Extremum-seeking control for harmonic mitigation in electrical grids of marine vessels

Transcription

1 Extremum-seeking control for armonic mitigation in electrical grids of marine vessels Mark Haring, Espen Skjong, Tor Arne Joansen, Senior Member, IEEE, and Marta Molinas, Member, IEEE Abstract Tis work focuses on te minimization of te armonic distortion in multi-bus electrical grids of marine vessels using a single active power filter. An active power filter is commonly used for local armonic mitigation. However, local filtering may lead to a wack-a-mole effect, were te reduction of armonic distortion at te point of installation is coupled to an increase of distortion in oter grid nodes. Te few existing filtering metods tat consider system-wide mitigation are based on an accurate model of te power grid, wic may not be available if te complexity and te scale of te grid are large. In tis work, we investigate te use of an extremum-seeking control metod to optimize te injection current of an active power filter for system-wide armonic mitigation. Because te extremum-seeking control metod is model free, it can be used witout knowledge of te electrical grid. Moreover, te metod can be implemented on top of existing approaces to combine te fast transient response of conventional armonic-mitigation metods wit te optimizing capabilities of extremum-seeking control. Index Terms Active power filter, Extremum-seeking control, Harmonic mitigation, Power grids I. INTRODUCTION EXTREMUM-SEEKING control is an adaptive-control metodology tat optimizes te performance of dynamical plants based on measurements by automated tuning of plant parameters [], [2]. Te main advantage of extremum-seeking control compared to many oter optimization tecniques is tat often no plant model is required. Tis makes extremumseeking control suitable to optimize te performance of systems for wic an accurate model is not available or costly to obtain. In tis work, we apply extremum-seeking control to compute te injection current of an active power filter tat minimizes te armonic distortion in an electrical grid on board of a marine vessel. Manuscript received June 2, 27; revised February 6, 2; accepted April 2, 2. Tis work is sponsored by te Researc Council of Norway and Ulstein Power & Control AS, project number 225, by te KMB project D2V, project number 267, and troug te Centres of Excellence funding sceme, project number NTNU AMOS. M. Haring and T. A. Joansen are wit te Centre for Autonomous Marine Operations and Systems, Department of Engineering Cybernetics, NTNU, Norwegian University of Science and Tecnology, 79 Trondeim, Norway pone: ; fax: ; mark.aring@ntnu.no, tor.arne.joansen@ntnu.no. E. Skjong is wit Ulstein Blue Ctrl AS, 6 Ålesund, Norway espen.skjong@ulstein.com. M. Molinas is wit te Department of Engineering Cybernetics, NTNU, Norwegian University of Science and Tecnology, 79 Trondeim, Norway marta.molinas@ntnu.no. Tere are many callenges related to te power quality on board of marine vessels; see for example [3], []. One of tese callenges is armonic distortion. Harmonic distortion in alternating-current power grids is te presence of armonic components in current and voltage signals oter tan te fundamental frequency. Harmonic distortion is caused by nonlinear loads in te power grid. Altoug low levels of armonic distortion are often tolerated, ig levels of armonic distortion can result in significant power losses and an increased wear of mecanical components in te grid. Severe armonic distortion may even lead to overeating and failure of components. Several armonic mitigation metods suc as passive filtering, active filtering and pase multiplication are discussed in [5] [7]. In practice, often a combination of mitigation devices is employed to enance power quality. An active power filter injects a current to counteract te armonic distortion generated by te nonlinear loads in te power grid. Te control of active power filters for local filtering as been studied extensively in [7] [] and references terein. Altoug local filtering decreases te armonic distortion at te point of installation, it may simultaneously increase te distortion in oter buses of te grid, leading to a wack-amole effect []. Tere are several metods tat avoid te wack-a-mole using a system-wide approac. To avoid te wack-a-mole, te armonic distortion in multi-bus electrical grids may be mitigated by connecting an active power filter to eac grid node. However, tis is often not a viable solution for marine vessels due to te large economic cost and te limited available space on board te vessel. Contrary to local filtering, tere are metods tat apply a system-wide approac using a limited number of active power filters often only a single active power filter is used to avoid wack-a-mole issues. Tese metods are based on computing te relation between te current injections of te active power filters and te corresponding armonic distortion in te grid nodes wit te elp of a grid model. To find te optimal current injection of an active power filter for system-wide mitigation under static load conditions, a cost function is introduced in [2], [3] to weig te armonic voltage distortion in te buses of te grid. Te impedance matrix of te power grid is used to link te voltage distortion to te current injection of eac active power filter. Te optimal current injection is subsequently obtained by minimizing te cost function. In [] [], a model-predictive control metod is presented for system-wide armonic mitigation. Te metods in [2], [3] and in [] [] ave two major drawbacks tat limit

