MPC for Wind Power Gradients Utilizing Forecasts, Rotor Inertia, and Central Energy Storage

Transcription

1 23 European Control Conferene (ECC) July 7-9, 23, Zürih, Switzerland. MPC for Wind Power Gradients Utilizing Foreasts, Rotor Inertia, and Central Energy Storage Tobias Gybel Hovgaard, Lars F. S. Larsen, John Bagterp Jørgensen and Stephen Boyd Abstrat We onsider the ontrol of a wind power plant, possibly onsisting of many individual wind turbines. The goal is to maximize the energy delivered to the power grid under very strit grid requirements to power quality. We define an extremely low power outt gradient and demonstrate how deentralized energy storage in the turbines inertia ombined with a entral storage unit or deferrable onsumers an be utilized to ahieve this goal at a minimum ost. We propose a variation on model preditive ontrol to inorporate preditions of wind speed. Due to the aerodynamis of the turbines the model ontains nononvex terms. To handle this nononvexity, we propose a sequential onvex optimization method, whih typially onverges in fewer than iterations. We demonstrate our method in simulations with various wind senarios and pries for energy storage. These simulations show substantial improvements in terms of limiting the power ramp rates (disturbane rejetion) at the ost of very little power. This apability is ritial to help balane and stabilize the future power grid with a large penetration of intermittent renewable energy soures. I. INTRODUCTION Today, wind power is the most important renewable energy soure. For the years to ome, many ountries have set goals for further redued CO 2 emission, inreased utilization of renewable energy, and phase out of fossil fuels. In Denmark one of the means to ahieve this is to inrease the share of wind power to 5% of the eletriity onsumption by 22 and to fully over the energy supply with renewable energy by 25 []. Installing this massive amount of wind turbines introdues several hallenges to reliable operation of power systems due to the flutuating nature of wind power. To mitigate flutuations, modern wind power plants (WPP) are equipped with variable speed wind turbine (VSWT) tehnologies, whih are interfaed with power eletronis onverters that are required and designed to fulfil inreasingly demanding grid odes (see, e.g. [2], [3]). The Grid Code (GC) is a tehnial doument setting out the rules, responsibilities and proedures governing the operation, maintenane and development of the power system. It is a bli doument periodially updated with new requirements and it differs from operator to operator. Countries with large amount of wind power have issued dediated GCs for its onnetion to transmission and distribution levels, foused mainly on power ontrollability and power quality [4], [5]. Partiularly, Denmark establishes some of the most T. G. Hovgaard and L. F. S. Larsen are with Vestas Tehnology R&D, DK-82 Aarhus N, Denmark. {togho,lfsla}@vestas.om J. B. Jørgensen is with DTU Comte, Tehnial University of Denmark, DK-28 Lyngby, Denmark. jbjo@dtu.dk S. Boyd is with the Information Systems Laboratory, Department of Eletrial Engineering, Stanford University, USA. boyd@stanford.edu demanding requirements regarding ative power ontrol [6]. One of the regulation funtions required is a power gradient onstraint that limits the maximum rate-of-hange of nonommanded variations in the power outt from the WPP to the grid. As of today, this onstraint is softened if the power prodution in the WPP drops due to the lak of wind. This is merely out of neessity, and the GCs are expeted to tighten further regarding this requirement. Ensuring slow power gradients redues the risk of instability on the grid, allows the TSO time for ounterating the hange, and improves the preditability of power outt, enabling the WPP owner to t less onservative bids on the power market. Energy storage strikes the major problems of wind power and joining energy storage with WPPs to smoothen variations and improve the power quality is not a new idea. In, e.g., [7] [] the benefits, eonomis, and hallenges of using different means of storage, i.e., batteries, hydrogen, flywheels et., in ombination with wind power are investigated. [] uses a Lithium-iron-phosphate battery to ahieve power foreast improvement and outt power gradient redution. However, the additional ost of batteries or other energy storages is usually the showstopper, at least as the market is today. In our previous works, we have shown how thermal apaity, e.g., in supermarket refrigeration, an be utilized for flexible power onsumption [2], [3]. It is very likely that suh tehniques (where the apaity is a bi-produt of fulfilling another need) an play a major role instead of adding