • No Resistance
  • Fixed Voltages near Unity
  • A linear model
• AC Model
  • Non-linear
  • Losses are implicit
  • Presence of congestion

3-Bus Example (Case 1)

• A three bus system

3-Bus Example (Case 2)

• A three bus system

3-Bus Example (Case 3)

• A three bus system

3-Bus Example (Case 4)

• A three bus system

LMP Summary

• LMPs are a function of different system characteristics:
  • Topology
  • Power limits
  • Outages
• Vary quickly with unpredictable patterns

DC OPF Formulation 1

• Objective and solution

3 Bus Example: Formulation 1

DC OPF Formulation 2

• Objective and solution/formula

3 Bus Example: Formulation 2

Generation Shift Factor (GSF) in Power Systems

• Can be found with DC power flow calculation
• Can be used in general form

Implication of Nodal Pricing

• Network rental
• Market/Merchandizing Surplus
• Congestion Charge

Zonal Pricing Introduction

• Used for congestion management

Decentralized Markets

• Have Uniform pricing system wide
• Assume done transaction
• No transmission network
• Isolated congestion problems
• Are not economical

Decentralized Market: Details

• Have complex systems
• Require optimization strategies to solve
• Require high level approximation
• System based divided pre assigned zones
• Can lead in price term