### Adaptive Voltage Control of DFIG in Weak Grids

#### Following a Quadratic Reference of Unknown Parameters

Wind turbines typically convert the wind energy into electric energy using induction generators. At every wind speed, the energy captured from the wind depends on the speed by which the blades rotate. Therefore, in order to capture the maximum available wind energy at different wind speeds, the rotation speed must be appropriately matched to the wind speed. This can be accomplished by regulating the electric power generated by the induction generator of the wind turbine. Earlier wind turbine technology did not provide this capability, which made the wind turbines optimal only at a certain wind speed. A more advanced technology, called doubly-fed induction generator (DFIG), does provide this capability. The wind turbine generators consist of a rotor that is mechanically connected to the blades through a gearbox, and a stator that is electrically connected to the electric grid. Power regulation in DFIG is achieved by controlling the voltage applied on the rotor circuits, whereas before these circuits were short-circuited such that no voltage was applied. For a given rotation speed, the power generated at the stator in steady state is a product of the voltage applied on the rotor and the voltage applied on the stator. The voltage applied on the stator is a function of the properties of the transmission line connecting the wind turbine to the main grid, the electric current produced by the wind turbine that flows through this line, and the voltage at the other end of the transmission which is usually kept constant by the large power plants generators. With a low-impedance transmission line, as a result of close proximity of the wind turbine to the main grid or when a high-capacity transmission line is installed, the stator voltage deviates very little from its nominal value as a function of the current. In this case the power generated by the wind turbine becomes almost a linear function of the voltage applied on the rotor. The widely used proportional-integrator (PI) controller is then able to regulate the generated power to the desired value. However, areas with major wind resources can be found very far from population centers, requiring the installation of long transmission lines to connect the wind turbines to the main grid. With such high-impedance lines, the voltage deviates more significantly as a function of the electric current, creating what is referred to as weak-grid conditions. This voltage variability creates non linearity that must be taken into account.

We formulate a general control problem in which the output signal to be regulated is a nonlinear function of the state. We assume the parameters of this nonlinear function are unknown, but that both the state and the output signal can be measured. We then design a new controller that simultaneously estimates the parameters of the nonlinear function and drive the state such that the output signal is regulated as desired. Even though estimation and control are in general two contradicting requirements, using a new adaptive law and a multilevel controller we are able to prove that the tracking error converges to zero in the absence of measurement noise. In the presence of bounded noise we show that the tracking error can be driven to a neighborhood around the origin. We apply this controller to DFIG control in weak-grid conditions and present its advantage over a PI-controller.

- Y. Sharon, A. Annaswamy, A. Motto, and A. Chakraborty,

“Adaptive Control for Regulation of a Quadratic Function of the State,”*IEEE Transactions on Automatic Control*, vol. 59(10), pp. 2831 - 2836, 2014 - Y. Sharon, A. M. Annaswamy, and A. L. Motto,

“Network-Constrained Adaptive Control with a Nonlinear Tracking Function,”

in*Proc. 6th Int. Conf. on Network Games, Control and Optimization*, 2012

### Topology Identification in Distribution Network

#### Limited Measurements with Stochastic Information

The electric grid consists of high-voltage transmission networks and low-voltage distribution networks. Transmission networks carry the electric power from the major power plants to the load centers. Distribution networks then distribute this power to the residential and commercial consumers. Because so many depend on the power flowing through every node of a transmission network, these networks are highly monitored and controlled remotely from a control center. Distribution networks, in contrast, are scarcely monitored and are managed manually by sending crews to reconfigure them. Until now, and despite having outages that sometimes take hours to identify and restore, there has been little demand to change the way distribution networks are operated. A major reason for this is that until now power flow in distribution networks always flowed one way from the substation, the point of connection to the transmission network, to the consumers. With the expected proliferation of distributed energy resources (DER), based mainly on wind and solar energy, we expect this to change. As more DERs are introduced, faults and subsequent line disconnections will affect not only those downstream but also those upstream. These networks, traditionally operated radially, may need to start to be operated with topologies that include loops in order to minimize power losses. All of these necessitate operating distribution networks more like transmission networks. However, the fundamental economics of distribution networks remain, and prevent equipping distribution networks with redundant real-time measuring capabilities covering every node in the network. Recognizing that the full state estimate required for transmission networks may not be required to perform the new tasks required in distribution networks, we ask whether we can add a minimal number of real-time sensors in order to provide just the information needed for these tasks.

We concentrate on the problem of circuit breakers status detection in non-radial distribution networks. We ask whether we can detect with high probability the correct status in real time, but without equipping every breaker with a sensor and a transmitter. We follow a state estimation approach, but since we do not have sufficient real-time data for state estimation, we complete the missing information using historical data. Because predicting the present conditions from historical data is probabilistic in nature, this leads to a machine learning problem. We compare two tools in machine learning, namely the maximum likelihood (ML) and support vector machine (SVM), and find the former to perform better. We provide a computationally efficient method to predict the success rate given the topology of the network, the location of the circuit breakers, and the placement of the few real-time sensors.

- Y. Sharon, A. M. Annaswamy, A. L. Motto and A. Chakraborty,

“Topology identification in distribution network with limited measurements,”

in*Proc. 2012 IEEE PES Innovative Smart Grid Technologies Conference*, 2012

### Stabilizing Controller with Coarse Quantization

Quantization in control systems is receiving a great deal of attention particularly since the late 90’s. The driving force is the need to operate control systems over networks with limited bandwidth communication between the plant, or the sensors, and the controller. Quantization is the mapping of a continuous-time real signal to a discrete-time signal, with a finite resolution and limited range. Such a discrete-time signal can be transmitted using a finite number of bits per second, denoted as the data-rate. Earlier works established the minimum data-rate that is required to stabilize an open-loop unstable undisturbed plant. We show that the same minimum data-rate is also required to stabilize an unstable plant in the presence of an unknown external disturbance. We design a controller that stabilizes the disturbed plant, making very few assumptions on the structure of the quantizer. We then show that the response of the closed-loop system depends of the magnitude of the disturbance, using the nonlinear input-to-state stability property.

