1 system planning 2013 lecture l8: short term planning of hydro systems chapter 5.1-5.3 contents:...

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System planning 2013

• Lecture L8: Short term planning of hydro systems

• Chapter 5.1-5.3• Contents:

– General about short term planning– General about hydropower– Hydropower generation, discharge and efficiency– Hydrological coupling– Example: hydro planning problem

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Course goals

• To pass the course, the students should show that they are able to formulate short-term planning problems of hydro-thermal power systems.

• To receive a higher grade the students should also show that they are able tocreate specialized models for short-term planning problems.

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General about short term planning

What is short term planning?

• Time frame: 24 hours – 1 week• In this course: Hourly planning.• Minimize cost, maximize profit• Results:

– Operation plans for the power plants– Trading on the market

• Exists limitations when planning operation:– Technical– Economical– Legal

• In this course: Deterministic modeling

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General about short term planning

Optimization problem!

maximize income during period + future income – costs during period – future costs

regarding physical limitationseconomical restrictionslegal restrictions

General short term planning problem:

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General about short term planning• In this course: Hydro-thermal systems.• Quite different characteristics and modeling.

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General about hydropower

• Generates power by using the difference in potential energy between an upper and a lower water level.

• Often reservoirs, but also run-of-the-river hydro plants

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• Hydropower station with low head


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• Hydropower station with high head


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• Variables in hydropower models:– Discharge, Q– Spillage, S– Reservoir contents, M

• All measured in hour equivalents, HE– Discharge, spillage: 1 HE = 1 m3/s during 1 h.– Reservoir contents: 1 HE = Volume

represented by the flow 1 m3/s during 1 hour

• Hydro generation as a function of the discharge is denoted H(Q)


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What do we want?

• Want to optimize the operation of our stations in the hydro system.

• Want to use linear equations so that we can use LP when solving the optimization problem.

• Want to model the characteristics of hydro systems.

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Power generation

• Production equivalent:Measured in MWh/HE

• Marginal production equivalent:Measured in MWh/HE

Definitions:

( )( )

H QQ

Q

( )dH Q

dQ

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Power generation

• Relative efficiency:Measured in percent

• Relative efficiency shows how much energy can be extracted from each m3 water compared to the maximal possible.

max

( )( )

QQ

max max ( )Q

Q

Definitions:

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Power generation

Power generation in hydropower station

H(Q)

Q

Relative efficiency in hydropower station

Q

(Q)

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Power generation

H(Q)

Q

Power generation curves are not linear!– Approximated by piece-wise linear curves

(segments)– Break points at best local efficiency points– Each segment has constant marginal

production equivalent (the slope of the curve)

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Power generation• Power generation in power station i:

Hi(Q)

Qi

i,1

i,2

i,3

i,4

Segment 1 Segment 2 Segment 3 Segment 4

Qi,1

Qi,2

Qi,3

Qi,4

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Power generation• Total discharge in power station i hour t:

, , ,1

in

i t i j tj

Q Q

Qi,j,t = Discharge in station i, segment j, during hour tni = Number of segments in station i

• Total production in station i hour t:

, , , ,1

in

i t i j i j tj

H Q

i,j = Marginal production equivalent for station i, segment j

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• Q: How make sure that discharge in segment 2 doesn’t occur before maximum discharge in segment 1 is reached?

• A: If the discharge will first occur in segment 1, then in segment 2, etc.

,1 ,2 ,...ii i i n

H(Q)

Q

Q1Q2

Q3Q4

1

2

3

4

Power generation

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Power generation

• Low discharges often have bad efficiency, but in the linear model discharges up to the first break point are assumed to have best production equivalent!

• Want to avoid low discharges!• Can be solved by introducing binary variables

Forbidden discharges:

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Power generation

• Production equivalent according to figure• Low efficiency for discharges lower than 50 HE• Want to have discharges larger than 50 HE or no discharge at

all• Create a linear model

50 100 150 200 250 300

0.1

0.2

0.3

0.4

i(u)

ui

MWh/HE

HE


Qi

i(Qi)

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Power generation

50 100 150 200 250 300

0.1

0.2

0.3

0.4

i(u)

ui

MWh/HE

HE

QKi,t

• Define 2 segments:1. Origin2. Discharges between

50 and 300 HE, QKi,t

• Assume constant marginal production equivalent in the interval 50-300 HE, Ki. Can e.g. chosen as the average value of i(Qi) in the interval.


Qi

i(Qi)

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Power generation

• The total discharge and production can now be calculated as:

, , ,

, , , ,

50

50i t i t Ki t

i t Ki t i t Ki Ki t

Q z Q

H z Q

Define binary variable representing which segment discharge occurs during the hour


,

0, if 0 HE

1, if 50 HEi

i ti

Qz

Q

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Power generation

• Variable limits:

,

,

0

0,1

Ki t

i t

Q

z

, ,Ki t i tKiQ Q z

• Must define constraints to assure that the discharge in the second segment is 0 when zi,t = 0


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Planning problem

• Short term hydropower planning problem:

maximize income during period -production costs during

period +assets after the end of the

period

regarding hydrological couplinglegal agreementsphysical limitations

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Income & cost

• The production costs can be neglected for hydropower production! They are very small.

