Experimental Validation of Feedback Optimization in Power Distribution Grids

Experimental V alidation of Feedback Optimization in Po wer Distrib ution Grids Lukas Ortmann, Adrian Hauswirth, Ivo Caduf f, Florian Dörfler , Sav erio Bolognani ETH Zurich, 8092 Zurich, Switzerland Abstract —W e consider the pr oblem of contr olling the voltage of a distribution feeder using the reactive power capabilities of in verters. On a real distribution grid, we compare the local V olt/V Ar droop contr ol recommended in r ecent grid codes, a centralized dispatch based on optimal power flow (OPF) pr o- gramming, and a feedback optimization (FO) controller that we propose. The local droop control yields suboptimal regulation, as predicted analytically . The OPF-based dispatch strategy requir es an accurate grid model and measurement of all loads on the feeder in order to achieve proper voltage regulation. Howev er , in the experiment, the OPF-based strategy violates voltage constraints due to inevitable model mismatch and uncertainties. Our proposed FO controller , on the other hand, satisfies the constraints and does not requir e load measurements or any grid state estimation. The only needed model knowledge is the sensitivity of the voltages with respect to reactiv e power , which can be obtained from data. As we show , an approximation of these sensitivities is also sufficient, which makes the approach essentially model-free, easy to tune, compatible with the current sensing and control infrastructure, and remarkably rob ust to measurement noise. W e expect these pr operties to be fundamental features of FO for power systems and not specific to V olt/V Ar regulation or to distribution grids. Index T erms —autonomous optimization, distribution grid, feedback optimization, reactiv e power , voltage control. I . I N T R O D U C T IO N The shift tow ards distributed microgeneration and the change in power consumption (electric mobility , storage, flex- ible loads) poses unprecedented challenges to power distribu- tion grids. One important concern is the occurrence of over - and undervoltages in distribution feeders, which may force the distribution system operator to curtail generation or to shed loads, respectively . The flexibility of the power in verters of distributed energy resources (DERs), and more precisely their reactiv e power capabilities, can be used to av oid these extreme remedial actions. Control of reactiv e power flows is a relativ ely inexpensi ve way to regulate the feeder voltage and This research has been performed using the ERIGrid Research Infrastructure and is part of a project that has receiv ed funding from the European Union’ s Horizon 2020 Research and Inno- vation Program under the Grant Agreement No. 654113. The support of the European Research Infrastructure ERIGrid and its partner T echnical Univ ersity of Denmark is very much appreciated. This paper reflects only the authors’ view and the EU Commission is not responsible for any use that may be made of the information it contains. The research leading to this work was supported in part by the Swiss Federal Of fice of Energy grant #SI/501708 UNICORN. Corresponding author: Lukas Ortmann, email: ortmannl@ethz.ch. should therefore be fully exploited in order to avert taking action on the activ e power flo ws in the grid. Many local contr ol strate gies hav e been proposed towards this goal. In these strategies, each DER only measures the voltage at its point of connection in order to decide its own reactiv e power set-point. No communication infrastructure is needed because the controllers are fully decentralized. An example of local control strategies are static control laws like droop control with dead band and saturation, which have been included in the recent grid codes [1]–[3]. Incremental local control strategies have also been proposed, where the reactive power set-point is calculated as a function of the voltage magnitude and the past reacti ve po wer set-point [4], [5]. The main advantage of local control strategies is that they are easy to implement due to being fully decentralized. Howe ver , it was recently shown that they are suboptimal [6]. Namely , they do not necessarily regulate the v oltage to the admissible range, ev en with suf ficient reactiv e power capability