2 teir applicability. First, an accurate grid model is required to effectively mitigate te armonic distortion in te electric grid. Obtaining an accurate grid model may require modeling of many components in te grid as well as teir interconnections. Hence, te effort and expenses of applying tese metods may be substantial, especially if te complexity and te scale of te grid are large. Second, te underlying optimization problems on wic tese metods are based need to be solved at every sampling instance if te metods are to be implemented in real time. Depending on te scale and complexity of te grid, one may ave to settle for a relatively coarse grid model to avoid tat te computational effort exceeds te available computational capacity to solve te optimization problem in te limited available time. In turn, a coarse grid model may impair te performance of te metods. Te contributions of tis work can be summarized as follows. First, we present a discrete-time extremum-seeking control metod to optimize te injection current of a single active power filter for system-wide armonic mitigation in electric grids of marine vessels. We note tat te presented metod can easily be extended to include several active power filters using a multivariable extremum-seeking control approac similar to [9], [2]. Te main advantage of te presented extremum-seeking metod is tat it does not require a model of te grid. Contrary to alternative model-based metods, it is computationally ceap and easily scalable to a grid wit an arbitrary number of nodes. Second, te extremumseeking controller can be implemented on top of local active filtering approaces to combine te fast transient response of conventional armonic-mitigation metods wit te systemwide optimizing capabilities of extremum-seeking control. Te organization of tis work is as follows. We formulate te armonic-mitigation problem in Section II. Te extremumseeking control metod is introduced in Section III. A case study of a diesel-electric sip wit a tree-bus electrical grid wit distributed generators is presented in Section IV. Te conclusion of tis work is given in Section V. We introduce te following notations. I is te identity matrix. is te zero matrix. M T denotes te transpose of te matrix M. II. HARMONIC-MITIGATION PROBLEM FORMULATION Consider a stable balanced tree-pase tree-wire multibus power grid. An active power filter is connected to one of te buses of te grid. Suppose we want to use te active power filter to minimize te armonic distortion in n buses of te electrical grid. Let tese buses be numbered one to n. Moreover, let te tree pases be denoted by a, b and c. For constant loads and steady-state conditions, a simplified representation of te voltages in bus j for te pases a, b and c is given by V j,a t = V j,b t = V j,c t = = = = 2πt A j sin 2πt A j sin + φ j, 2π 3 + φ j, 2πt A j sin + 2π 3 + φ j, for j = {, 2,..., n}, were A j and φ j are te amplitude and te pase offset of te t-order armonic of te voltages in bus j, were t denotes te time, and were is te period of te fundamental frequency. We note tat te voltage contributions for interarmonic frequencies may be substantial in some marine applications [2]. However, tese are neglected ere in order to focus on te armonic mitigation problem. To balance te objective of minimizing te armonic distortion in te n buses, we introduce te following cost function consisting of te sum of squared voltage amplitudes of te dominant distortion armonics in te electrical grid, similar to [2], [3]: n JA, A 2,..., Am n = βj A 2 j, 2 j= H were βj is a cosen positive weigting constant for te voltage amplitude A j, and were H = {, 2,..., m } is a set consisting of te orders of m dominant armonics in te electrical grid to be mitigated, were eac element of H is unique and larger tan one. As pointed out in [3], te cost function in 2 is suited to incorporate several armonicdistortion measures, including te total armonic distortion, te telepone influence factor and te motor-load loss function. To minimize te armonic distortion in te buses, we provide te following current reference to te active power filter for te tree pases a, b and c: i r,a t = H i r,c t = H i r,ct, i r,at, i r,b t = H i r,bt, wit 2πt 2πt i r,at = u sin + u 2 cos, T f 2πt i r,bt = u sin 2π 2πt 3 + u 2 cos 2π 3, 2πt i r,ct = u sin + 2π 2πt 3 + u 2 cos + 2π 3 and parameters u, u 2,..., um 2. By feeding te references in 3 to te active power filter, te active power filter generates a current injection for te tree pases wit feedback from te power grid. Assuming tat te generated current injection is equal to te reference current and tat te bus connections in te grid can be modeled by linear impedance, te impedance matrix of te grid can be used to determine te effect of te current reference on te voltages in te buses; see [2], [3]. 3