• Income during the period can consist of:Sale of power on spot marketSale of power bilateral to customer

,1

IncomeT

t i tt

H

t = price hour t

For power station i:

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Assets after the end of period• Assets after the end of the planning period consists of

the value of stored water in the reservoirs. Depends on:– Future power price– How much power that you can expect to produce with

the stored water

• Value of stored water at reservoir i:

,( )i

i i e i T jj M

B M M

e = expected future power priceMi,T = reservoir contents in reservoir i at the end of the planning periodMi = set of indices for all station downstream station i (including station i)j = expected prod. equivalent for station j

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Hydrological coupling

• The hydropower stations in a river are not independent, they are a part of a system

• The operation of one station affects the other stations in the river

• To operate the river system in an efficient way, the whole system needs to be considered

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• Balance equation for a reservoir:

new reservoir contents = old reservoir contents + water flowing into reservoir – water flowing out from reservoir

Discharge from stations directly

upstream

Spillage from stations directly

upstream

Inflow from surrounding

area

Discharge Spillage

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Hydrological coupling• Hydrological coupling equation for station i:

, ,, , 1 , , , , ,j i j i

i i

i t i t i t i t j t j t i tj Κ j Κ

M M Q S Q S V

Discharge & spillage in station i

during hour t

Spillage from stations directly

upstream coming down to station i during

hour t

Contents in

reservoir i at the end of hour t

Contents in reservoir i at the end of hour t-1

Discharge from stations directly

upstream coming down to station i during

hour t

Inflow to reservoir i during hour t

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• Take some time for the water to flow from one station to the next, so called delay time:

ji = Delay time between station j and the closest station downstream i

ji is a complicated function of water flow, reservoir levels, etc. Assume constant delay time and let the delay time between station j and i be hj hours and mj minutes

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tt-hj

hjmj

t-hj-1

Discharge & spillage this hour will arrive hour t

,, , 1 ,

60

60 60j i j j

j jj t j t h j k h

m mQ Q Q

Weighted mean value of discharge and spillage hj+1 and hj hours earlier

t+1

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Legal agreements & physical limitations

• Physical limitations & legal agreements limits the operation of the power stations

• Results in variable limits in the optimization problem:

,

,

,, ,

i t ii

ii i t

i Ti T i T

Q Q Q

M M M

M M M

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Legal agreements & physical limitations

• Contracts with costumers

tIi

ti DH

,

Hi,t = production in station i hour tDt = contracted load hour t

Can also be ”=” depending on the problem

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Planning problem - example

• Two hydropower stations (1 & 2) located after each other in a river.

• All produced power is sold on the power exchange• Plan the operation for the 6 next-coming hours

• Known:– Price forecast for the 6 hours: λ(t), t=1,...,6– Stored water can be used for generation of power

that can be sold for the price λf

– The reservoirs are half full at the beginning ofthe planning period

1

2

Example:

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• Hydropower data:– Installed capacity: – Maximum discharge:– Maximum reservoir contents:– Local hydro inflow:

• Assume constant efficiency, i.e. constant production equivalent. Installed production capacity is reached at maximum discharge

2,1, iH i

2,1, iQ i

2,1, ivi

2,1, iM i

Example

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Solution:

The problem:

maximize income from selling power on power exchange + value of stored water

s.t. hydrological coupling is fulfilled

Variables:

Discharge in station i during hour t: Qi(t), i = 1,2, t = 1,...,6 Spillage in station i during hour t: si(t), i = 1,2, t = 1,...,6 Reservoir contents, station i, at the end of hour t: Mi(t), i = 1,2, t = 1,...,6

Example

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• Constant production equivalent • Max. discharge gives installed capacity 2,1, i

Q

H

i

i

i

Value of sold power =

6

1

2

1

)(t i

iiQtlambda

Value of stored water = )6()6( 22121 MMlambdaf

Objective function:

Thus:

)6()6()( 22121

6

1

2

1

MMlambdaQtlambdaz tt i

ii

Example

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Hydrological constraints for the stations:

6,...,1,)()()()()()1(

6,...,1,)()()()1(

2112222

11111

tvtstQtstQtMtM

tvtstQtMtM

Initial reservoir contents:

ii MM 5.0)1(

Variable limits:

6,...,1,2,1,)(0

6,...,1,2,1),(0

6,...,1,2,1,)(0

tiMtM

tits

tiQtQ

ii

i

ii

Example

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• Operation planning of a system of hydropower plants in a river

• Planning per hour• Planning period 12-24 hours• In this course we use linear or MILP

models linear or MILP programming problem

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PROBLEM 12 - Symbols for Short-term Hydro Power Planning Problems

State whether the following symbols denoteoptimization variables or parameters.a) γi = expected future production equivalent for

water stored in reservoir ib) Mi, 0 = contents of reservoir i at the beginning

of the planning periodc) Mi, t = contents of reservoir i at the end of

hour t

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PROBLEM 12 - Symbols for Short-term Hydro Power Planning Problems

State whether the following symbols denoteoptimization variables or parameters.

d) Qi, t = discharge in power plant i during hour t

e) Si, t = spillage from reservoir i during hour t

f) Vi, t = local inflow to reservoir i during hour t

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PROBLEM 13 - Value of Stored Water

Which expression can be used for the value ofstored water in a 24 hour planning of three hydropower plants in line along a river?*

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PROBLEM 14 - Hydrological Constraints

Which expression can be used for the hydrologicalconstraint of the third hydro power plantalong a river?*

1 system planning 2013 lecture l8: short term planning of hydro systems chapter 5.1-5.3 contents:...

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