expensive tehnologies whih have storage as their sole rpose. In the rest of this paper, we onsider energy storage in general without distinguishing atual storage from flexible power onsumption. Traditionally, the rotor speed of modern wind turbines is ontrolled for traking the tip-speed ratio (TSR = angular rotor speed rotor radius / wind speed) for maximum power extration, onstrained by the maximum rated speed. However, due to the inertia of the rotating masses in the turbine, there is a potential to improve the quality of the power outt by atively letting the rotor speed deviate from the optimal setting. This might of ourse ome at a ost of slightly redued power outt. In, e.g., [4], [5] turbine inertia is used for frequeny response and power osillation damping. In addition, a vast amount of works exist that address power optimization, fatigue load redution and pith ontrol for individual turbines in the more traditional sense, e.g., [6] [8]. Some of these take optimization and model preditive ontrol approahes to solve the problems and many rely on a known operating point (e.g., loal wind speed and power set-point) for deriving linearized models / 23 EUCA 47

2 Other works onsider the ontrol of large wind farms where the power extrated by upwind turbines redues the power that is available from the wind and inreases the turbulene intensity in the wake reahing other turbines (see, e.g., [9] [2]). In [3], we demonstrate the appreiation of a sequential onvex programming (SCP) approah [22] for a model preditive ontrol problem, ontrolling the power onsumption for ommerial refrigeration with linear dynamis, onvex onstraints, and a nononvex objetive. Inspired by this, we now turn to the power produers side of the grid and apply the same tehnique to a nonlinear wind turbine model. Our method, like sequential quadrati programming (SP) [23], involves the solution of a sequene of (onvex) quadrati programs (Ps), but differs very muh in how the Ps are formed. In SP, an approximation to the Lagrangian of the problem is used; the linearization required in eah step an end up dominating the omtation [24]. In our SCP method, the onvexifiation step is quite straightforward. We demonstrate how model preditive ontrol using foreasts of the wind speed an ensure very low power gradients (e.g., less than 5% of the rated power per minute). We do this with a entral energy storage added to the WPP and show how we an utilize the inertia in the individual turbines to further improve this and minimize the extra storage apaity needed. Our method gives no guarantee in terms of onvergene or optimality but is observed to perform well in pratie. [25] uses onvex optimization to operate a portfolio of eletrial storage devies. In [26], we present a hange of variables that renders the problem fully onvex and demonstrate effiient losed-loop simulations with real wind data. II. WIND POWER PLANT In this setion, we desribe the dynami model used for the WPP in the paper. The WPP an have a number of individual wind turbines arranged in a ertain geographial topology and one entral storage unit. We desribe the simplified dynamis of rotational motion, the onstraints of the system and the funtion refleting the objetive of operating the plant. A. Wind Turbine Model The WPP in the examples onsists of turbines using the NREL 5MW model sine this is openly available, but, ould easily be substituted with any speifi turbine model. The model is desribed in detail in, e.g., [27], [28]. We simplify the model and derive the system equations as follows. Negleting the shaft torsion, we desribe the turbine itself by two dynamial states, the generator speed,, in rad/s and the generator torque, T g, in Nm. = I g +I r /N 2 ( Tr N T g ), () T g = τ g (T g,ref T g ), (2) where I g and I r are the inertias of generator and rotor respetively, N is the gear ratio, τ g is the time onstant of the generator and T g,ref is the torque set-point. The torque, T r, delivered to the rotor by the wind is given by T r = 2 ρac P(λ,β) v3 ω r, where ρ is air density, A is swept area of the rotor, v is wind speed in m/s, ω r is angular rotor speed in rad/s, and the oeffiient of power, C P, is a look-up table (see Fig. ) derived from the geometry of the blades as a funtion of TSR and blade pith angle (β) in degrees. TSR is defined as λ = Rω r /v, where R is the rotor radius in m. We use ω r = /N to eliminate ω r and desribe the power produed in the generator by = η g T g, where η g is the generator effiieny. B. Energy Storage Model We use a simple integrator for illustrating the entral energy storage and desribe its state-of-harge ( [J]) in disrete-time by (t+t s ) = (t)+(t)t s, (3) where is the harge rate in W and t denotes time. We assume that the energy storage is lossless. However, in reality batteries have losses just as, e.g., refrigeration systems inrease the heat load, and thereby the power loss, as the temperatures are lowered to store extra ooling energy. A loss term ould be modeled as η loss (t) whih is added to the equation above, but, as our time-sale for storing energy is in the range of seonds to minutes, we neglet this. We an now find the power supplied from the WPP to the grid =. β ( ) λ Fig. : Coeffiient of power C P. The peak power oeffiient is

