Wind Farm Energy Optimization using Nested-Loop

Extremum Seeking Controls and Load Reduction

Turaj Ashuri, Ebenesh Rabiraj, Yaoyu Li and Yan Xiao

Department of Mechanical Engineering, The University of Texas at Dallas, Richardson, TX75080, USA

Zhongzhou Yang

EXA Corporation, 210 Six Mile Rd, Livonia, MI 48152

E-mail: turaj.ashuri@utdallas.edu

Abstract. A load opimization algorithm is developed to compliment the nested loopextreemum seeking control. An objective function of the controller is formulated toaccommodate the variations in the load as penalties. The formulated objective function allowsthe controller to find the optimal power while mitigating excessive loading on the turbine thatare caused due to the NLESC. The results show a considerable reduction in structual loads andfatigue loads while preserving the power output of the turbine. Under steady wind conditionsa reduction of upto 25% decrease in peak loads compared to the nested loop extremum seekingcontrol. Under turbulent wind with a turbulence intensity of 5% we can see a reduction of upto20% in fatigue loads and upto a 15% decrease in mean loads. This paper also aims to study thestability and effectiveness of the controller with increase in turbulence intensities.

1. IntroductionIn 2015, the US coal-fired power plants experienced a reduction of 12.9 GW in power generation,while wind energy power generation increased by 9.8 GW 1. This shows that the effects of climatechange have pushed the energy sector to transition into renewable energy with an emphasis onwind. Although wind energy power generation seems promising, its cost is in general higherthan that from the conventional energy resources. Improving blade design, new manufacturingtechniques, upscaling wind turbines, wind farm optimization, better operation and maintenancestrategies, and advance control algorithms to maximize the energy production are among theefforts to reduce the cost [1–7].

Maximizing energy production using controls can be performed either at wind turbine or windfarm level. In the case of wind farms, maximizing the energy output of individual wind turbinesdoes not guarantee the maximum energy output of the entire farm [8–11]. This is because of thecomplex wake interaction among wind turbines that leads to sub-optimal performance of thewind farm. Talking into account such complex wake interaction is difficult using model-basedcontrol algorithms.

1 US energy information administration, Scheduled 2015 capacity additions mostly wind and natural gas;retirements mostly coal, http://www.eia.gov/todayinenergy/detail.cfm?id=20292, Retrieved June 9, 2016

Model-free control strategies have the advantage of requiring minimal knowledge of the systemunder operation. Marden et al. [12] proposed a game theoretic optimization algorithm foran array of three wind turbines to optimize the energy production of the wind farm. Dattaand Ranganathan [13] proposed tracking the optimal power point, independent of the turbinecharacteristics and the air density, by varying the generator speed dynamically using the activepower as a reference. Guo et al. [14] proposed a model predictive controller to assist anadaptive controller to smoothly track the maximum power point for a wind farm. Gebraad andWingerden [15] proposed maximum power point tracking (MPPT) algorithm of the gradient-ascent and quasi-Newton types to optimize the power output of a three turbine array.

Extremum seeking control (ESC) is a variation of MPPT that tracks the optimum byperturbing the system with a sinusoidal probing signal to extract gradient information. Johnsonand Fritsch [16] assessed the effectiveness of ESC in a wind farm to maximize the power outputin a low turbulence condition. Yang et al. [17] have optimized the power output of a cascadedwind turbine array along the prevailing wind direction using the nested-loop extremum seekingcontroller (NLESC). These algorithms focus on maximizing the power output of the turbinewithout considering the increase in structural loads.

This paper presents a NLESC to maximize the energy production of an array of wind turbinesconsidering the structural loads imposed on the tower and shaft. The modified NLESC uses thecollective array power coefficient to maximize the power output. Structural loads are applied asa continuous penalty function for each individual wind turbine power coefficient.

The remainder of this paper is structured as follows. First, the methodology to model thewind farm wake interaction, and the structure of the NLESC is discussed. Next, the results ofimplementing the NLESC on an array of wind turbines is presented. Finally, the conclusion ispresented.

Figure 1: Cascaded NLESC implementation

2. MethodologyA Simulink model of the NLESC is used to control and optimize an array of three windturbines [17]. It is dynamically generated using a MATLAB script, based on input parameterssuch as mean wind speed, turbulence intensity, and the number and coordinates of each windturbine. The model is linked with SimWindFarm (SWF)[18] computational code that is capableof modeling aerodynamics and wake interactions in a wind farm. Three different wind farm

controllers are used for this research, a baseline controller distributed with SWF, a NLESC tomaximize the wind farm power output, and a NLESC to maximize the wind farm power outputwith load penalties. These are explained next.