of the in verters. An alternative solution to the v oltage regulation problem is to use an optimal power flo w (OPF) solver to calculate the optimal reactiv e power set-points (see [7] and references therein). This optimization-based method requires an accurate grid model and full observability of the grid state, neither of which are usually av ailable in distrib ution grids. Estimating the real-time state of a distribution grid is only possible if enough sensors are deployed which adds significant complexity and cost to this approach. A third and more promising option is feedback optimiza- tion (FO) or autonomous optimization . FO has been recently proposed as a strategy to adjust DER set-points in real-time and to drive the system to an optimal operating point without measuring or estimating the power demands [8]–[15]. T o the best of the authors’ kno wledge, there is no publicly a vailable report of testing of these solutions on a real grid, and their robustness to model mismatch and measurement noise has been conjectured but nev er verified in experiments. This paper presents an experimental verification of the effecti veness of FO for V olt/V Ar control on a simple, yet plausible testbed. The experiment shows that the grid state con verges to the optimal reactive power flow , and it allo ws to assess the performance in the presence of: • model mismatch , especially in comparison to standard OPF-based dispatch, showing that FO performs well with an extremely rudimentary model of the grid; • realistic measurement accuracy , based on off-the-shelf sensors and without any state-estimation stage. Published on Electric P ower Systems Research, vol. 189, December 2021. https://doi.org/10.1016/j.epsr .2020.106782 Additionally , the experiment illustrates the suboptimality of the local V olt/V Ar control strategies included in recent grid codes. As predicted in [6], they can be ineffecti ve and even detrimental in regulating under- and overv oltages, leading to more loads being shed or renew able generation being curtailed than necessary . The rest of the paper is structured as follows: In Section II the general concept of FO is presented and the assumptions are introduced that make the implementation more tractable. Af- terwards a FO controller is designed for the V olt/V Ar problem. The experimental setup and the controller implementation are explained in Section III and Section IV, respectively . Finally , the experimental results are presented in Section VI and the paper is concluded in Section VII. I I . F E E D B AC K O P T I M I Z A T I O N Consider the problem of determining the v alues of some set- points u (e.g, reacti ve power injections) in order to minimize a given cost function (typically a cost of the control ef fort) while satisfying some constraint on an output signal y (e.g, voltage bounds). The output y is also affected by an e xogenous uncontrollable input w (e.g, po wer demand of the loads), and depends on these inputs via a nonlinear map y = h ( u, w ) . The aforementioned decision problem is mathematically represented by the possibly non-con vex optimization problem min u f ( u ) cost of actuation effort s.t. g ( y ) ≤ 0 constraints on the output y = h ( u, w ) u ∈ U actuation bounds. (1) For a more general approach with f ( u, y ) and consideration of underlying dynamics, see [16]. One way to approach this decision problem is to solve (1) using the model y = h ( u, w ) and then apply the resulting set-points to the system in a feedforward manner . This approach comes with se veral disadvantages, such as the need for an accurate model h of the system and for full measurement or an estimate of the exogenous input w . An alternativ e approach is called feedback optimization , and is based on the assumption that the output y of the system can be measured in real-time, while the exogenous input w is unmonitored. Real-time measurements are used to iteratively adjust the set-points u , based on reduced model information, in such a way that the closed-loop system con verges to the solutions of the optimization problem (1) (hence the name). A. F eedback Optimization Principle The core idea behind FO is to exploit the measurements y instead of relying on the model y = h ( u, w ) . One way to do so is to dualize the output constraints