3 For example, te voltage difference in bus j for pase a due to te current injection of te active filter can be written as V j,a t = Z R,ju + ZI,ju 2πt 2 sin H + 5 ZR,ju 2 ZI,ju 2πt cos for j {, 2,..., n}. Here, ZR,j and Z I,j denote te real and imaginary part of te impedance tat links te t armonic of te current injection of te active power filter to te voltage of bus j. Similar expressions for te voltage differences for te pases b and c can be obtained by applying appropriate pase sifts as in and 3. Now, let te voltage in bus j for pase a prior to te current injection be denoted by Vj,at = A, 2πt j sin + φ, j, 6 = suc tat te voltage after te current injection is given by V j,a t = V j,at + V j,a t. 7 From and 5-7, it follows tat A 2 2 j = A, j cosφ, j + ZR,ju + ZI,ju A, j sinφ, j + ZR,ju 2 ZI,ju for all j {, 2,..., n} and all H. By combining and te cost function in 2, we obtain tat te output of te cost function is a function of te parameters u, u 2,..., um 2, tat is, JA, A 2,..., Am n = F u, u 2..., u m, wit F u, u 2..., u m n = βj A, j cosφ, j + ZR,ju + ZI,ju 2 j= H + A, j sinφ, j + Z R,ju 2 Z I,ju 2 9 and u = [u, u 2] or all H. We refer to te function F as te objective function. To minimize te cost function in 2, we aim to find te values of te parameters u, u 2,..., um 2 for wic te value of te objective function is minimal. To simplify te task at and, we note tat F u, u 2..., u m = H F u, wit u = [u, u 2] T and n 2 F u = βj A, j cosφ, j + ZR,ju + ZI,ju 2 j= 2 + A, j sinφ, j + ZR,ju 2 ZI,ju. Hence, minimizing te objective function F in 9 is equivalent to minimizing eac quadratic function F in. Contrary to [2], [3] and [] [], we assume tat detailed knowledge of te electrical grid is not available. Te active power filter and te power grid are regarded as a black box; see Fig.. Tis 2 Current reference + Active power filter Controller Current injection Feedback Black box Cost function 2 Cost 2 Controller function. m Controller Fig.. Harmonic-mitigation sceme.. m Cost function Power grid Voltage measurements Fast Fourier transform implies tat te impedance matrix of te grid and tus te constants ZR,j and Z I,j are unknown. Hence, te functions F are unknown and teir minima cannot be computed analytically. To minimize te cost function in 2, we introduce m extremum-seeking controllers to find te values of te parameters u and u 2 tat minimize te function F and generate te corresponding partial current reference in for eac H. Te current reference in 3 tat minimizes te cost function in 2 is subsequently obtained by summing te partial current references produced by te controllers. In order to determine te parameters u and u 2 tat minimize te function F, eac extremum-seeking controllers minimizes te cost function J A, A 2,..., A n = n j= β j A j 2 2 for one armonic H, were te voltage amplitudes A, A 2,..., A n are obtained by applying a fast Fourier transform algoritm to measurements of te voltage signals of te buses. A comparison of different algoritms to compute coefficients in Fourier series for electric power system applications is provided in [22]. We note tat, similar to [2], [3] and [] [], te presented armonic-mitigation solution requires communication between te buses to process te voltage measurements. An overview of te armonicmitigation sceme is given in Fig.. Te assumptions in tis section tat are used to obtain te expression of te objective function in 9 may not old in practical applications: te power grid may be unbalanced, te active power filter may generate a current injection tat differs from te current reference, modeling of te bus connections by linear impedance may be inaccurate, etc. Noneteless, te assumptions in tis section often appear to be be good approximations in practice and are common in many text books about armonic mitigation; see for example [7]. As we will see in te case study in Section IV, extremumseeking control is a robust optimization tecnique tat may

4 be successfully applied even if te assumptions in tis section do not entirely old. III. EXTREMUM-SEEKING CONTROL METHOD For eac H, we introduce a discrete-time extremumseeking controller to find te values of te parameters u = [u, u 2] or wic te objective function F in exibits a minimum. Let te sampling time of te extremum-seeking controller be denoted by te positive constant T s. At eac sampling instance t = kt s wit counter k =,, 2,..., te extremum-seeking controller updates te values of te parameters u and te corresponding partial current reference to te active power filter in. Let u k = [u,k, u 2,k ]T denote te vector of parameter values at time t = kt s. Moreover, let yk = J A,k, A 2,k,..., A n,k, were te inputs A,k,..., A n,k of te cost function J in 2 are obtained by taking te fast Fourier transform of te measured voltage signals of te buses for te time interval kt s, kt s ]. In Section II, we implicitly assumed tat u and u 2 are constant suc tat J A, A 2,..., A n is equal to F u under steady-state conditions. For a stable power grid, as we assume ere, te output yk of J remains close to F u k for suitable initial conditions of te active power filter and te electrical grid if u k is sufficiently slow compared to te dynamics of te active power filter and te electrical grid. Tis can be formally proved using a singular-perturbation metod