2.1. Simulation PlatformSWF is a powerful toolbox that provides an environment to develop new control algorithms. Itis capable of simulating wakes for turbines with an in-built NREL5MW turbine model [19]. Thisis done by dynamically generating a wind field based on the input parameters specified in theMATLAB script. The model allows extraction of two moments, including the shaft and towermoment. The shaft moment (Mshaft) can be measured from a third order drive train model.The drive train is modeled as a pair of rotating shafts through a gearbox.

Ω =1

Irot

(Mshaft − φKshaft − φBshaft

)(1)

ω =1

Igen

(−Mgen +

1

N

(φKshaft + φBshaft

))(2)

φ = Ω − 1

Nω (3)

where, Kshaft is the torsional spring constant and Bshaft is the viscous friction of the gearboxof gear ratio N . φ is the torsion angle of the shaft and, Irot and Igen are the rotor and generatorinertias.

Further, the tower deflection(z) is modeled as a second order spring-damper system(equation 4)from which the tower moment(Mtow) can be measured.

z =1

Mtow(Ftow −Ktowz −Btowz) (4)

where, Ktow is the spring constant of the tower and Btow is the damping term.SWF is also capable of performing fatigue calculations. The fatigue postprocessor utilizes

Mcrunch to perform calculations including, rainflow counting and damage equivalent load forthe tower and the shaft [20].

2.2. ControllerThe performance of the controller that incorporates load optimization is justified by comparingit against the baseline controller of the NREL5MW turbine and the NLESC algorithm. Thedifferent control algorithms that are used are mentioned below.

2.2.1. Baseline ControlDuring region 3 operation, the baseline control has a constant generator power reference. Usinga gain scheduled proportional integrator control algorithm, the blade pitch is actuated to controlthe rotor speed. During region 2 operation, the blade pitch is kept constant, while the generatorpower is estimated using the generator speed as an input. The estimation is done using a simplelookup table. In the simulation performed, only the region 2 control scheme is active.

2.2.2. Nested-Loop Extremum Seeking ControlThe ESC strategy as shown in Figure 2, employs a gradient based search technique to find amaximum in the input signal. A dither signal (fdither = asin(ωt)) is added to the input of theplant as a sinusoidal probing signal (ud = u + fdither) to excite it. THe output of the plant isrepresented as,

y = l(ud) = l(u+ asin(ωt)) = l(u) +∂l

∂uasin(ωt) + ... (5)

× Low pass Filter Integrator

High pass Filter Wind Turbine

fdem = sin(ωt) fdither = asin(ωt)

+u

udy

+

Figure 2: Block diagram of the ESC for a single wind turbine

A high pass filter is used to remove the DC term (l(u))while retaining the harmonics. Ademodulation signal (fdem = sin(ωt)) converts the first harmonic to a DC component. Thisdemodulation signal is proportional to the gradient. The higher order terms are then removedusing a low pass filter while retaining the DC component. An integrator is used to eliminateany steady state error present to reach zero gradient.

In the case of steady wind, the input chosen is the aerodynamic power. As there are novariations in the wind speed, the change in power is a direct consequence of the control signal.However, in the case of turbulent wind, the aerodynamic power is a function of the variation inthe wind speed over time. This makes it difficult to isolate the variation in the power due to thecontrol signal. Thus we aim to optimize the coefficient of power instead. The coefficient of poweris extended to operate in a nested-loop configuration as the Array Power Coefficient(APC). Thisis based on a similar approach by Corten et al. [21] and justified by Yang et al. [17]. APC isdefined as the ratio of sum of aerodynamic power of the turbine(i) and all the n turbines in it’swake, to the estimated power. This can be represented as,

Kip =

P i +∑j

n=1 Pna

12ρAU

3(6)

where, Kip is the APC of the ith turbine, P j

a is the aerodynamic power of the most downstreamturbine in the wake of turbine i. A trasportation delay is implemented to compensate for thetime taken for the wake to travel to the downstream turbine. The estimated power is calculatedusing the measured wind speed in front of the upstream turbine i and for the following j numberof turbines in its wake, a transport delay, Ti is applied to the wind speed of the upstream turbine.Therefore for any time t, the APC can be written as,

Kip(t) =

P i(t− T ) +∑j

n=1 Pna (t− T )

12ρAU(t− Ti))3

(7)

Any residual fluctuations in the APC signal due to the changes in wind speed appear as highfrequencies in the signal which can be mitigated using a moving average filter. This signal isused as an input to the controller optimize the power output of the wind farm.