and get the Lagrangian L ( u, λ ) = f ( u ) + λ T g ( h ( u, w )) , (2) where λ is a vector of dual variables in which each dual variable corresponds to one constraint. Instead of (1) we consider the optimization problem max λ ≥ 0 φ ( λ ) , (3) where the dual function φ ( λ ) is defined as φ ( λ ) := min u ∈U L ( u, λ ) . (4) Assuming that the feasible space of (1) has a non-empty interior , (1) and (3) hav e the same solution (Strong Duality Theorem, [17, Proposition 5.3.1]). T o solve (3) we use a gradient ascent with a fixed step size, in which the multiplier λ is repeatedly updated in the direction of steepest ascent of φ ( λ ) , while ensuring λ ≥ 0 . By introducing the element-wise projection operator [ a ] ≥ 0 := max { a, 0 } and the tuning parameter α we can write λ ( t + 1) = [ λ ( t ) + α ∇ λ φ ( λ )] ≥ 0 . (5) In [17, Proposition 6.1.1] it was shown that ∇ λ φ ( λ ) = g ( h ( u, w )) . In other words, the gradient of φ is gi ven by the violation of the dualized constraints g ( h ( u, w )) at the solution of the optimization problem (4), leading to: λ ( t + 1) = [ λ ( t ) + αg ( y ( t ))] ≥ 0 . (6) Instead of computing g ( y ( t )) based on model information, we exploit the physical system to enforce the constraint y = h ( u, w ) and measure the output y = h ( u, w ) as feedback from the plant. The v ariable λ integrates the output constraint violation with a step size of α . Note, that this corresponds to the integral part of a PI-controller . Using λ ( t + 1) , we update the set-points u with the solution of (4), i.e., u ( t + 1) = arg min u ∈U L ( u, λ ( t + 1)) = arg min u ∈U f ( u ) + λ ( t + 1) T g ( h ( u, w )) . (7) Whether this optimization problem is easier to solve than the original one in (1) is not apparent at this point. In the next subsection we will see ho w , under mild assumptions, this optimization problem admits an approximation which is numerically very tractable. T o summarize, the FO controller is realized by running the following algorithm at ev ery time t = 0 , 1 , . . . Algorithm 1 Feedback optimization controller 1: Measure the system output y ( t ) 2: Calculate λ ( t + 1) as in (6) 3: Solve the optimization problem in (7) 4: Apply the calculated set-points u ( t + 1) to the system See Figure 1 for a block diagram of a FO controller for the V olt/V Ar problem, that we deriv e in Section II-C. B. Practical F eedback Optimization Design W e now make two assumptions that are not necessary , but make the FO controller numerically more tractable. First, we assume the cost is a quadratic function f ( u ) = 1 2 u T M u with M being square, symmetric and positive semidefinite. Second, we make the mild assumption that the constraints on the input and output are linear . W e therefore get U = { u | C u ≤ d } and g ( y ) = Ay − b . Linearity of the constraints is often given, as in many cases the limits are upper and lower bounds of the form u min ≤ u ≤ u max . This leads to (1) taking the form min u 1 2 u T M u quadratic cost of actuation s.t. Ay ≤ b linear constraints on the output y = h ( u, w ) C u ≤ d linear actuation bounds. (8) Notice that the output is still a possibly nonlinear and non- con vex function of the input y = h ( u ) . The dual update (6) for the special case (8) of (1) takes the form: λ ( t + 1) = [ λ ( t ) + α ( Ay ( t ) − b )] ≥ 0 . (9) Howe ver , the major advantage of (8) over (1), lies in the ev aluation of (7) which can now be explicitly solved. There are se veral ways to solve (7). W e choose to do this in two steps that we feel are easy to understand. First, we ignore the constraint u ∈ U and calculate the critical point u for which ∇ u L ( u, λ ( t + 1)) = 0 (first order optimality condition). Then, we project this unconstrained critical point onto U . The deriv ativ e of the Lagrangian L ( u, λ ( t + 1)) is ∇ u L ( u, λ ( t + 1)) = ∇ u f ( u ) + ∇ u  λ T ( t + 1) g ( h ( u ))  = M u + ∂ h ( u, w ) ∂ u T A T λ ( t + 1) . (10) The factor ∂ h ( u,w ) ∂ u is the sensitivity of the output y with respect to the input u . This sensitivity is in general dependent on u and w , but in many practical applications can be approxi- mated by a constant matrix H . Furthermore, the approximation error will be compensated by the feedback nature of this scheme. The theoretical analysis of this robustness