BITS Pilani, K K Birla Goa Campus

Optimal Operation of Power System

Unit Commitment

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Introduction

Gen 1

Gen 2

Gen N

--

--

--

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PD

F1

F2

FN

P1

P2

PN

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This problem is divided into two sub-problems

1) Unit Commitment sub-problem

2) Economic Dispatch sub-problem

Definition

The process of determining a schedule of

generating units that yields the minimum total

production cost and satisfies all constraints is

called Unit Commitment (UC).

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Formulation of UC problem

A. Objective function

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Constraints

The UC problem is subjected to

1) Equality constraint

2) Inequality constraints

3) Spinning reserve constraint

4) Ramp rate limits

5) Minimum up and minimum down time

6) Must run units

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Existing methods

1. Priority List method

2. Dynamic Programming

3. Lagrangian relaxation method

4. Stochastic search methods

5. Hybrid methods

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Priority list method

The units will be committed based on the average full load cost.

The average Full Load Cost=

The generating units are arranged in ascending order based on the average full-load cost (AFLC) of the generating units.

Steps

Compute the ALFC for all units and find priority list

Commit the units according to priority list until reach the power demand

Example

Cost data of four units system

The priority order for the four

units is

unit 3, unit 2, unit 1, unit 4

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Limitations

However, the UC solution may not be the optimal schedule because start-up cost and

ramp rate constraints are not included in

determining the priority commitment order.

AFLC does not adequately reflect the operating cost of generating units when they do not

operate at the full.

The dynamic programming (DP) method consists in implicitly enumerating feasible schedule alternatives and comparing them in terms of operating costs. Thus DP has many advantages over the enumeration method, such as reduction in the dimensionality of the problem.

DYNAMIC PROGRAMMING

There are two DP algorithms. --Forward dynamic programming --Backward dynamic programming FDP is often adopted in the unit commitment Why FDP? Initial state and conditions are known. The start - up cost of a unit is a function of the time.

In the FDP approach, the previous information of the unit can be used to compute the transition cost between hour t-1 and hour t such as the start-up cost as well as to check the unit constraints like the unit minimum uptime and down-time.

The recursive formula to compute the minimum cost during interval k with combination l is given by

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Least total cost to arrive at state (k,l)

Fuel cost of generation at state (k,l)

Start up cost from (k-1,l) state to (k,l)

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Start

K=1

Perform economic dispatch for all

possible combination at interval k

K=K+1

Save N lowest cost strategy

K=M

(last hour)

Trace the optimal schedule

Stop

)],:,1(),([),( coscoscos lKLKSlKPMinklF ttt

)],1(),:,1(),([),( coscoscoscos LKFlKLKSlKPMinklF tttt

Dynamic Programming

The DP does not take into account the coupling of adjacent time periods. The DP does not handle the minimum up and down time constraints unless some heuristic procedures were included. The DP suffers from Curse of dimensionality.

LIMITATIONS

Lagrangian Relaxation method

Lagrangian Relaxation (LR) method can eliminate the dimensionality problem encountered in the Dynamic Programming by temporarily relaxing coupling constraints and separately considering each unit. The LR method decomposes the Unit Commitment problem into one dual sub-problem and one primal sub-problem based on the dual optimization theory. The primal sub-problem is the objective function of the UC problem.

The dual sub-problem incorporates the objective function and the constraints multiplied with the Lagrange multipliers. The dual and primal sub-problems are solved independently in an iterative process. Instead of solving the primal sub-problem to receive the minimum cost, the dual sub-problem is usually solved to receive the maximum cost by maximizing the Lagrangian function with respect to the Lagrange multipliers.

Dual optimization theory

Min f = (0.25 x21+15)U1 + (0.255 x2

2+15)U2 subject to:

x1U1 +x2U2 =5 0 < x1 < 10 0 < x2 < 10 U1 ana U2 may be only 0 or 1

L = (0.25 x21+15)U1 + (0.255 x2

2+15)U2 + l(5 x1U1 - x2U2)

Pick a value for l and keep it fixed

Minimize for U1 and U2 separately

0 = d/dx1(0.25x2

1 + 15 - x1l1)

0 = d/dx2(0.255x2

2 + 15 - x2l1)

0 = d/dx1(0.25x2

1 + 15 - x1l1)

if the value of x1 satisfying the above falls outside the 0 < x1 < 10, we force x1 to the limit.

If the term in the brackets is > 0, set U1 to 0, otherwise keep it 1

0 = d/dx2(0.255x2

2 + 15 - x2l1)

same as above

Now assume the variables x1, x2, U1, U2 fixed

Try to maximize L by moving l1 around

dL/dl = (5 x1U1 - x2U2)

l2 l1 dL/dl (a)

if dL/dl 0, a 0.2

if dL/dl < 0, a 0.005

After we found l2, repeat the whole process

starting at step 1

Lagrangian Relaxation for Unit Commitment problem

Primal sub problem The primal sub-problem

is to find the minimum cost of committed units

subjected to various constraints.

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Dual sub-problem The dual sub problem is to find the maximum cost received by maximizing the Lagrangian function with respect to the Lagrange multipliers

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To incorporate the power balance equation and spinning reserve constraint into the UC problem, the Lagrangian function is defined as

The data for the three - unit, four - hour unit commitment problem are as below, which is solved with Lagrange relaxation technique

Inappropriate method of updating the

Lagrangain multipliers may cause the solution

adjustment to oscillate around the global

optimum.

LR method tends to over-commit generating

units when identical generating units with the

same cost characteristics exist in the system.

Limitations