as in [23], [2]. Due to te use of te fast Fourier transform, a distributed time delay is introduced between u k and y k. Tis delay is especially large if T s. Bounded time delays in extremum-seeking scemes can be andled by making te parameter vector u k sufficiently slowly time varying; see for example [25], [26]. In tis work, owever, we aim to compensate for te time delay, wic may elp to enable a faster convergence of te extremum-seeking controller [27], [2]. We model te relation between u k and y k as y k = N N r= F u k r, 3 were we assume tat N = T s is a positive integer. Linear interpolation can be applied to obtain a similar expression if T s is not a positive integer. We note tat 3 implies tat yk = F u k if u k is constant and tat a similar argument as before can be invoked to prove tat 3 is an accurate approximation for suitable initial conditions if u k is sufficiently slowly time varying. Next, we introduce a perturbation-based extremum-seeking controller. Te controller steers te parameters u k towards te point of optimal armonic mitigation using a gradient-descent approac based on an estimate of te gradient of te objective function F. We define [ ] T 2πk 2πk u k = û k + αωω k, ω k = sin, cos, N ω N ω were ω k is a vector of perturbations wit amplitude α ω >. Te tuning parameter Nω > is an integer related to te frequency of te perturbations in. From and Taylor s teorem, we ave tat F u k = F û k + αω df du û kω k + αω 2 R,k, 5 were αω 2 R,k is te remainder of te Taylor series expansion. R,k is a function of te uniformly bounded vector ω k in and te Hessian of te quadratic function F in. Hence, it is independent of u k and uniformly bounded, wic implies tat te remainder term αω 2 R,k can be made arbitrarily small by coosing sufficiently small values of αω. Define û k = û k+ û k. Similar to 5, it follows from Taylor s teorem tat F û k+ = F û k + αω df du û k û k αω + αω 2 R2,k, 6 and αω df du û k+ = αω df du û k + αω 2 R T 3,k. 7 Assuming tat û k α ω is uniformly bounded wit a bound tat is independent of αω, by using similar reasoning as for R,k, it follows tat R 2,k and R 3,k are uniformly bounded, wic implies tat te remainder terms αω 2 R2,k and αω 2 R T 3,k can be made arbitrarily small by coosing sufficiently small values of αω. From -7, we obtain tat 3 can be accurately approximated by yk = F û k + αω df du û k N N r= u k r α ω û k α ω û k α ω for sufficiently small values of αω if is uniformly bounded wit a bound tat is independent of αω. Now, let us define te vector [ F û m k ] k = T αω df û du k. 9 By combining 6-9, we obtain tat te dynamic model wit and C k = [ m k+ = A km k y k = C km k, û k A k = αω I N N r= u k r α ω T 2 2 û k α ω T ], 22 is accurate if u k is sufficiently slowly time varying, if û k is α ω uniformly bounded wit a bound tat is independent of αω, and if αω is sufficiently small. We note te vector m k in 9 contains te gradient of te objective function F scaled by te tuning parameter αω. Terefore, an estimate of te gradient of te objective function can be obtained by estimating te state vector m k of te model. We note tat te perturbations in

5 are essential for estimating te gradient of te objective function because tey ensure tat te model 2 is uniformly observable under appropriate tuning conditions. We introduce te following tree-step observer [29] to estimate te state vector m k : Step 2 correction step: wit ˆm k 2 = ˆm k + L k yk C k ˆm k Q k 2 I = L k C k Q k T Lk, + λ m L k Step 2 3 regularization step:, I L k C k ˆm k 3 = ˆm k 2 L k 2 D ˆm k 2, T Q k 3 = I L k 2 D Q k 2 I L k 2 D + σ r λ m L k 2 Step 3 prediction step: ˆm k+ = A k ˆm k 3, Q k+ = λ A kq k 3 m T 23 2 T L k 2, A k T, 25 L k = Q k C T k λ + C kq k C T k, m 26 L k 2 = Q k 2 DT σr λ m I + DQ k 2 DT and D = [ I ]. 27 Te observer in is comparable to a Kalman filter, were ˆm k 3 is an estimate of te state vector m k, and were Q k 3 resembles te positive-definite state covariance matrix of te Kalman filter. Te vectors ˆm k and ˆm k 2 and te positivedefinite matrices Q k and Q k 2 are intermediate variables. Te observer is initiated by selecting te values of ˆm k and Q k, after wic can be used to obtain an estimate of te state vector for eac subsequent time step. Te tuning parameter λ m, is sometimes referred to as te forgetting factor [3]. Its value determines te convergence speed of te observer: a value close to zero implies a fast convergence, wile a value close to one implies a slow convergence. Commonly, te value of λ m is set to be close to one. Contrary to te Kalman filter, te observer contains a regularization step tat prevents te elements of te matrix Q k 3 from becoming excessively large if te parameter vector u k is momentarily not sufficiently ric to accurately estimate te state vector m k. Because regularization deteriorates te estimate of te state vector m k, te regularization constant σr > is commonly cosen to be small. Noting tat ˆm k 3 is an estimate of te state vector m k, we obtain tat D ˆm k 3 is an estimate of te gradient of te objective function F scaled by te tuning parameter αω. Wit tis in mind, we define te following gradient-descent optimizer to guide u k towards te minimum of te objective function F : ηud ˆm û k+ = û k λ k 3 u ηu + λ u D ˆm 2 k 3, wit linear gain λ u > and normalization gain ηu >. Te linear gain λ u influences te convergence speed of te optimizer: a large value results in a fast convergence, wile a low value results in a slow convergence. Its value sould be cosen sufficiently small to enable convergence towards te minimum of te objective function F and to preclude cattering. Te normalization gain ηu limits te maximal convergence convergence rate of te optimizer. By normalizing te optimizer gain, we obtain tat û k η u suc tat û k α ω is uniformly bounded wit a bound tat is independent of αω for any value of η u tat is proportional to αω. A. Tuning of te controller As mentioned, te values of αω and σ r sould be sufficiently small for a successful controller implementation. Te remaining tuning parameters of te controller are cosen suc tat te resulting closed-loop system exibits te following time scales, similar to [2] and also [23], [2]: fast active power filter, power grid; medium fast perturbation of te controller; medium slow observer of te controller; slow optimizer of te controller. Tis can be acieved by coosing te tuning parameters suc tat N ω, lnλ mnω, λ u α ω lnλ m and η u α ω lnλ m are sufficiently small; see also [29]. As mentioned, te active power filter and te power grid are faster tan te controller to ensure tat te model in 2 is accurate. Te perturbations of te controller are faster tan te observer of te controller so tat te time window of te observer is sufficiently long for estimating te state vector ˆm k by observing te perturbations in te time signal of yk. Finally, te observer of te controller is faster tan te optimizer of te controller to provide an accurate state estimate witout muc lag. More details about te stability and tuning of te controller can be found in [29]. IV. CASE STUDY: THREE-BUS ELECTRICAL GRID WITH DISTRIBUTED GENERATORS We consider te tree-bus electrical grid wit distributed generators in Fig. 2. Te electrical grid portrays a simplified sipboard power system. It consists of two generators, tree buses wit propulsion loads, an active power filter, an LCL filter and RC sunts. Te loads are modeled as variable-speed drives wit 2-pulse rectifiers. Due to te 2-pulse rectifiers, te dominating armonics are of te orders 2r± for positive integer values of r. Te parameters of te model are presented in Table I. Te per-unit model is given relative to te generator power rating. Te current tat can be produced by te active filter is limited. To avoid unwanted effects due to saturation of te filter current tat is, current clipping, te current

6 TABLE I PARAMETERS OF THE ELECTRICAL GRID WITH ω f = 2π Parameter Value Unit Nominal voltage 69 V Fundamental frequency 5 Hz Generator power rating MVA L G, L G2.2 pu R G. L G ω f pu R G2. L G2 ω f pu L M. pu R M. L M ω f pu L M2. pu R M2. L M2 ω f pu C S, C S2 2 µf R S, R S2 2 Ω L L, L L2.3 mh R L, R L2.3 Ω C C 3 µf R C Ω R D 6 Ω references given to te active power filter are cut off if tey exceed te maximal current tat te active power filter control can produce. Because te models of te grid and te active power filter are similar to te ones used in [7], te reader is kindly referred to [7] for more information. Te simulations in tis section are conducted in MATLAB/Simulink using te Simscape Power Systems toolbox. We use te extremum-seeking control metod in Section III to compute te optimal parameters u and u 2 of te current reference 3 of te active power filter for te armonics {, 3, 23, 25}. Te sampling time of te extremum-seeking controller is given by T s = 3 s. Te tuning parameters =. pu, Nω =, λ m =.7, λ u = λ 3 u = pu, λ 23 u = λ 25 u = pu, ηu = ηu 3 = 3, ηu 23 = ηu 25 = 3 and σr = 3 for all {, 3, 23, 25}. Te applied cost function in 2 is J = 3 j= A j,a 2 + A j,b 2 + A j,c 2, were A j,a is te amplitude of te t armonic of te voltage in bus j tat is obtained from te fast Fourier transform of te voltage signal of pase a. Te amplitudes A j,b and A j,c are defined similarly suc tat J = 3 3 j= A j 2 if A j = A j,a = A j,b = A j,c under steady-state balanced conditions. It is essential tat te perturbation amplitude is sufficiently large so tat te effect of te perturbations can be observed in te voltage amplitude signals in order to estimate te gradient of te objective function; see Section III. However, because te resulting oscillations in te voltage amplitude signals impair te obtained steadystate performance, te perturbation amplitude is cosen to be relatively small. To illustrate te difference between local and system-wide armonic mitigation, we compare our results wit tose of a local-filtering metod. Te local-filtering metod extracts te t, 3t, 23rd and 25t armonic from current measurements of te local load Load 2 using a fast Fourier are set to αω = αω 3 =.2 pu, αω 23 = αω 25 transform and provides te same armonics wit an opposite pase to te active power filter as current reference, similar to [3]. Te extremum-seeking control metod can easily be combined wit oter metods. To demonstrate tis, we additionally present results for a combination of te extremumseeking metod and te local-filtering metod. For tis combined metod, te current reference tat is supplied to te active power filter is te sum of te