2.2.3. Load optimized NLESC

Load optimization is done by penalizing the objective function of the NLESC based on thestructural loads. The penalty function is formulated such that the loads are a variation ofthe APC. The objective function with load optimization is thus formulated as a multiplicativepenalty and is represented as,

K = Kp(1 −Kload) (8)

where K is the load optimized objective function which is a function of the APC (equation 7)and the load coefficient, KLoad.

To formulate the load coefficient, we intend to normalize the structural loads that we consideras a part of the penalty function. The structural loads include the tower moment and the shaftmoment. The tower moment is usually in the order of 107Nm and the shaft moments are in theorder of 106Nm. The normalization ensures that we give an equal weight to both the structuralloads initially. As we optimize of power output below rated speed, the maximum loads at ratedspeed are chosen as a factor to normalize the loads. The load coefficient for any turbine i canbe represented as,

Kiload =

[(wi1 ×M i

shaft

M∗shaft

)−(wi2 ×M i

tower

M∗tower

)]w3 (9)

where M ishaft and M i

tower represent the shaft and tower moments. M∗shaft and M∗

tower are themaximum shaft and tower moment at rated wind speed (11.4 m/s). The values of these momentsare 4.2e6Nm and 9e7Nm respectively. Each normalized load is then weighted individually usingwi1 and wi

2 and also together using wi3.

Initially the weights wi1 and wi

2 are chosen to be 1 for all turbines and, the wi3 is adjusted

such that the load coefficient penalizes the power is effective. The independent weights are thentuned such that all the turbines see a decrease in loads.

At any time t, from equations 7 and 9 we get,

Ki(t) =P i(t− T ) +

∑jn=1 P

na (t− T )

12ρAU(t− Ti))3

[1 −

[(wi1 ×M i

shaft(t)

M∗shaft

)−(wi2 ×M i

tower(t)

M∗tower

)]w3

](10)

In figure 3 the objective K1(t) is shown for a mean wind speed of 8m/s with a turbulenceintensity of 5%. The signal, especially for upstream turbines contain noise that is a consequenceof the turbulence in the wind speed and its summation over the nested loop and even a suddendecrease in the load coefficient. There are also spikes in the signal (marked with circles) whichcould compromise the stability of the controller. To mitigate this a saturation block is introducedto extract region of most relevance from the objective function before the introduction of amoving average filter.

3. ResultThe first subsection deals with load optimization is carried out for the steady wind condition.In the steady wind condition, wind speeds of 6m/s and 8m/s are analysed. In the consequentsubsection turbulent conditions are analysed. The results of 8m/s wind with a 5% and 10%turbulence intensity are studied.

3.1. Steady windIn the case of the 8m/s steady wind, the gains are 1.5e-8, 2.5e-8, 5e-6 respectively for the first,second and third turbine for the NLESC. The dither amplitudes are 0.05, 0.03, 0.10 respectivelyThe time periods for the dither frequencies chosen are 2800s, 1400s and 80s respectively. Thegains of the load optimized NLESC are 3e-9, 7e-9, 5e-6 respectively. All the other parameters

Figure 3: The noise in the objective function of turbine 1 is shown for a mean wind speed of8m/s and a turbulence intensity of 5% . The signal is conditioned using a saturation with arange[1.05, 1.18]. A moving average filter is applied to the saturated signal.

remain the same. The optimal torque gain for the NLESC is about 3.05, 2.77, 2.4KNm/rpm2

respectively and for the load optimized NLESC is 2.9, 2.83, 2.4KNm/rpm2 respectively as seenin figure 4. The results from figure 5 shows that there is a negligible loss in power output but asignificant decrease in ultimate loads on the turbine. The percentage change in ultimate loadson the turbine are shown in figure 6.

In the case of the 6m/s the gains chosen are 3.7e-7, 3.1e-8, 5e-6 for the NLESC and the loadoptimized NLESC. The rest of the parameters are the same as that of the 8m/s condition. Atlower wind speeds the NLESC is able to extract much more power from the wind. The resultsare similar to the 8m/s case. Though the percentage decrease of power is higher, it is stillcapable of reducing a significant amount of loads on the turbine.

