HYDRO-THERMAL COORDINATION

Rakesh RoshanElectrical-B0701209329

16TH SEPT 2010.

Presentation Outline1) Introduction2) Types of hydro-thermal co-ordination3) Methods of hydro-thermal co-ordination4) Mathematical formulation5) Solution for long-term6) Solution for short-term7) Advantages8) Example9) Conclusion10)References

Sep 2010 .

Introductiono Hydro-plants can be easily started and

assigned load in very short time.o Slow response of thermal power due to

boiler, super heater, and turbine system.o Thermal plant base load plantso Hydro plant peak load plants.

Sep 2010 .

Sep 2010 .

The optimal scheduling problem in a hydro-thermal system stated as to minimize the fuel cost of thermal plants under the constraint of water availability for hydro-generation over a given period of operation.

During high stream flows period hydro plant base load thermal plant peak load. During lean flow period thermal plant base load hydro plant peak load

Fundamental hydro-thermal system.

Water discharge G

Reservoir (or)storage

Water inflow, J

Types of hydro-thermal coordination1) Long term co-ordination – one week to one year

or several years. Unknown such as load ,hydraulic inflows, and unit availability(i.e. , steam and hydro-plants).

2) Short term co-ordination –one day or one week, which involves the hour-by-hour scheduling. The load, hydraulic inflows, and unit availabilities are assumed to be known.

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Methods of hydro-thermal co-ordination1) Constant hydro-generation method.2) Constant thermal generation method.3) Maximum hydro-efficiency method.4) Kirchmayer’s method.

Sep 2010 .

MATHEMATICAL FORMULATION To mathematically formulate the optimal scheduling,

the following assumptions are to be made for a certain period of operation:

(i)storage of a hydro-reservoir at the beginning and end of period T are specified.

(ii)After accounting for the irrigation purpose, water inflow to the reservoir and load demand on the system are known deterministically as functions of time with certainties.Sep 2010 .

Sep 2010 .

The optimization problem here is to determine

the water discharge rate q(t) so as to minimize the cost function of thermal generation.

Objective function is Min ……(1)

Sep 2010.

Subject to the following constraints:(i)The real power balance equation i.e., for t (0,T) ……(2)

where is the real power thermal generation at

time ‘t’ , the real power hydro generation at time ‘t’ ,

real power loss at time ‘t’, and the real power demand at time ‘t’ .

Nov 2008 .

(ii) water availability equation: ……(3)Where is the water storage at time ‘t’ , the water storage at the beginning of operation time ,T, the water storage at the end of operation time, T, the water inflow rate, and the water discharge rate.

Nov 2008 .

(iii)Real power hydro-generation The real hydro-generation is a function of

water storage X’(t) and water discharge rate q(t)

……(4)

solution of problem-discretization principle

The optimization interval T is sub-divided into N equal sub-intervals of t time length and over each sub-interval ,it is assumed that all the variables remain fixed in value.

The same problem can be reformulated as

……(5)

Sep 2010 .

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Subject to the following constraints:(i)Power balance equation ……(6) where is the thermal generation in kth interval, the hydro generation in Kth interval,  the transmission power loss in kth interval

  and expressed as

,and is the load demand in the Kth interval.

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(iii)Water availability equation: ………(7) where is the water storage at the end of interval K,

the water inflow rate in interval K, and the water discharge rate in interval K. Dividing the equation(7) by t, it becomes

for K=1,2, … N ………(8) where is the water storage in discharge units.

and are specified as water storage rates at the beginning and at the end of the optimization interval, respectively.

sep 2010 .

(iii) The real power hydro-generation in any sub-interval can be written as

……(9) where ; is the basic water head which is corresponding

to dead storage,

is the water correction factor to account for the variation in head with storage ,and

the non-effective discharge (due to the need of which a hydro generation can run at no-load condition).

Sep 2010 .

The objective problem is mathematically stated for any sub-interval ‘K’ by the objective function given by equations (5), which is subjected to equation constraints given by equations (6), (8), (9).