3 C. Control Manilated variables: Our optimizer manilates the setpoints to the generator torque, T g,ref, and the pith angle, β ref, for eah individual turbine in the WPP. Normally, the pith is ontrolled by an inner loop exerising a gainsheduled PI ontroller. This ontroller samples up to times faster than our MPC and we set β ref = β as long as the slew rate limit on β ref is observed. Additionally, we manilate the harge rate,, to/from the entral energy storage. Measured variables: The ontroller bases its deisions on measurements of the rotational speed and generator torque in eah turbine, on the known urrent wind speed, and on the filtered, predited future values of the latter overing the entire predition horizon, N p. D. Constraints The extrated power,, must be equal to or less than the available power in the wind, P w, whih is a funtion of the wind speed, v. The turbine is build for a rated power and when the available power in the wind exeeds this level, the blades gradually pith out of the wind to keep the extrated power at the rated level and redue loads on the turbine. Likewise, the extrated power an only follow the available power urve down to a ertain level, P min, due to the mehanial design. Thus, P min P rated, (4) P w (v), (5) For seurity reasons, the turbine is turned off for wind speeds above 25m/s. Therefore, P w =, and the onstraint (4) is not relevant for suh high wind speeds. In addition, four physial onstraints are given by the system β 9, (6) and 8 /s β(t+t s) β(t) T s 8 /s, (7), (8) µp rated, (9) where we introdue the variable µ whih is the maximum needed storage apaity in per unit (), i.e., normalized by rated power. The rotational speed is usually limited by a maximum rated speed, mainly due to too high loads on the turbine at higher speeds. However, sine we want to t the turbine inertia in play, we allow for higher speeds and introdue the parameter ω os whih is the fration of the rated maximum speed that we aept as over-speed.,rated,min (+ω os ),rated,max. () The power supplied to the grid must fulfill the power gradient (t+t s ) (t) P rated T s, () where [,] is the grid ode for maximum power gradient in per unit with respet to rated power. In this study, we do not inlude the wake effets that ouple the individual turbines through the downwind wind flow whih is affeted by the amount of power extrated by upwind turbines. This type of onstraint is a fous of our future work. We define the set Ω as all (T g,ref,β ref,) that satisfy the system dynamis () (3) and the onstraints (4) (). E. Cost We assume in this study that the objetive of the WPP is to maximize the average power supplied to the grid. Alternative operating modes suh as delta prodution (keeping a reserve by produing less than possible) or frequeny response (reating on frequeny deviations on the grid to support stabilization at nominal grid frequeny) are thus not onsidered. The supplied energy, E, over the period [T,T final ] is E = Tfinal T dt. Furthermore, we have a ost on the available storage apaity. For a period [T,T final ] this is S = max[µ] T final T storage. We an onsider the storage prie, storage, as a tuning parameter or as diretly refleting, e.g., rhase prie of batteries divided by their lifetime, a servie agreement with a flexible onsumer, et. Thus, S is a ost in the design phase only (or for simulations as we will show here). F. Nominal Controller We ompare the performane of our proposed method to the solution from the nominal wind turbine ontrol strategy, also defined in the NREL 5MW model. For natural reasons this system is only apable of obeying the power gradient onstraint in three ases: ) When the rate-of-hange of the available power in the wind is less than the power gradient onstraint. 2) When the available power in the wind only hanges from one point to another, where both are above rated power. 3) When suffiiently large amounts of storage is added and its harge/disharge is ontrolled by some kind of preditive ontrol with knowledge of the future wind speed. III. MPC CONTROLLER The WPP is influened by disturbanes from the wind speed whih we an predit (with some unertainty) over a time horizon into the future. The ontroller must obey ertain onstraints, while maximizing the power supply and limiting additional osts for storage. Eonomi MPC an address all these onerns. Whereas the ost funtion in MPC traditionally penalizes a deviation from a set-point, the proposed eonomi MPC diretly reflets the atual osts of operating the plant. Like in traditional MPC, we implement the ontroller in a reeding horizon manner, where an optimization problem over N p time steps (the ontrol and predition horizon) is solved at eah step. The 473