remains an open question, and is one of the main motiv ations for the experimental validation reported in this paper . Under this modeling assumption we hav e ∇ u L ( u, λ ( t + 1)) ≈ M u + H T A T λ ( t + 1) , (11) and a critical point of L ( u, λ ( t + 1)) in u is approximated by u unc := − M − 1 H T A T λ ( t + 1) . (12) This is the unconstrained critical point. The solution to the constrained case is obtained by projecting u unc onto the set of feasible control inputs U , that is u ( t + 1) = arg min u ∈U k u − u unc k 2 M = arg min u ∈U ( u − u unc ) T M ( u − u unc ) . (13) The feasible set U is known and described by linear inequality constraints. Therefore, this minimization is a simple conv ex quadratic program. Notice how both the unconstrained and the constrained solution do not depend on the unmeasured exogenous input w , as desired. C. F eedback Optimization for V olt/V Ar Regulation In this section we specialize FO to the V olt/V Ar regulation problem. This problem is defined as follows: Determine the reactiv e power q h at every DER h such that q min ≤ q h ≤ q max and that v min ≤ v h ( q , w ) ≤ v max . Here, v h ( q , w ) is the steady state map of the nonlinear power flow equations that defines voltages v h as a function of both reactiv e powers q h and external influences w (e.g., acti ve and reactiv e demands, activ e generation). Mathematically speaking, we try to solve a feasibility problem: q ∈ F F := { q | q min ≤ q ≤ q max , v min ≤ v ( q , w ) ≤ v max } , where q and v are the vectors of reactiv e power set-points and voltage magnitudes that we obtain by stacking the individual q h and v h of the DERs, respecti vely . W e choose not to control activ e power with our algorithm. Due to the different cost of the two control actions one should first utilize reactiv e power and only afterwards use active power to control the voltage. Therefore, these two control actions can be applied individually and do not need a unified control approach. How- ev er, activ e power could easily be included in the controller without adding technical difficulties. In order to apply the proposed methodology , we cast this feasibility problem into the optimization problem min q 1 2 q T M q s.t. v min ≤ v h ( q , w ) ≤ v max ∀ h q min ≤ q h ≤ q max ∀ h. (14) This is a special case of (8), where M can be used to weight the reactiv e power contribution of the different in verters h . W e introduce the dual v ariables λ min and λ max corresponding to the voltage (output) constraints. W e adapt (9) to this specific case (namely , A =  − I I  , b = [ v min − v max ] ) and we get λ min ( t + 1) = [ λ min ( t ) + α ( v min − v )] ≥ 0 (15) λ max ( t + 1) = [ λ max ( t ) + α ( v − v max )] ≥ 0 . (16) As we can see, we are integrating the voltage violations, which can be measured, with a gain of α . As discussed before, in order to calculate (12), we need a constant approximation of the sensitivity of the voltages with respect to the reacti ve po wer injection akin to power transfer distribution factors for activ e power generation on the trans- mission level. Under no-load conditions and the assumption of negligible cable resistances we have the approximation ∂ v ( q , w ) ∂ q = X , (17) where X is the reduced bus reactance matrix that can be deriv ed from the grid topology and the data in T able I. The approximation is accurate for lightly loaded systems, because the nonlinearity of the power flow equations is mild near this operating point [18]. In our application the system can be heavily loaded, but in Section VI we verify that the proposed FO is sufficiently robust against this model mismatch. In verters Distribution Grid q ( t + 1) V oltage Magnitude Measurements Con troller Ph ysics λ min ( t + 1) = [ λ min ( t ) + α ( v min − v ( t ))] ≥ 0 λ max ( t + 1) = [ λ max ( t ) + α ( v ( t ) − v max )] ≥ 0 λ ( t + 1) Reactiv e Po wer Set-Poin ts v ( t ) External Influences q unc = M − 1 X T ( λ min ( t + 1) − λ max ( t + 1)) q ( t + 1) = arg min q ∈Q ( q − q unc ) T M ( q − q unc ) w ( t ) Fig. 1: Block diagram of the controller with (15) and (16) (left block) and (18) and (19) (right block). The controller gets the voltage magnitude measurements from the in verters and determines the reactiv e power set-points, which are send to the inv erters. The