Independent variables are for K=2,3, … , N and for K≠1.

Dependent variables are for K=1,2, … ,N.

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Equation (8) can be written for all values of K=1,2,…,N:

i.e. , K=1 K=2 K= intervalBy adding the above set of equations, we get ……(10)Equation (10) is known as the water availability

equation.For K=2, 3, … ,N, there are (N-1) number of water

discharges (q ’s), which can be independent variables and the remaining one, i.e. , ,dependent variable and it can be determined from equation(10) as

……(11) Sep 2010

Solution technique

…(12 )

where are the Lagrangian multipliers that are dual variables

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……(13)

……(14)

……(15)

……(16)

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The partial derivatives of the Lagrangian function with respect to independent variables give the gradient vector:

…(17)

For optimally, the gradient vector should be zero

Algorithm

Step 1: Assume an initial set of independent variables, for all sub-intervals expect the first sub- interval

Step 2: Obtain the values of dependent variables

using equations (8),(9),(6),and (11), respectively.

Step 3: Obtain the Lagrangian multipliers using equations (13),(14),(16), and (15), respectively.

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Step 4: Obtain the gradient vector and check

whether all elements are close to zero within a specified tolerance, if so the optimal value is reached; if not, go to the next step.

Step 5: Obtain new values of control variables using the first-order gradient method,

……(18)

where is a positive scalar, which defines the step

length, and having a value depends on the problem on hand, then go to Step 2 and repeat the process.

KIRCHMAYER’S METHOD be the power generation of thermal plant in MW,

be the power generation of hydro-plant in MW,

be the incremental fuel cost of thermal plant in Rs./MWh, be the quantity of water used for power generation at

hydro-plant in ,

be the incremental water rate of hydro-plant in

be the incremental transmission loss of thermal plant,

be the incremental transmission loss of hydel plant,sep 2010 .

Sep 2010 .

be the Lagrangian multiplier,

be the constant which converts the incremental water rate of hydel plant j into an incremental cost,

n be the total number of plants,

be the number of thermal plants,

n- be the number of hydro-plants, and

T be the time interval during which the plant operation is considered.

Sep 2010 .

The objective function is to minimize the cost of generation:

i.e., ……(19)

subject to the equality constraints

……(20)

and ……(21) Now, the objective function becomes ……(22)

Sep 2010 .

……(23)

……(24)

where is the incremental fuel cost of the thermal plant and the incremental water rate of the hydro-plant.

Equation (23) and (24) are co-ordinate equations, which are used to obtain the optimal scheduling of the hydro-thermal system considering losses.

Advantages of operation of hydro-thermal coordinationFlexibilityGreater economySecurity of supplyBetter energy conservationReserve capacity maintenance

Sep 2010 .

ExampleQ. A two-plant system that has a thermal station near

the load center and a hydro-power station at a remote location is shown in fig.

The characteristics of both stations are

The transmission loss coefficient, Determine the power generation at each station and

the power received by the load when Sep 2010 .

Thermal plant Hydro-plant

Load

Sep 2010 .

solution: Here, n=2 Transmission loss,

Since the load is near the thermal station, the power flow is from the hydro-station only;

therefore, :

For the thermal power station, the co-ordination equation is

Sep2010 .

For a hydro-power station, the co-ordination equation is

By solving the above equation, we get Transmission loss,

Therefore, the power received by the load, =433.33+622.38-193.68=533.327 MW

ConclusionHydro-thermal scheduling is done when power demand is less than the maximum capacity of

power generation by the hydro and thermal plants.

Sep 2010 .

Referances1) Power system engineering 2nd edition –D P Kothari I J Nagrath2) Power system optimization –D P Kothari J S Dhillon3) Power system operation and control -S.Sivanagaraju G.Sreenivasan

4) www.ieeexplorer.org5) www.google.com6) www.nhpcindia.com

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Sep 2010 .

Thank you