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DC grids are prone to small-signal instabilities due to the presence of tightly controlled loads trying to keep the power consumption constant over range of terminal voltage variations. Th, so-called, constant power load (CPL) represents a classical example of this destabilizing behavior acting as an incremental negative resistance. Real-life DC loads represented by controlled power converters exhibit the CPL behavior over a finite frequency range. There exist a number of methods for stability certification of DC grids which are primarily concerned with the source-load interaction and do not explicitly account for the influence of network. In the present manuscript, we develop a method for stability assessment of arbitrary DC grids by introducing the Augmented Power Dissipation and showing that it's positive definiteness is a sufficient condition for stability. We present an explicit expression for this quantity through load and network impedances and show how it could be directly used for stability certification of networks with arbitrary configuration., 10 pages

Many traditional emergency control schemes in power systems accompany with power interruption, yielding severely economic damages to customers. Aiming at alleviating this remarkable drawback, this paper sketches the ideas of a viable alternative for traditional remedial controls for power grids with high penetration of renewables, in which the renewables are integrated with synchronverters to mimic the dynamics of conventional generators. In this novel emergency control scheme, the power electronics resources are exploited to control the inertia and damping of the imitated generators in order to quickly compensate for the deviations caused by fault and thereby bound the fault-on dynamics and stabilize the power system under emergency situations. The control design is based on solving convex optimization problems tractable for large scale power grids. This emergency control not only saves investments and operating costs for modern and future power systems, but also helps to offer seamless electricity service to customers. Simple numerical simulation will be used to illustrate the concept of this paper., National Science Foundation (U.S.) (Award No. 1508666), Masdar Institute of Science and Technology, Russia (Federation). Ministry of Education and Science (grant No. 14.615.21.0001), Russia (Federation). Ministry of Education and Science (grant code: RFMEFI61514X0001)

Ad hoc electrical networks are formed by connecting power sources and loads without pre-determining the network topology. These systems are well-suited to addressing the lack of electricity in rural areas because they can be assembled and modified by non-expert users without central oversight. There are two core aspects to ad hoc system design: 1) designing source and load units such that the microgrid formed from the arbitrary interconnection of many units is always stable and 2) developing control strategies to autonomously manage the microgrid (i.e., perform power dispatch and voltage regulation) in a decentralized manner and under large uncertainty. To address these challenges we apply a number of nonlinear control techniques---including Brayton-Moser potential theory and primal-dual dynamics---to obtain conditions under which an ad hoc dc microgrid will have a suitable and asymptotically stable equilibrium point. Further, we propose a new decentralized control scheme that coordinates many sources to achieve a specified power dispatch from each. A simulated comparison to previous research is included., Comment: To be presented at the 2016 Conference on Decision and Control. Changes from v1: refined proposed control strategy, updated Lyapunov function to generalize stability analysis to broader class of networks, fixed errors and typos

Interconnected networks describe the dynamics of important systems in a wide range such as biological systems and electrical power grids. Some important features of these systems were successfully studied and understood through simplified model of linear interconnection of linear subsystems, where provably global properties, e.g. global convergence to a specific state, usually hold true. However, in severely disturbed conditions many of those systems exhibit strongly nonlinear behaviour. Particularly, multiple equilibrium points may coexist and make the dynamical behavior of the system difficult to predict. Aiming at understanding the fragility of interconnected systems, we will provide a hierarchical framework to assess the metastability and resilience of such systems. This framework is based on independently characterizing stability of individual subsystems when they are uncoupled from the network, and then enforcing the diagonal dominance property on a structure matrix capturing the subsystems stability and the input-to-output gains of interconnection network. Since the subsystems are usually of low order and the structure matrix has size equal to the number of subsystems, this framework is easy to implement and thus scalable to large scale interconnected systems. Possible application of this framework in assessing stability of microgrids will be discussed at the end of this paper.