parameter α is the controller gain and is the only tuning knob . Note, that the left block corresponds to the integral part of a PI-controller . The expression in (12) for the optimal unconstrained reac- tiv e power set-points q unc becomes q unc = M − 1 X T ( λ min ( t + 1) − λ max ( t + 1)) , (18) while the solution of the constrained optimization problem (13) becomes q ( t + 1) = arg min q ∈Q ( q − q unc ) T M ( q − q unc ) , (19) where Q = { q | q min ≤ q ≤ q max } . In practice, these reactive po wer set-points q ( t + 1) are to be communicated to the different DERs, which will adjust their reactiv e power accordingly and collect the measurement of the consequent steady state voltage magnitudes, which need to be communicated to the central control unit. Therefore, at every time step the measurement and set-point need to be communicated by and to ev ery in verter , respectively . The resulting centralized controller is represented in Figure 1 and consists of equations (15) and (16) (left block in the figure) and (18) and (19) (right block in the figure). W e can see that the FO controller uses the same mea- surements as local controllers, but these measurement are processed by a central unit which coordinates the actions of the dif ferent DERs and steers the system to the optimal steady state. In comparison to the OPF-based dispatch, no nonlinear model nor knowledge of the power consumption or generation (modelled as external influences w ) is needed. I I I . E X P E R I M E N TA L S E T U P The experiment has been implemented in the SYSLAB distribution grid at DTU Risø, Denmark. A small yet realistic distribution feeder has been configured in order to observe an overv oltage condition caused by local generation. The same setup was used in [19] to analyze a distributed FO controller for the V olt/V Ar problem. W ithout proper reactiv e power control, the feeder’ s ability to host renewable energy injections is limited and generation has to be curtailed. This scenario was chosen because it constitutes a non-tri vial voltage regulation problem which cannot be solved without a coordi- nated V olt/V Ar control strategy , as will be demonstrated in Section VI-A1. Note, that the applicability of the proposed FO strategy is not limited to the chosen topology . PCC v 1 v 2 v 3 R 1 , L 1 R 2 , L 2 R 3 , L 3 p 1 , q 1 p 2 , q 2 p 3 , q 3 PV1 PV2 Battery ± 8 kV Ar Static load ± 6 kV Ar ± 6 kV Ar 0 kV Ar 10 kW 0 kW 0 kW − 15 kW V oltage [p.u.] 1 0.99 1.06 1.05 0.95 Fig. 2: Sketch of the voltage profile and the distribution feeder . The colors of the voltage profile match the colors of the sketched feeder . The setup consists of a vanadium battery , two photovoltaic systems (PV), a resistiv e load, and the distribution substation (PCC) connecting the distribution feeder to the remaining grid, see Figure 2. The dif ferent nodes are connected via cables with non-negligible resistance (T able I). The cable connecting the battery to the grid has a particularly large resistance. The activ e power injection p 3 of the battery can represent a renew able source, which should not be curtailed. In our experiments we choose the acti ve power of the battery to be p 3 = 10 kW . The high cable resistance and acti ve power in- jection deteriorates the approximation of the sensiti vity matrix in (17). In Section VI-A3 we will show that the FO controller can cope with the model mismatch. The static load is set to an activ e po wer consumption of 15 kW ( p 1 = − 15 kW) which is larger than the local production, therefore requiring a positiv e activ e power flow from the substation. PVs are fluctuating power sources. Therefore, to facilitate repeatability of the experiments and to allow for a comparison between different controllers, the PVs do not inject activ e power ( p 2 = 0kW). The resulting voltage profile with no reactive power flo ws is represented in Figure 2, where the overv oltage at the end of the feeder is apparent. Both the PVs and the battery can measure their voltage mag- nitudes, and their reactiv e power injections can be controlled. T ABLE I: Overvie w of the resistances and inductances in the grid. R 1 [Ω] L 1 [Ω] R 2 [Ω] L 2 [Ω] R 3 [Ω] L 3 [Ω] 0.195 0.124 0.11 0.027 0.97 0.093 The PV in verters hav e a reactiv e power range of ± 6 kV Ar and the battery