3.2. Turbulent windIn the case of turbulent wind, the weights are set to w1 = [1, 1, 1] and w2 = [1, 1, 1] initially.w3 is varied until the effect of the load is significant. By following this procedure w3 is chosento be 0.1. For a wind speed of 8m/s at a turbulence intensity of 5%, the objective function, asshown in figure 3, uses a moving average filter that have time periods of 1800s, 850s and 35s forturbines 1, 2 and 3 respectively. Similarly, the time periods for the dither frequencies chosenare the same as those used for the previous cases. The Integrator gains used are larger, whencompared to those used for the steady wind case due to the fact that we use the coefficient ofpower instead of the aerodynamic power. The gains chosen are 0.023, 0.01 and 1 respectively.

The DEL for the shaft and tower are shown in figure 12a and 12b respectively. In this casewhere the weights are evenly split between the turbines the DEL of the load optimized controlscheme lies between the baseline and the NLESC. Another consideration is that the first turbineexperiences higher loading due to the higher wind speeds. To reduce the load on the first turbineeven further, with a primary focus on the shaft, we can alter the weight w1 = [1.03, 1, 1] whilekeeping the other weights the same.

The torque gain obtained for the modified weight is shown in figure 14. The DEL, as shown infigure 15a and 15b sees a reduction in the loads on the first turbine preserving the performance

and the load reduction on the second and third turbine. The DEL gives a look at fatigue causedby the load while the mean loads as shown in figures 3.2 and 3.2 show the overall change inthe loads imposed on the turbine. The mean loads are calculated after 5000s assuming that theNLESC and the load optimized NLESC have reached a stable condition.

The challenging aspect of the NLESC, with and without the load optimized algorithm, ishigh turbulence conditions. In the case of higher turbulence the input of the controller, asdiscussed earlier in figure 3, is to remove the variation in the power due to the wind. Higherturbulence causes larger variation in the wind and even larger variation in the power. This tendsto compromise the integrity of the controller.

For a mean wind speed of 8m/s and a turbulence intensity of 10% the moving average ischosen to be more aggressive. The average is taken over 190s, 100s, 4.3s respectively for theNLESC and 190s, 100s, 5.3s respectively for the load optimized NLESC. This affects the time thecontroller takes to react to a change in the loads and thus the effectiveness is reduced. In figure18, at about 1400s there is a considerable dip in the windspeed. Due to a narrower averagingof the input of the third turbine, the change in the torque gain, as seen in figure 19b is muchmore noticeable. These variations may cause the controller to become unstable if not taken intoaccount.

For the NLESC the gains are chosen to be 0.020,0.005,0.2 respectively. The gains are lowerthan those chosen for the 5% turbulence to maintain the stability of the controller. The ditherfrequency is the same as the previous cases while the dither amplitudes are chosen to be0.5,0.3,0.8 respectively. Similarly, the gains for the load optimized NLESC are 0.008,0.004,0.2and the dither amplitudes are the same as those chosen for the NLESC. the weights are chosenas w1 = [1.2, 1, 1.1], w2 = [1.05, 1, 1] and w3 = 0.1.

The results show a similar trend to that seen for the 5% turbulence. The DEL are in somecases marginally higher than the NLESC but maintains it close to or upto 6% less than thebaseline loads. The mean loads are maintained at upto 35% lower tha the baseline load. Thepower output is also marginally lower than the NLESC but is still about 2% greater than thebaseline power.

(a) Torque gain for NLESC (b) Torque gain for load optimized NLESC

Figure 4: Optimal torque gain at a steady wind speed of 8m/s for 16000 seconds

(a) Total power of 3 turbines (b) Percentage change of power of each turbine

Figure 5: Power for a steady wind speed of 8m/s for 16000 seconds for different controllers andthe percentage change in power between them.

(a) Tower moment (b) Shaft moment

Figure 6: Percentage change of the maximum moments for a steady wind speed of 8m/s for16000 seconds for different controllers.

(a) Torque gain for NLESC (b) Torque gain for load optimized NLESC

Figure 7: Optimal torque gain at a steady wind speed of 6m/s for 16000 seconds

(a) Total power of 3 turbines (b) Percentage change of power of each turbine

Figure 8: Power at a steady wind speed of 6m/s for 16000 seconds for different controllers andthe percentage change in power between them.

(a) Tower moment (b) Shaft moment

Figure 9: Maximum moment at a steady wind speed of 6m/s for 16000 seconds for differentcontrollers.

Figure 10: Tower moment of turbine 1 over a period of 30000 seconds for a wind speed of 8m/sand a turbulence intensity of 5% using the different control schemes.

Figure 11: Shaft moment of turbine 1 over a period of 30000 seconds for a wind speed of 8m/sand a turbulence intensity of 5% using the different control schemes.

