Optimised set point distribution for row wind farms with

A suitable objective function has been dened and fminunc() is used to nd the optimal set point distribution. The rst wind farm where the optimisation

algorithm is applied is the row wind farm with two turbines. Only the set point of Turbine 1 is optimised as Turbine 2 should be run in nominal operation.

The problem is solved for wind speeds form 3 m/s to 15 m/s with steps of 0.1 m/s, meaning 121 problems are solved for the small row wind farm. When solving the problem the optimiser needs an initial guess. The initial guess should obviously be as close to the true optimum as possible. The problems are solved in the order of their wind speeds. This makes it possible to use the last found solution as the new initial guess. This approach should give good initial guesses as long as the steps in the wind speeds are suciently small.

The wind speed decides the order in which the problems are solved, meaning that the problems can be solved in ascending or descending order. It shouldn't matter in which order the problems are solved, if the problems are convex and have a sucient curvature all over the solution space. To test if this indeed is the case the problems are solved twice. The rst time the problems are solved it will be in ascending order, meaning the initial guess will be slightly too low compared to the true solution. This method will be known as the "forward scan"

(FS). Afterwards the problems are solved in descending order meaning the initial guess are too high. As the problems are in reversed order this method is known as the "reversed scan" (RS).

In Figure 4.1 the results from solving the 121 problems from dierent wind speeds using the forward scan method are shown. At the left plot the solutions for all the wind speeds are shown. As a reference the set points from the MPPT strategy is also shown. It can be seen on the blue and black dotted line that the average load on the turbines are higher using the optimised set point from 3.8 m/s to around 11 m/s compared to the MPPT strategy. Above 11 m/s it is optimal for both strategies to have Turbine 1 running on full load, meaning that the strategies are the same for wind speeds above this level .

Below 3.8 m/s both strategies are also the same as both of them let the Turbine 1 extract as much energy from the wind as possible. After 3.8 m/s the Turbine 1 is down regulated when using the optimised set points. As Turbine 1 is down regulated the wake decit doesn't reduce the wind speed in front of Turbine 2 below the cut-in speed allowing the turbine to start producing power. Since the average load is higher for the optimised set points the loss in power from down regulating Turbine 1 is less than the gain in power from starting Turbine 2.

Looking at the average load for the optimised set points, at the right plot on Figure 4.1, it appears that a gain in average power could come from down regulating Turbine 1 for wind speeds lower than 3.8 m/s. These issues will be investigated further after the results from the reversed scan has been analysed.

4 6 8 10 12 14 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Wind Speed (m/s)

Power (MW)

Optimised set points for row wind farm of 2 turbines using forward scan

FS Avg.

MPPT Avg.

FS Turb 1 MPPT Turb 1 FS Turb 2 MPPT Turb 2

3 3.5 4 4.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Wind Speed (m/s)

Power (MW)

Zoomed on low wind speeds

FS Avg.

MPPT Avg.

FS Turb 1 MPPT Turb 1 FS Turb 2 MPPT Turb 2

Figure 4.1: Optimised set points using the forward scan method compared with MPPT. Left side is the solution for 3 to 15 m/s. Right side is zoomed on the solution from 3 to 4.5 m/s.

At Figure 4.2 the optimised set points for the same 121 problems are shown and the problems are solved using the reversed scan method. Comparing the solutions from the forward and reversed scan only small dierences are noticed.

Looking at the plot for lower wind speeds it is seen that Turbine 2 is switching between running and not running for wind speeds around 3.5 m/s to 4 m/s. It is seen on the blue curve that for 3.6 m/s the total power production is higher when down regulating Turbine 1 and having two turbines running. Looking at the solution for 3.7 and 3.9 m/s the optimiser found it optimal to have only one turbine running, which obviously is wrong as the curve for the average production clearly indicates it is benecial to have two turbines running in this case.

An issue with optimising the problems was found when analysing the forward and reversed scan solutions. Local optimums seems to appear for wind speeds around 3.5 to 4 m/s. Since only one set point is optimised for this wind farm it is easy to show the solution space for a given wind speed. This will shed light on when its optimal to have two turbines running and why the optimiser doesn't nd the same solutions using forward and reversed scan.

4 6 8 10 12 14 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Wind Speed (m/s)

Power (MW)

Optimised set points for row wind farm of 2 turbines using reversed scan

RS Avg.

MPPT Avg.

RS Turb 1 MPPT Turb 1 RS Turb 2 MPPT Turb 2

3 3.5 4 4.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Wind Speed (m/s)

Power (MW)

Zoomed on low wind speeds

RS Avg.

MPPT Avg.

RS Turb 1 MPPT Turb 1 RS Turb 2 MPPT Turb 2

Figure 4.2: Optimised set points using the reversed scan method compared to MPPT. Left side is the solution for 3 to 15 m/s. Right side is zoomed on the solution from 3 to 4.5 m/s.