4 result is an optimal int sequene for the entire horizon, out of whih only the first step is implemented. The optimization problem is thus formulated as maximize E S, subjet to (T g,ref,β ref,) Ω, (2) where the variables are T g,ref, β ref and (all funtions of time). Instead of (2) we solve a disretized version with N p steps over the time interval [T,T final ], {T g,ref,β ref,} = { Tg,ref,β k ref, k k} N p. (3) k= The MPC feedbak law is the first move in (3). The ontroller uses the initial state as well as preditions of the wind speed for the time interval. The preditions ould ome from any good soures available, see e.g., [29] where - minute ahead preditions are implemented. A. Sequential onvex programming method As we saw in II, neither the feasible set Ω nor the ost funtion term P are fully onvex. Instead of using a generi nonlinear optimization tool, we hoose to solve the optimization problem iteratively using onvex programming, replaing the nononvex terms with onvex approximations. In eah iteration, i, we perform a first-order Taylor expansion of the nononvex parts around the operating point found in iteration i, estimating the derivatives that involve table look-ups by perturbing the parameters. As the wind speed v is predited we an use v 3 as int to our model instead. We establish the following linear approximations ˆT i r =T r i +[ Tr ω r, T r C P ] i [ ω i r ω i r C i P Ci P ˆ=P i i Pg g +[, P ] i [ g ω i g ωg i T g Tg i Tg i Thus, in eah iteration we solve a onvex optimization problem, whih an be done very reliably and extremely quikly [3]. While our proposed method gives no theoretial guarantee on the performane, we must remember that the optimization problem is nothing but a heuristi for omting a good ontrol and that the quality of losed-loop ontrol with MPC is generally good without solving eah problem aurately. B. Regularization We use two different types of regularization in the optimization problem. To avoid osillations from iteration to iteration, we add proximal regularization of the form N ϕ prox = ρ prox k= ], ]. X k X k,prev 2 2, (4) for eah of the ontrol variables X = {T g,ref,β ref,}. The supersript prev indiates that it is the solution from the previous iteration and ρ prox is a onstant weight hosen to damp large steps in eah iteration. In addition, we add a quadrati penalty on the rate-of-hange (ro) of the manilable variables, N ϕ ro = ρ ro k= X k X k 2 2. (5) This regularization term serves two rposes: It improves the onvergene of the sequential programming method, and also disourages rapid hanges, whih helps redue osillations and fatigue loads. C. Algorithm Algorithm outlines the method. The term nominal refers to the solution obtained from the nominal ontroller. We use this as a baseline for initializing the algorithm. In MPC, the solution from the previous time step is usually well suited for warm-starting the algorithm. Algorithm Iterative optimization. Initialize { T g,t r,ω g,ω r,c P,T g,ref,βref} = {nominal(vk )} Np k=, i =. Comte ˆT i g, ˆP i g and Ĉi P, from {T g,t r,,ω r,c P,T g,ref,β ref } i and v. Solve maximize E i S i +ϕ prox +ϕ ro, subjet to (T i g,ref,β i ref,) ˆΩ, Update { } T i g,t i r,ω i g,ω i r,c i P,T i g,ref,β i ref, and i = i+ Repeat until onvergene. IV. RESULTS In this paper, we apply the proposed method to a oneptual study limited to only one wind turbine. We implement and solve our ontroller for different senarios using CVX [3], [32]. In this setion, we report on results with a power gradient onstraint as low as 3% of the rated power per minute ( = 5 4 /s) and with an allowed overspeed of 5% above rated speed for short time intervals. We sample with T s = s intervals and use a horizon of 5 minutes (N p = 3) in this ase. Obviously, a wide range of solutions an be obtained depending on the speifi ramp rate of the wind speed, the wind speeds before and after the hange ours, the allowed amount of overspeed and the definition of storage prie versus power sales prie. In this paper, we give proofof-onept of the method, using a few seleted trajetories, and for the next version of this work, we will derive a more generalized measure of the relation between wind ramp rates, overspeed ratio, power onstraint, and storage apaity. Fig. 2 shows examples of how our proposed method performs in different ases, while satisfying the power gradient onstraint. For all four ases shown in the figures, we an alulate the total power delivered to the grid from t =...8s. For the senario in figures 2(a) 2(b) (wind speed hanges from m/s to 8m/s), the available power in the wind is below the rated power for the entire interval. Thus, 474