current references of te extremum-seeking control metod and te local-filtering metod. Te current reference of te local-filtering metod acts as a feedforward to te extremum-seeking controller in order to respond faster to canging grid conditions. Moreover, we also compare our results wit tose of te model-predictive control metod in [] []. Te model-predictive control metod uses a simplified two-bus grid model, were Load 2 and Load 3 are replaced by a single load wit an equivalent combined power. Te two-bus model as fewer states and parameters tan a tree-bus model, wic makes model identification easier and te computational burden lower. However, because Bus 3 is not contained in te simplified model of te model-predictive controller, te resulting current injection of te active power filter only targets te armonic distortion in Bus and Bus 2. Because modeling and monitoring all loads tat are connected to and disconnected from te electrical grid of a sip is often practically infeasible, it is commonly necessary to resort to model simplifications similar to te one ere. A. Constant load conditions We use te total armonic distortion THD as a measure for te mitigation performance, were te total armonic distortion of te voltage in Bus j, wit j {, 2, 3}, is given by Vrms,j Vrms,j Vrms,j THD j = V rms,j, 29 were Vrms,j is te root-mean-square value of te t voltage armonic in Bus j. Table II present te total armonic distortion of te voltage in te buses under different constant load conditions, were te power of Load, Load 2 and Load 3 is denoted by P, P 2 and P 3, respectively. From Table II, we obtain te total armonic distortion of te modelpredictive control MPC metod and te extremum-seeking control ESC metod are comparable for low-load conditions. For ig-load conditions, te electrical grid is more sensitive to te applied armonic compensation. Because te model imperfections are more predominant under ig-load conditions, te model-predictive controller performs sligtly worse tan te extremum-seeking controller under tese conditions. Because te model-predictive controller is designed to mitigate te armonic distortion in Bus and Bus 2 only, te armonic distortion in Bus 3 may be muc larger tan te distortion in te oter two buses under certain load conditions, as sown in Table II. Te extremum-seeking control metod, on te oter and, uses voltage measurements from all tree buses and is terefore able to mitigate te distortion in te buses

7 L L2 R L2 L L R L Gen R G L G Gen 2 R G2 L G2 R D C C R C Active power filter Bus Bus 2 Bus 3 R M L M R M2 L M2 C S C S2 R S R S2 Load Load 2 Load 3 Fig. 2. Model of tree-bus sipboard power system. more evenly. Compared to tese two system-wide armonicmitigation metods, te local-filtering metod Local performs significantly worse. Combining te extremum-seeking control metod and te local-filtering metod Local + ESC gives a performance tat is similar to tat of te extremumseeking control metod. Te amount of armonic distortion of te voltages in te buses mildly oscillates if te extremumseeking controller is applied due to te use of perturbations; see Section III. To obtain te constant THD values in Table II, te root-mean-square values of te corresponding voltage armonics are computed by taking te mean over a sufficiently long time interval. B. Dynamic load conditions To compute te total armonic distortion under dynamic load conditions, te lengt of te time window for te THD calculation is set to te wavelengt of te fundamental frequency. In Fig. 3, te THD dynamic responses to a step in te power of te loads are displayed; te power of Load and Load 2 is increased from.3 pu to. pu at time zero, wile te power of Load 3 is kept constant at zero. Te oscillations in te voltage THD signals of te extremum-seeking control metod are due to te used perturbations. Due to te increased sensitivity of electrical grid to te applied armonic mitigation for ig-load conditions, te amplitude of te oscillations is larger for ig-load conditions tan for low-load conditions. Compared to te model-predictive control metod and te local-filtering metod, it takes te extremum-seeking control metod longer to adapt to te new power levels of te loads. Te convergence is faster for te combined extremum-seeking control and local-filtering metod, but not as fast as for te model-predictive control or te local-filtering metods. To simulate te sipboard system during dynamicpositioning operation under roug sea conditions, we apply a sinusoidal oscillation to te power of Load and Load 2 wile te power of Load 3 is kept at zero. Te power of TABLE II PERCENTAGE OF TIME-AVERAGED VOLTAGE THD IN THE BUSES FOR CONSTANT LOAD CONDITIONS IN pu MPC Local ESC Local + ESC P P 2 P 3 THD [%] THD 2 [%] THD 3 [%] THD [%] THD 2 [%] THD 3 [%] THD [%] THD 2 [%] THD 3 [%] THD [%] THD 2 [%] THD 3 [%] THD [%] THD 2 [%] THD 3 [%] THD [%] THD 2 [%] THD 3 [%] Load and Load 2 oscillates between.3 pu and. pu wit a wavelengt of five seconds. Te voltage THD signals in te buses during one oscillation are presented in Fig.. Similar to Fig. 3, te THD values and te magnitude of te perturbationrelated oscillations correlate to te load power. Altoug te response of te extremum-seeking controller is too slow to effectively mitigate te armonic distortion around times.5 s and 3 s, te combined extremum-seeking control and localfiltering metod is able to better track te load canges due to a iger convergence rate, wic leads to a lower THD around tese times. We note tat te convergence rate and