can be actuated with ± 8 kV Ar . The inv erters at SYSLAB are ov ersized such that their full reactiv e po wer range is av ailable independently of their concurrent active power injection. The PVs and the battery can communicate with a central computational unit via a general-purpose Ether- net network, while the load is uncontrolled and unmeasured. The voltage limits are defined to be 0.95 p.u. and 1.05 p.u. W e set these limits tighter than most grid codes in order to be able to observe persistent ov ervoltages without hardware protections being activ ated. I V . C O N T R O L L E R I M P L E M E N T A T I O N The FO controller is implemented in Matlab at a central computation unit (Figure 1), where it is provided with the voltage magnitude measurements from the different inv erters and computes the reactiv e po wer set-points. These are send to the in verters e very 10 seconds, because the PV systems in the laboratory were not to be actuated more frequently , due to special hardware constraints. In general, the controller can run more frequently . A. Contr oller T uning The controller has one tuning parameters which is the scalar control gain α in (15) and (16). The higher its value, the faster a voltage constraint violation is integrated and the faster the DERs’ reactive power set-points counteract the violation. Ho wev er , as known from the optimization literature the stability of the gradient ascent we perform in (6) is lost if α is chosen too large (see [17, Proposition 1.2.3]). B. Anti-W indup If the activ e power injections are too high (ov ervoltage) or too low (undervoltage) there do not exist feasible reactiv e power injections that lead to voltages which are inside the allowed voltage band. Therefore, the V olt/V Ar problem is infeasible and at least one voltage violation is persistent. In this case the dual variable ( λ min or λ max ) corresponding to the violated constraint keeps integrating, yielding a windup of this variable. W e implemented the follo wing simple anti-windup solution in which the integration of the constraint violation is inhibited if all DERs are saturated: λ h, min ( t +1) =      λ h, min ( t ) if v min − v h ( t ) > 0 and q k = q k, max ∀ k λ h, min ( t ) + α ( v min − v h ( t )) otherwise, λ h, max ( t +1) =      λ h, max ( t ) if v h ( t ) − v max > 0 and q k = q k, min ∀ k λ h, max ( t ) + α ( v h ( t ) − v max ) otherwise. Furthermore, an active power curtailment could be triggered once all DERs are saturated. V . B E N C H M A R K C O N T RO L L E R S W e implement a local droop controller and an OPF-based dispatch as two benchmark solutions to compare with the pro- posed FO strate gy . These approaches ha ve almost opposite fea- tures: The droop controller only needs local voltage magnitude measurements, no communication, and no model of the grid; the OPF-based dispatch is centralized, requires communication of full state measurements (all power generation and demand), and relies on an accurate nonlinear grid model. A. Dr oop Contr ol The droop controller that we implement complies with the recommendations by recent grid codes [1]–[3]. Every DER measures the magnitude of the voltage at their point of connection and absorbs/injects reacti ve po wer following the piecewise linear control law q h =                    q h, max v h < v 1 q h, max v 2 − v h v 2 − v 1 v 1 ≤ v h ≤ v 2 0 v 2 ≤ v h ≤ v 3 q h, min v h − v 3 v 4 − v 3 v 3 ≤ v h ≤ v 4 q h, min v 4 < v h . v h q h q h, max q h, min v 1 v 4 v 3 v ref v 2 Based on the voltage band specifications of our experiment, we tune the droop curve to v 1 = 0 . 95 p.u., v 2 = 0 . 99 p.u., v 3 = 1 . 01 p.u. and v 4 = 1 . 05 p.u.. B. OPF-based Dispatch W e implement an OPF-based dispatch by communicating all reactive and active po wer consumption and generation to a centralized computation unit. There, we solve (14) using the OPF solver provided by Matpower [20], which we provide with a nonlinear grid model that we obtain from the grid topology and the data from T able I. The reactiv e power set- points which are the solution of (14) are then giv en to the in verters. This approach guarantees optimality of the set-points under perfect model knowledge, but all po wer generation and consumption needs to be measured or estimated. This information is a vailable at