Chapter 3 hydro thermal scheduling

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Contents

Problems definition

Mathematical model of long and short termproblems

Dynamic and incremental dynamic programming Methods of local variation

Hydro thermal system with pumped hydro units

Solution by local variation

Treating Pumped Hydro unit for loadmanagement and spinning reserve


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Ethiopian power system

Generation potential

Supply system

ICS supplied from hydro power plants

SCS mini hydro plants and diesel generators

Type of energy source Potential

Hydropower Above 4,000MW

Wind Above 1,350,000MW

Geothermal Above 5000MW


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Generation trend

Generation trend is increasing


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Electrification status

According to EEPCO report 48.3%


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Installed capacity


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Installed capacity


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Percentage contributions

Largest portion is supplied from hydro 89.8%

hydro

diesel

geothermal

wind


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Transmission system

Total length 11,796.32Km with 5 voltage levels

There are 143 substations with 127 supplying

distribution systems


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Distribution system

Distribution system consists of 148745.5km

and 18,888 transformers


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Customers


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Energy sales

Total number of customers by tariff group is as

shown below


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Historic energy sales


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Total energy soled


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Problem definition

Generation scheduling

Determining optimal operation strategy for the next

scheduling period

subject to a variety of constraints

Limited energy storage capability of water reservoirs

Stochastic nature of availability of water

Can be broken down into sub problems Hydrothermal coordination (yearly, monthly and weekly )

Unit commitment (weekly or daily)

Economic dispatch (hourly)


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Problem definition

Generation planning problem is

Nonlinear optimization problem

With continuous and discrete variable s

Non explicit objective function

With equality and inequality constraints


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Problem formulation

Hydro thermal coordination (HCP)

Determining optimal amount of hydro andthermal generation to be used in a scheduling

period Can be classified into short, mid and long term

scheduling

Constraints

Irrigation Recreation

Effect on downstream users


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Problem formulation

Hydrothermal scheduling problem

Long range

This involves

Long range forecasting of water availability Scheduling of water releases

Affected by

assumption used to calculate water replacement (Usestatistical mean or Worst case )

Load, hydraulic inflows, unit availabilities

Medium range

Short range


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Mathematical model of long and short

term problems

Mid term hydro scheduling (1week to 1 day)

This involves

Long range forecasting of water availability

Scheduling of water releases

Affected by

assumption used to calculate water replacement

Use statistical mean

Worst case

Load, hydraulic inflows, unit availabilities


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Problem formulation

Short range ( 1 day to 1 week)

Hour by hour scheduling of all generators to

achieve minimum production cost

Load, hydraulic inflow and unit availability is

known


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Operational factors affecting

hydrothermal scheduling

Type of turbine used

Francis turbine may operate at 65 to 125% of ratednet head

Loading At light unit loading, efficiency may drop to 70% and

at full load may rise to 87%

Hence

unit loading must be at best efficiency gate position Water releases schedule must coordinate with inflow

Note: better one full than two halves


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Types of hydrothermal scheduling

All hydro Difficult to meet the demands and operate

economically

Simulate the water system

Find a schedule which leaves maximum stored

energy in reservoir

Hydro dominated

Schedule the system for minimum cost of thermal

Balanced or with most thermal

Minimize thermal production costs


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Scheduling of mostly hydro case

Problem formulation

One hydro and one thermal

Rated hydro power is enough to cover load but

available hydro energy is not enough

Use thermal energy to fill the gap


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Problem formulation

Symbols used

EH- total available hydro energy

Time period to use all energy is TMAX

Hence

Where PHj is hydro generation limit at time j


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Problem formulation

ES=total thermal energy used

PSj- thermal generation level at time j

TS- total thermal run time


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Problem formulation

Total energy used by the load

Where PLj is load at time j

The thermal energy used is then


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Problem formulation

Then the problem statement is to minimize

the cost of generation of Es


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Solution using Lagrangian method

Form the Lagrangian

Apply necessary condition


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Solution using Lagrangian Method

Solving the two equations, from the first one

Which means the thermal must run at

constant generation level for all time

If the generation level is assumed to be PS*


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Solution by Lagangian Method


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Solution by Lagrangian Method

Substituting into the second equation

If the cost function of the thermal unit is givenbye

(*)


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The total cost of running the thermal unit is

then

Substituting from equation (*) above

Now we have nonlinear unconstrained

optimization with single variable


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Solving for Ps

This means the thermal unit must be operated

at the maximum efficiency point for as long as

it produced the demanded energy Es


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Example 7A page 217

Given the following characteristics for the

hydro and thermal plants, they have to supply

a load of 90MW for a week (15120MWhr)