8 THD [%] THD2 [%] THD3 [%] MPC Local ESC Local + ESC Time [s] Fig. 3. Percentage of voltage THD in te buses as a function of time as te values of P and P 2 jump from.3 pu to. pu at time zero wile P 3 remains zero. THD [%] THD2 [%] THD3 [%] MPC Local ESC Local + ESC Time [s] Fig.. Percentage of voltage THD in te buses as a function of time as te values of P and P 2 oscillate between.3 pu and. pu wit a wavelengt of five seconds wile P 3 remains zero. te steady-state performance including te amplitude of te oscillations due to te perturbations of te extremum-seeking control metod and te combined metod depend on te tuning of te extremum-seeking controllers. A faster convergence will generally deteriorate te steady-state performance due to te tuning trade-off discussed in [2]. V. CONCLUSION In tis work, we ave presented an extremum-seeking control metod tat optimizes te injection current of an active power filter for te system-wide minimization of armonic distortion in electrical grids of marine vessels. Te main advantage of te presented metod compared to alternative metods is tat no grid model is required. Te presented metod is computationally ceap compared to model-based system-wide armonic mitigation metods, can easily be applied to an electrical grid wit an arbitrary number of nodes, and can be implemented on top of existing metods. A case study of a tree-bus electrical grid displays tat an equally good or superior steady-state armonic mitigation can be acieved wit te presented metod compared to a modelpredictive control metod and a local-filtering metod. Te convergence rate of te extremum-seeking control metod is lower, but can be improved by combining te extremumseeking control metod and te local-filtering metod witout significant loss of steady-state performance. REFERENCES [] K. B. Ariyur and M. Krstić, Real-time optimization by extremum-seeking control. Hoboken, NJ: Wiley-Interscience, 23. [2] Y. Tan, W. H. Moase, C. Manzie, D. Nešić, and I. M. Y. Mareels, Extremum seeking from 922 to 2, in Proceedings of te 29t Cinese Control Conference, pp. 26, Beijing, Cina, July 29-3, 2. [3] J. Mindykowski, Case study-based overview of some contemporary callenges to power quality in sip systems, Inventions, vol., no. 2, [] S. G. Jayasinge, L. Meegaapola, N. Fernando, J. Z., and J. M. Guerrero, Review of sip microgrids: system arcitectures, storage tecnologies and power quality aspects, Inventions, vol. 2, no., 27. [5] T. Key and J.-S. Lai, Analysis of armonic mitigation metods for building wiring systems, IEEE Transactions on Power Systems, vol. 3, no. 3, pp. 9 97, 99. [6] G. K. Sing, Power system armonics researc: a survey, European Transactions on Electrical Power, vol. 9, no. 2, pp. 5 72, 29. [7] H. Akagi, E. H. Watanabe, and M. Aredes, Instantaneous power teory and applications to power conditioning. Hoboken, New Jersey: Jon Wiley & Sons, 27. [] M. El-Habrouk, M. K. Darwis, and P. Meta, Active power filters: a review, IEE Proceedings - Electric Power Applications, vol. 7, no. 5, pp. 3 3, 2. [9] W. M. Grady, M. J. Samotyj, and A. H. Noyola, Survey of active power line conditioning metodologies, IEEE Transactions on Power Delivery, vol. 5, no. 3, pp , 99. [] B. Sing, K. Al-Haddad, and A. Candra, A review of active filters for power quality improvement, IEEE Transactions on Industrial Electronics, vol. 6, no. 5, pp , 999. [] K. Wada, H. Fujita, and H. Akagi, Considerations of a sunt active filter based on voltage detection for installation on a long distribution feeder, IEEE Transactions on Industry Applications, vol. 3, no., pp. 23 3, 22. [2] W. M. Grady, M. J. Samotyj, and A. H. Noyola, Minimizing network armonic voltage distortion wit an active power line conditioner, IEEE Transactions on Power Delivery, vol. 6, no., pp , 99. [3] W. M. Grady, M. J. Samotyj, and A. H. Noyola, Te application of network objective functions for active minimizing te impact of voltage armonic in power systems, IEEE Transactions on Power Delivery, vol. 7, no. 3, pp , 992. [] E. Skjong, M. Ocoa-Gimenez, M. Molinas, and T. A. Joansen, Management of armonic propagation in a marine vessel by use of optimization, in IEEE Transportation Electrification Conference and Expo ITEC, pp., 25. [5] E. Skjong, M. Molinas, and T. A. Joansen, Optimized current reference generation for system-level armonic mitigation in a diesel-electric sip using non-linear model predictive control, in IEEE International Conference on Industrial Tecnology ICIT, pp , 25. [6] E. Skjong, M. Molinas, T. A. Joansen, and R. Volden, Saping te current waveform of an active filter for optimized system level armonic conditioning, in Proceedings of te st International Conference on Veicle Tecnology and Intelligent Transport Systems, pp. 9 6, 25. [7] E. Skjong, J. A. Suul, A. Rygg, M. Molinas, and T. A. Joansen, System-wide armonic mitigation in a diesel electric sip by model predictive control, IEEE Transactions on Industrial Electronics, vol. 63, no. 7, pp. 9, 26.