SYSLAB with a significant lev el of accuracy . In most distribution grids, the cable data and grid topology are not known exactly , nor are all reactive and acti ve power consumption and generation measurements av ailable. V I . E X P E R I M E N TA L R E S U L T S In the follo wing experiment, we analyze two crucial fea- tures: the tracking performance when solving a time-varying voltage regulation problem, and the robustness against model uncertainty . W e also contrast the proposed FO strategy with the local droop controller and the OPF-based dispatch. A. T racking P erformance W e repeat the following 21-minute experiment for the three aforementioned strategies: droop control, OPF-based dispatch, 0 . 97 1 1 . 03 1 . 05 1 . 07 − 5 0 5 Battery 0 5 10 15 20 0 . 97 0 . 98 0 . 99 1 T ime [min] 0 5 10 15 20 − 1 0 1 2 PV1 0 . 97 0 . 98 0 . 99 1 V oltage [p.u.] 0 1 2 PV2 Reactiv e Po wer [kV Ar] Fig. 3: Performance of the Droop Control. and FO. All power inv erters are initialized with zero reactive power injection. 1 After three minutes the controllers are acti vated and start regulating the voltage. After 11 minutes the activ e power in- jection of the battery is reduced to 0 kW (effecti vely removing the cause of the overv oltage and the need for reacti ve power regulation). At minute 14 the acti ve power injection is stepped up again to 10 kW for the remaining seven minutes of the experiment. 1) Dr oop Contr ol: The performance of the droop controller can be seen in Figure 3. Once the controller is activ ated the reactiv e po wer of the battery drops to its lower limit which reduces the overvoltage. Howe ver , the limited reactiv e power capability of the battery cannot drive the voltage into the desired voltage range. The PV systems do not absorb reacti ve power to help reduce the ov ervoltage because they do not sense an ov ervoltage condition at their point of connection, and they will not lower their voltage below the nominal value of 1 p.u. Using a lower nominal voltage is also not possible as it will increase the occurrence of undervoltage ev ents. This behavior is general for all local control strategies, and cannot be prev ented without introducing some form of coordination between the inv erters. Local control strategies are therefore inherently suboptimal; as established from a theoretical perspectiv e in [6]. During minutes three to fiv e, PV1 ev en injects reactiv e power to increase its voltage, because it has fallen under its deadband voltage of 0.99 p.u. This worsens the overv oltage at the battery , showing that droop control can ev en be detrimen- tal. 2) OPF-based Dispatch: An OPF-based strategy guaran- tees optimality under perfect model knowledge. This is a strong requirement which cannot be met in practice. Even in the SYSLAB distribution grid, where the setup, the cables and their parameters are accurately known, the OPF solution does not lead to feasible voltages (see the persistent voltage 1 The plots show that the battery is injecting a small amount of reactiv e power at the beginning of the experiments. This is due to a measurement error . An inaccurate sensor is used for the internal reactive po wer controller of the battery , and a small tracking error is therefore present. The reported measurements in the figures are accurate. 0 . 97 1 1 . 03 1 . 05 1 . 07 − 5 0 5 Battery 0 5 10 15 20 0 . 97 0 . 98 0 . 99 1 T ime [min] 0 5 10 15 20 − 2 − 1 0 PV1 0 . 97 0 . 98 0 . 99 1 V oltage [p.u.] − 2 − 1 0 PV2 Reactiv e Po wer [kV Ar] Fig. 4: Performance of the OPF-based dispatch. violation in Figure 4). Standard techniques such as disturbance observers, model adaptation, and state estimation could be used to alle viate the ef fect of model uncertainty . Also, robust optimization techniques could be used to solve the OPF prob- lem. Nevertheless, an OPF-based dispatch requires a nonlinear grid model and knowledge of all acti ve and reacti ve po wer consumption and production on the feeder . 3) F eedback Optimization: The control gain α of the FO controller is chosen to be 100, and the matrix X was calculated using the data from T able I. The weighting matrix of the optimization problem M is a diagonal matrix with the entries being the in verse of the reactiv e power limits ( q − 1 max ): X =   0 . 10 0 . 09 0 . 09 0 . 09 0 . 