HYENA Hydro Energy Simulator. Piecewise Linear Approximation of Future Costs

HYENA (HYdro ENergy simulAtor) Piecewise Linear Approximation of Future Costs Skuli Johannsson, Annad veldi ehf, Reykjavik, Iceland skuli@veldi.is 6-Mar-2006

1. Piecewise Linear Approximation

In [1] the contribution of the reservoirs content in the objective function is the future cost and is represented by:

 k ,t 1 



Min    k (x) s,t1(x)  dx

(1)

 k  0



In the linear case k (x) and s,t1(x)

simplifies to:

Min  k   s,t1  k,t1 

k

are taken out of the integral and (1) (2)

The production coefficients of the kth reservoir k and the water values of the sth water area s,t1 are estimated for every time step in the simulation.  k,t1 is the stored volume in reservoir k at end of stage t.

Using the nonlinear method (1) results in enormous computer time compared to the linear method (2). By using the linear method (2) the solution tends to “swing up and down” between time steps which can be very inconvenient and unrealistic especially for operation of the smaller reservoirs. This has been a long lasting problem for Icelandic hydro power models.

In this report we use a piecewise linear approximation of (1) by introducing the new variables  k, j,t1 :



Segments



Min  0  k,t1     k, j   k, j,t1

(3)

 k

k  j1



With the additional constraints:

Segments

  k, j,t1   k,t1

(4)

j 1

vk ,t1 max

0  vk, j,t1 

(5)

Segments

k, j is estimated numerically according to (1).

The variable k,t1 is kept in the model to be consistent with former results and to simplify programming work.

2. Numerical Results

Figure 1 shows the k, j in one time step of a simulation run for 7 reservoirs and figure 2 shows results of the simulation for 4 water years. In figure 1 Haganga reservoir has the same values as Þorisvatn and is therefore hidden beneath.

Price in Segments [kr/kWh]

Figure 1

Piecewise Linear Approximation of Reservoir Price Function 0 1 3 5 7 9 11 13 15 17 19 21 23 25 -20

Háganga -40 Þórisvatn

Sigalda

-60

Sultartangi

Blöndumiðlun

-80

Hraunaveita

Kárahnjúkar

-100

-120 Segments

Figure 2 Hydro Power Reservoirs

Reservoir Content [Gl]

2000 1500 1000

Háganga Þórisvatn Sigalda Sultartangi Blöndumiðlun Hraunaveita Kárahnjúkar

0 1950

1952 Water Years (1. Sep - 31.Aug)

The results in Figure 2 are much more streamlined than former results using (2) with around twice as much computer time, when using 25 segments in the piecewise linear approximation. The nonlinear case (1) is still almost impossible to use in practice because of excessive computer time. 3. References [1] HYENA (HYdro ENergy simulAtor). Some Issues in a Simulation Model of a Hydro Thermal Power System. Skuli Johannsson, Annad veldi ehf, Reykjavik, Iceland skuli@veldi.is and Elias B Eliasson The National Power Company, Iceland , elias@lv.is . 01-April-2005. [2] Elias B Eliasson and Ulfar Linnet, both from Landsvirkjun The National Power Company, contributed to the report.