In the series of papers listed below, the first paper is the most comprehensive and contains most of our results. The second paper, building on the theorems established in the first paper, extends the results to time-delay systems. The third paper focuses on quantized output feedback systems. It includes additional state estimators using quantized output feedback that are not discussed in the first. The fourth paper is the earliest conference publication of our results.

- Y. Sharon and D. Liberzon,

“Input-to-state stabilizing controller for systems with coarse quantization,”*IEEE Transactions on Automatic Control*, vol. 57(4), pp. 830-844, 2012 - Y. Sharon and D. Liberzon,

“Stabilization of linear systems under coarse quantization and time delays,”

in*Proc. 2nd IFAC Workshop on Distributed Estimation and Control in Networked Systems*, 2010 - Y. Sharon and D. Liberzon,

“Input-to-state stabilization with quantized output feedback,”

in*Proc. 11th International Workshop on Hybrid Systems: Computation and Control*,

*Lecture Notes in Computer Science*, vol. 4981, pp. 500-513, 2008 - Y. Sharon and D. Liberzon,

“Input-to-state stabilization with minimum number of quantization regions,”

in*Proc. 46th IEEE Conference on Decision and Control*, pp. 20-25, 2007

### Adaptive Control with Quantized Measurements

#### Vision-based Landing Control

Human pilots are capable of landing airplanes using nothing more than their vision and an air speed indicator for feedback. Automatic pilots traditionally require an inertial measurement unit to measure their orientation and an external signal to measure their location with respect to the runway. Here we design an automatic landing controller that uses only a front, fixed-mounted, limited-resolution camera for feedback. We assume the air speed is controlled by another controller, not considered here. Extracting the present orientation from the video sequence is usually done through a structure-from-motion algorithm. However, that requires knowing the parameters of the motion dynamics. We assume these parameters, which depend on the type of airplane and environmental conditions, are unknown. We therefore employ an algorithm where we simultaneously estimate both the parameters and the current orientation of the airplane, and prove its convergence using tools from convex and non-smooth analysis. This algorithm is designed specifically for quantized measurements, the result of using a limited-resolution camera. The estimated parameters and the current orientation are then used to compute, based on finite horizon optimization, the next control that will guide the airplane to follow the glide slope.

- Y. Sharon, D. Liberzon and Y. Ma,

“Adaptive control using quantized measurements with application to vision-only landing control,”

in*Proc. 49th IEEE Conference on Decision and Control*, pp. 2511-2516, 2010

### Robust Estimation

Estimating an unknown state from a redundant set of noisy measurements is a common problem in many engineering applications. The most widely used solution to this problem is the least-square fitting. Under certain assumptions on the measurement noise distribution, this method minimizes the estimation error between the true unknown state and its estimated value. When these assumptions fail, however, the least-square fitting is prone to unbounded estimation error due to the corruption of just a few measurements. A known alternative is to use the robust least-absolute-value (LAV) fitting. It is a generalization of the median estimator, which as opposed to the average estimator is able to cancel the influence of a few extreme measurements. With the median estimator, which is only defined for a scalar state, up to almost 50% of the measurements can be arbitrarily corrupted before the estimation error becomes unbounded. That is, the breakdown point of the median, or the LAV applied to a scalar state, is 50%. While the LAV estimator had been used in numerous applications for estimating a multidimensional state, in general the breakdown point was not computed.

Building on results from compressive sensing we reestablish the robustness of the LAV estimator by providing a new efficient algorithm for computing the breakdown point when using the LAV to estimate a multidimensional variable. We also establish its stability for the first time by deriving a sharp bound on the estimation error which is a function of the measurement noise of the uncorrupted measurements and the number of corrupted measurements. Naturally this bound grows as the number of corrupted measurements increases, but for any given number of corrupted measurements less than the breakdown it only depends on the uncorrupted measurements. As a motivational example we find the location and orientation of a vehicle using inertial and GPS measurements, where some of the GPS measurements may be corrupted. We then show the significantly improved performance of the LAV estimator compared to that of the Kalman filter.

- Y. Sharon, J. Wright and Y. Ma,

“Minimum sum of distances estimator: Robustness and stability,”

in*Proc. 2009 American Control Conference*, pp. 524-530, 2009

### Stability of Switched Systems

We consider an affine control system whose vector fields span a third-order nilpotent Lie algebra. We show that the reachable set at time T using measurable controls is equivalent to the reachable set at time T using piecewise-constant controls with no more than four switches. The bound on the number of switches is uniform over any final time T. As a corollary, we derive a new sufficient condition for stability of nonlinear switched systems under arbitrary switching. This provides a partial solution to an open problem posed by D. Liberzon

- Y. Sharon and M. Margaliot,

“Third-order nilpotency, nice reachability and asymptotic stability,”*Journal of Differential Equations*, vol. 233, pp. 136-150, 2007 - Y. Sharon and M. Margaliot,

“Third-order nilpotency, finite switchings and asymptotic stability,”

in*Proc. 44th IEEE Conference in Decision and Control, CDC-ECC '05*, pp. 5415-5420, 2005