(a) DEL for shaft (b) DEL for tower

Figure 12: Percentage change in damage equivalent loads for turbulent wind with a mean windspeed of 8m/s and turbulence intensity of 5% for a period of 30000 seconds using the weightsw1 = [1, 1, 1] and w2 = [1, 1, 1].

(a) Mean shaft moment (b) Mean tower moment

Figure 13: Percentage change in mean moments for turbulent wind with a mean wind speed of8m/s and turbulence intensity of 5% for a period of 30000 seconds using the weights w1 = [1, 1, 1]and w2 = [1, 1, 1]. The mean loads are calculated from 5000s for all controllers.

(a) Torque gain of NLESC (b) Torque gain of load optimized NLESC

Figure 14: Torque gains for turbulent wind with a mean wind speed of 8m/s and turbulenceintensity of 5% for a period of 30000 seconds using the weights w1 = [1.03, 1, 1] and w2 = [1, 1, 1].

(a) DEL for shaft (b) DEL for tower

Figure 15: Percentage change in damage equivalent loads for turbulent wind with a mean windspeed of 8m/s and turbulence intensity of 5% for a period of 30000 seconds using the weightsw1 = [1.03, 1, 1] and w2 = [1, 1, 1]

(a) Mean shaft moment (b) Mean tower moment

Figure 16: Percentage change in mean moments for turbulent wind with a mean wind speedof 8m/s and turbulence intensity of 5% for a period of 30000 seconds using the weightsw1 = [1.03, 1, 1] and w2 = [1, 1, 1]. The mean loads are calculated from 3500s for all controllers.

(a) Total power output (b) Percentage change in power output

Figure 17: Power output shown on a windfarm level of the three control algorithms and a perturbine evaluation of the percentage change in power output using the load optimized NLESCfor turbulent wind with a mean wind speed of 8m/s and turbulence intensity of 5% for a periodof 30000 seconds using the weights w1 = [1.03, 1, 1] and w2 = [1, 1, 1]..

Figure 18: Wind speed over a period of 30000s for a wind speed of 8m/s and a turbulenceintensity of 10% at the rotor of each turbine.

(a) Torque gain of NLESC (b) Torque gain of load optimized NLESC

Figure 19: Percentage change in mean moments for turbulent wind with a mean wind speedof 8m/s and turbulence intensity of 10% for a period of 30000 seconds using the weightsw1 = [1.2, 1, 1.1] and w2 = [1.05, 1, 1]. The mean loads are calculated from 5000s for allcontrollers.

(a) DEL for shaft (b) DEL for tower

Figure 20: Percentage change in damage equivalent loads for turbulent wind with a mean windspeed of 8m/s and turbulence intensity of 10% for a period of 30000 seconds using the weightsw1 = [1.2, 1, 1.1] and w2 = [1.05, 1, 1].

(a) Mean shaft moment (b) Mean tower moment

Figure 21: Percentage change in mean moments for turbulent wind with a mean wind speedof 8m/s and turbulence intensity of 10% for a period of 30000 seconds using the weightsw1 = [1.2, 1, 1.1] and w2 = [1.05, 1, 1]. The mean loads are calculated from 5000s for allcontrollers.

(a) Total power output (b) Percentage change in power output

Figure 22: Power output shown on a windfarm level of the three control algorithms and a perturbine evaluation of the percentage change in power output using the load optimized NLESCfor turbulent wind with a mean wind speed of 8m/s and turbulence intensity of 10% for a periodof 30000 seconds using the weights w1 = [1.2, 1, 1.1] and w2 = [1.05, 1, 1].

4. Conclusion and future workBased on the results we can see that the utilizing NLESC does impose structural loads on theturbine while maximizing the power. The steady wind simulations provided us a proof for thetheory and this is further applied in the case of the turbulent wind. The introduction of theloads as a penalty function to the power coefficient mitigates the increase in loads due to theNLESC with negligible decrease in the power. Though the power output is on average upto0.5% lower than the NLESC, it comes with upto 30% decrease in the mean loads and upto 20%decrease in the fatigue loads imposed on the turbine. The results are however subject to thequality of the wind.

At higher turbulence intensity the decrease in the loads reduces. This is because theNLESC and it’s load optimization is tedious to control at these conditions. This brings upthe requirement to smoothen the input of the controller such that it does not a function of thewind. This requires the use of a moving average filter over larger sample times. This makes thecontroller stable at the cost of slower response to changes in the APC.

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