Case I- if the hydro is limited to 10, 000MWh

energy, solve the run time of the thermal

Required energy from thermal is 5120MWhr


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Example 7A

The steam plants maximum efficiency point is

at 50MW, hence it has to run for 102.4hr

Hence

Steam run at 50MW for the first 102.4 hr of the

week

Hydro runs at 40MW for the first 102.4hr and then

for the rest 65.6hr, it runs at 90MW


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Example 7A

Case II- if the limit is on the amount of water

drawn from the reservoir 250,000 acre-fit

The water drawn from the reservoir is

determined by the amount of power

generated

Giving Ts=36.27h


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Short term hydrothermal scheduling

Problem definition

Given a hydro and thermal unit supplying a load

and constraints of total water discharge, starting

volume, ending volume etc, find a schedule whichminimizes the total cost


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Problem formulation


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Forming the Lagangian

Applying the necessary conditions

Sh t t h d th l h d li


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Example 7B

Solve example 7B


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Assignment

Explain the idea of pumped hydro systems

List some of world famous pumped hydro

systems and explain how they are used, i.e

Load management

Spinning reserve

Discuss the hydrothermal scheduling of

generation units which are in series cascad like

that of Gibe dam units


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Dynamic programming

Is a systematic way of determining optimal

solution in a problem having multiple stages

and with many possible decisions at each

stage

Is developed by Bellman in 1957

Has nothing to do with computer

programming


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Dynamic programming

Consider the following graph, numbers

correspond to time to travel between nodes

Find a path from A toB, travelling on to the

right, such that the

total sum of numbers

is minimum


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DP algorithm

Forward DP- start with A and find the cost of

moving forward until B

Backward DP start with B and find backward

the optimal path

In both cases, we have to decisions at each

stages


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DP algorithm

There are 20 possible routes

The optimal path can be computed using DP

as follows

Start at B and move one step back

Find the minimum path to arrive at B from one

step back

Repeat this for each stage

Stop when arriving at A


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DP algorithm

Solution


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DP used for hydrothermal scheduling

Consider the following situation where a

single hydro and single thermal units supply

power to a load

Total scheduling time

Tmax is divided into

time periods j

Given: loads at time

periods, starting andending volume Vo

and Vmax


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DP for hydrothermal scheduling Assumptions

The thermal unit is on for the whole time period

Water use rate of hydro is given by

Maximum flow rate qmax is limitedcorresponding to the maximum power output ofthe hydro


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DP for hydrothermal scheduling

The volume in storage changes as

For two consecutive intervals Vj and Vj-1

Problem is

find the volume at each stage sothat the total cost is minimum


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DP for hydrothermal scheduling

DP algorithm


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DP hydrothermal scheduling Example

A thermal unit and hydro unit have the

following characteristics curves

1200P200and

2000

PP8.4700F

s

2

s

s


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DP hydrothermal scheduling Example

Initial and final volumes are 10,000m3 and

storage volume limits are 6000 and 18000m3

The scheduling period is 24h divided by 4h

intervals


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DP for hydrothermal scheduling Example

If the storage volume steps are 2000m3Discharge at any

time interval j is

given by


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DP for hydrothermal scheduling Example

The first and second stage computations

becomes


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DP example


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Linear programming

Is a mathematical method used to allocate

scarce resources to competing alternatives in

an optimal manner

Linear objective function and

Constraints are linear inequality


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Example Linear programming

Consider the following profit function

Subject to the constraints


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Example LP

Solution methods

Graphical method for two dimensional case

Draw the graph of constraints

The closed boundary formed from constraints is thefeasible solution region

Corner points of the boundary are max/min points

Check for these points

Simplex method

for multi variable case


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Hydrothermal scheduling with LP

To apply LP

Objective function has to be linear

Divide the cost function of the steam turbine into

sections and approximate it by linear function withknown slope

Hydro turbine output versus the discharge has to be

approximated by linear

Constraints have to be linear Approximate water


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Hydrothermal scheduling using LP

For thermal characteristics, if 3 steps are used

For the hydro


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Thermal and hydro linear approximation


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Spill characteristics approximated by linear

Water spills out if volume is above a limit


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Problem formulation

Minimize the linear cost function

Subject to

Chapter 3 hydro thermal scheduling

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