5 ,nom ,nom (a) v = m/s, v 2 = 8m/s, max() = 27.9, storage = low ,nom () v = 2m/s, v 2 = m/s, max() = 22.5, storage = low (b) v = m/s, v 2 = 8m/s, max() = 2.5, storage = high ,nom (d) v = 2m/s, v 2 = m/s, max() = 2., storage = high Fig. 2: Test of power gradient satisfation. We use as the unit for all quantities, exept the state-of-harge () whih is normalized first with respet to maximum storage apaity. In all senarios we let the wind speed drop from v to v 2 linearly from t = 4s to t = 43s, and we show ases with high and low storage ost.,nom is the power outt from the nominal ontroller. no extra power exists for aelerating the rotor beyond rated speed. If entral energy storage is relatively heap (Fig. 2(a)) this is used entirely as a buffer for ahieving the ommanded power gradient while the turbine behaves exatly as with the nominal ontroller. In this ase, the total amount of energy delivered to the grid is equal to the nominal ase too. As the prie of energy storage inreases the ontroller trades off the power prodution that is lost during the phase where the rotor is aelerated, in order to use that kineti energy during the power ramp to redue the peak of needed storage apaity. In Fig. 2(b) the storage apaity is redued by 26.5% at the ost of.3% of the energy delivered to the grid, ompared to Fig. 2(a). For the senario in figures 2() 2(d) (wind speed hanges from 2m/s to m/s), the available power in the wind goes from above rated to below rated power. In this ase, the rotor an be aelerated to reah the maximum allowed speed just when the available power in the wind begins to drop. This kineti energy is used during the power ramp no matter how heap storage is, as it only adds to the total delivered energy. In Fig. 2() the amount of energy delivered to the grid is.6% higher than with the nominal ontroller. When storage ost is inreased, the utilization of stored inertia is shifted towards the time when the storage needs peak, to redue the required additional apaity, and the extra prodution gained otherwise is now traded off with storage ost. In Fig. 2(d) the energy delivered is just.3% less than with the nominal ontroller while the storage need is redued by almost 7%. A. Convergene and Comtation When initialized with the trajetory from the nominal ontroller, the proposed method generally onverges in 5 iterations. In MPC, however, the open-loop trajetory 475

6 from the previous run of the optimizer, shifted one timestep, is an exellent guess on the next outome and is wellsuited for warm-starting the algorithm. Using this warm start initialization, the method generally just need a ouple of iterations to onverge. V. CONCLUSION In this paper, we present an approah to power gradient redution for fulfilling future, tighter grid odes and for improving the quality of power delivered to the grid from wind power plants. We utilize turbine inertia as a resoure of distributed energy storage, limited by the rotational speed, in addition to a entral storage unit whih is assoiated with an extra ost. Our method is based on onvex optimization, solved iteratively to handle the nononvexity of the aerodynamis. Simulation on realisti models reveal a signifiant ability to rejet the disturbanes from fast hanges in wind speed, ensuring ertain power gradients, while keeping the amount of produed power lose to nominal. We an easily trade off lost prodution versus prie of extra energy storage. VI. ACKNOWLEDGMENTS We thank Martin Ansbjerg Kjær from Vestas Tehnology R&D for helpful disussions, ideas and suggestions. REFERENCES [] Danish Ministry of Climate Energy and Building, Energy poliy report 22, Teh. Rep., May 22. [Online]. Available: Sider/danish-limate-and-energy-poliy.aspx [2] J. Morren, J. Pierik, and S. W. H. de Haan, Inertial response of variable speed wind turbines, Eletri Power Systems Researh, vol. 76, no., pp , 26. [3] J. F. Conroy and R. Watson, Frequeny Response Capability of Full Converter Wind Turbine Generators in Comparison to Conventional Generation, IEEE Transations on Power Systems, vol. 23, no. 2, pp , may 28. [4] F. Iov, A. Hansen, P. Sørensen, and N. Cutululis, A survey of interonnetion requirements for wind power, in Pro. of the Nordi wind power onferene (NWPC). Risø National Laboratory, 27. [5] B. Singh and S. 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