11 0 . 11 0 . 09 0 . 11 0 . 16   , M =   1 / 6 0 0 0 1 / 6 0 0 0 1 / 8   . The control performance can be seen in Figure 5. When the controller is activ ated the central unit is provided with the voltages at the PV systems and the battery . The dual variable λ max,3 that corresponds to the violation of the upper voltage limit of the battery starts integrating the violation. This then leads to all inv erters reducing their reactiv e power injections. As long as there is an overv oltage the dual v ariable keeps integrating, which leads to the in verter absorbing more reacti ve power which lowers the voltage. At steady state the voltage at the battery is at the upper voltage limit and the reactive power injections are at the optimal solution of (14). The temporal constraint violation before the system con- ver ges to the feasible v oltage band can be made shorter by using a faster sampling time. Furthermore, the power system is equipped to withstand short ov ervoltages. B. Robustness to Model Mismatch Due to its feedback nature, the proposed FO approach is expected to be robust to model mismatch. Howe ver , in spite of recent theoretical insights [21], the robustness of these strategies has not been analyzed experimentally before. In order to validate this claim in an experiment, we assume uncertainty in the knowledge of the grid sensitivity matrix X . W e consider the crude approximation in which all entries of the X matrix are believed to be 1 . This choice corresponds to 0 . 97 1 1 . 03 1 . 05 1 . 07 − 5 0 5 Battery 0 5 10 15 20 0 . 97 0 . 98 0 . 99 1 T ime [min] 0 5 10 15 20 − 6 − 4 − 2 0 PV1 0 . 97 0 . 98 0 . 99 1 V oltage [p.u.] − 4 − 2 0 PV2 Reactiv e Po wer [kV Ar] Fig. 5: Performance of the FO controller . 0 . 97 1 1 . 03 1 . 05 1 . 07 − 5 0 5 Battery 0 5 10 15 20 0 . 97 0 . 98 0 . 99 1 T ime [min] 0 5 10 15 20 − 6 − 4 − 2 0 PV1 0 . 97 0 . 98 0 . 99 1 V oltage [p.u.] − 4 − 2 0 PV2 Reactiv e Po wer [kV Ar] Fig. 6: Performance of the FO controller with no model information. assuming that all in verters are connected to the same point on the feeder . No other model information is used, making the controller design essentially model-free. The behavior of this FO controller with α = 10 can be seen in Figure 6. Notice, that the controller is still able to driv e the voltages to the feasible voltage band. The DERs are utilized differently than in Figure 5 due to the different X matrix. This leads to a different v alue of the cost function, which is within 12% of the optimal value. V I I . C O N C L U S I O N W e hav e implemented three V olt/V Ar control strategies on a real distribution feeder: local droop control, centralized OPF- based dispatch that guarantees optimal regulation under perfect model information, and a recently proposed FO scheme. While the droop control fails to regulate voltages in a satisfactory manner (as predicted analytically), the OPF-based dispatch exhibits substantial fragility with respect to model uncertainty . In contrast, the FO strategy drives the system to the feasible voltage range while relying only on voltage measurements collected from the inv erters (without measuring or estimating any power flows). Within our experimental setup, feedback optimization is extremely robust to model mismatch and its design and tuning is essentially model-free. This leads us to conclude that feedback optimization is a promising approach for the real-time coordinated control of DERs in future distribution grids. W e conjecture that these features of feedback optimization are not specific to this application and we plan to in vestigate them in the more general context of real-time control of power systems. V I I I . A C K NO W L E D G M E N T W e thank Alexander Maria Prostejovsky and Kai Heussen for their support during the implementation of the experiments at DTU and for many useful inputs and fruitful discussions. R E F E R E N C E S [1] IEEE 1547-2018, “Standard for interconnection and interoperability of distributed energy resources with associated electric po wer systems interfaces. ” [2] “VDE-AR-N 4105: generators connected to the L V distribution network - technical requirements for the connection to and parallel operation with lo w-voltage distribution networks, ” 2018. 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