9 [] E. Skjong, J. A. Suul, M. Molinas, and T. A. Joansen, Optimal compensation of armonic propagation in a multi-bus microgrid, Renewable Energy and Power Quality Journal, vol., pp , 26. [9] M. Guay and D. Docain, A time-varying extremum-seeking control approac, Automatica, vol. 5, pp , 25. [2] M. Haring and T. A. Joansen, Asymptotic stability of perturbationbased extremum-seeking control for nonlinear plants, IEEE Transactions on Automatic Control, vol. 62, no. 5, pp , 27. [2] J. Mindykowski and T. Tarasiuk, Problems of power quality in te wake of sip tecnology development, Ocean Engineering, vol. 7, pp. 7, 25. [22] T. Tarasiuk, Comparative study of various metods of DFT calculation in te wake of IEC standard 6--7, IEEE Transactions on Instrumentation and Measurement, vol. 5, no., pp , 29. [23] M. Krstić and H.-H. Wang, Stability of extremum seeking feedback for general nonlinear dynamic systems, Automatica, vol. 36, no., pp , 2. [2] Y. Tan, D. Nešić, and I. M. Y. Mareels, On non-local stability properties of extremum seeking control, Automatica, vol. 2, no. 6, pp. 9 93, 26. [25] H. Yu and U. Ozguner, Extremum-seeking control strategy for ABS system wit time delay, in Proceedings of te American Control Conference, Ancorage, AK, May -, 22. [26] M. Haring, N. van de Wouw, and D. Nešić, Extremum-seeking control for nonlinear systems wit periodic steady-state outputs, Automatica, vol. 9, no. 6, pp. 3 9, 23. [27] M. Krstić, Performance improvement and limitations in extremum seeking control, Systems & Control Letters, vol. 39, no. 5, pp , 2. [2] W. H. Moase and C. Manzie, Semi-global stability analysis of observerbased extremum-seeking for Hammerstein plants, IEEE Transactions on Automatic Control, vol. 57, no. 7, pp , 22. [29] M. A. M. Haring, Extremum-seeking control: convergence improvements and asymptotic stability, P.D. dissertation, Norwegian University of Science and Tecnology, 26. [3] R. M. Jonstone and B. D. O. Anderson, Exponential convergence of recursive least squares wit exponential forgetting factor, Systems & Control Letters, vol. 2, no. 2, 92. [3] S. M. Williams and R. G. Hoft, Adaptive frequency domain control of PWM switced power line conditioner, IEEE Transactions on Power Electronics, vol. 6, no., pp , 99. Mark Haring received is BSc degree and MSc degree in mecanical engineering from Eindoven University of Tecnology, Eindoven, Te Neterlands, in 2 and 2, respectively. In 26, e obtained is PD degree in engineering cybernetics at NTNU, te Norwegian University of Science and Tecnology, Trondeim, Norway. He is currently working as a postdoctoral researcer at te Department of Engineering Cybernetics, NTNU. His researc interests include automatic control, adaptive control, and estimation. Espen Skjong received is MSc and PD degree in Engineering Cybernetics at te Norwegian University of Science and Tecnology NTNU, Trondeim, Norway, in 2 and 27, respectively. During is PD e specialized in optimal control of sipboard electrical systems, and te control of active filters to obtain optimal system-level armonic mitigation in AC grids using Model Predictive Control MPC. During is MSc e specialized in MPCs for autonomous control of Unmanned Aerial Veicles UAVs. He is currently employed in Ulstein Blue Ctrl AS Ålesund, Norway as Researc, Development and Tecnology RD&T Manager. His main researc interest is centered around optimal control and control applications for marine veicles. Tor Arne Joansen M 9, SM received te MSc degree in 99 and te PD degree in 99, bot in electrical and computer engineering, from te Norwegian University of Science and Tecnology NTNU, Trondeim, Norway. From 995 to 997, e worked at SINTEF as a researcer before e was appointed Associated Professor at NTNU in Trondeim in 997 and Professor in 2. He as publised several undred articles in te areas of control, estimation and optimization wit applications in te marine, automotive, biomedical and process industries. In 22, e co-founded te company Marine Cybernetics AS were e was Vice President until 2. Prof. Joansen received te 26 Arc T. Colwell Merit Award of te SAE, and is currently a principal researcer witin te Center of Excellence on Autonomous Marine Operations and Systems AMOS and director of te Unmanned Aerial Veicle Laboratory at NTNU. Marta Molinas M 9 received te Diploma degree in electromecanical engineering from te National University of Asuncion, Asuncion, Paraguay, in 992; te Master of Engineering degree from Ryukyu University, Japan, in 997; and te Doctor of Engineering degree from te Tokyo Institute of Tecnology, Tokyo, Japan, in 2. Se was a Guest Researcer wit te University of Padova, Padova, Italy, during 99. From 2 to 27, se was a Postdoctoral Researcer wit te Norwegian University of Science and Tecnology NTNU and from 2-2 se as been professor at te Department of Electric Power Engineering at te same university. From 2 to 29, se was a Japan Society for te Promotion of Science JSPS Researc Fellow wit te Energy Tecnology Researc Institute, National Institute of Advanced Industrial Science and Tecnology, Tsukuba, Japan. In 2, se was Visiting Professor at Columbia University and Invited Fellow by te Kingdom of Butan working wit renewable energy microgrids for developing regions. Se is currently Professor at te Department of Engineering Cybernetics, NTNU. Her researc interests include stability of power electronics systems, armonics, oscillatory penomena, and non-stationary signals from te uman and te macine. Dr. Molinas as been an AdCom Member of te IEEE Power Electronics Society. Se is Associate Editor and Reviewer for IEEE Transactions on Power Electronics and PELS Letters.