Power systems normally operate at their stable operating conditions where the power supply and demand are balanced. In emergency situations, the operators proceed to cut a suitable amount of loads to rebalance the supply-demand and hopefully stabilize the system. This traditional emergency control scheme results in interrupted service with severely economic damages to customers. In order to provide seamless electricity service to customers, this paper proposes a viable alternative for traditional remedial controls of power grids by exploiting the plentiful transmission facilities. In particular, we consider two emergency control schemes involving adjustment of the susceptance of a number of selected transmission lines to drive either fault-on dynamics or post-fault dynamics, and thereby stabilize the system under emergency situations. The corresponding emergency control problems will be formulated and partly solved in some specific cases. Simple numerical simulation will be used to illustrate the concept of this paper., arXiv admin note: substantial text overlap with arXiv:1510.07325

Ease of miniaturization and minimal maintenance are among the advantages for replacing conventional batteries with vibratory energy harvesters in a wide of range of disciplines and applications, from wireless communication sensors to medical implants. However, the current harvesters do not extract energy from the ambient vibrations in a very efficient and robust fashion, and hence, there need to be more optimal harvesting approaches. In this paper, we introduce a generic architecture for vibration energy harvesting and delineate the key challenges in the field. Then, we formulate an optimal control problem to maximize the harvested energy. Though possessing similar structure to that of the standard LQG problem, this optimal control problem is inherently different from the LQG problem and poses theoretical challenges to control community. As the first step, we simplify it to a tractable problem of optimizing control gains for a linear system subjected to Gaussian white noise excitation, and show that this optimal problem has non-trivial optimal solutions in both time and frequency domains.

We develop a new approach for construction of convex analytically simple regions where the AC power flow equations are guaranteed to have a feasible solutions. Construction of these regions is based on efficient semidefinite programming techniques accelerated via sparsity exploiting algorithms. Resulting regions have a simple geometric shape in the space of power injections (polytope or ellipsoid) and can be efficiently used for assessment of system security in the presence of uncertainty. Efficiency and tightness of the approach is validated on a number of test networks.

We study the evolution of fast blackout cascades in the model of the Polish (transmission) power grid (2700 nodes and 3504 transmission lines). The cascade is initiated by a sufficiently severe initial contingency tripping. It propagates via sequential trippings of many more overheated lines, islanding loads and generators and eventually arriving at a fixed point with the surviving part of the system being power-flow-balanced and the rest of the system being outaged. Utilizing an improved form of the quasi-static model for cascade propagation introduced in our earlier study (Statistical Classification of Cascading Failures in Power Grids, IEEE PES GM 2011), we analyze how the severity of the cascade depends on the order of tripping overheated lines. Our main observation is that the order of tripping has a tremendous effect on the size of the resulting outage. Finding the "best" tripping, defined as causing the least damage, constitutes a difficult dynamical optimization problem, whose solution is most likely computationally infeasible. Instead, here we study performance of a number of natural heuristics, resolving the next switching decision based on the current state of the grid. Overall, we conclude that controlled intentional tripping is advantageous in the situation of a fast developing extreme emergency, as it provides significant mitigation of the resulting damage., 7 pages, 6 figures, 1 diagram, revision of earlier version, included "load shedding" into the model, main message unchanged, manuscript enhanced

Analysis of transient stability of strongly nonlinear post-fault dynamics is one of the most computationally challenging parts of Dynamic Security Assessment. This paper proposes a novel approach for assessment of transient stability of the system. The approach generalizes the idea of energy methods, and extends the concept of energy function to a more general Lyapunov Functions Family (LFF) constructed via Semi-Definite-Programming techniques. Unlike the traditional energy function and its variations, the constructed Lyapunov functions are proven to be decreasing only in a finite neighborhood of the equilibrium point. However, we show that they can still certify stability of a broader set of initial conditions in comparison to the traditional energy function in the closest-UEP method. Moreover, the certificates of stability can be constructed via a sequence of convex optimization problems that are tractable even for large scale systems. We also propose specific algorithms for adaptation of the Lyapunov functions to specific initial conditions and demonstrate the effectiveness of the approach on a number of IEEE test cases.

In this paper, we analyze the impact of communication failures on the performance of optimal distributed frequency control. We consider a consensus-based control scheme, and show that it does not converge to the optimal solution when the communication network is disconnected. We propose a new control scheme that uses the dynamics of power grid to replicate the information not received from the communication network, and prove that it achieves the optimal solution under any single communication link failure. In addition, we show that this control improves cost under multiple communication link failures. Next, we analyze the impact of discrete-time communication on the performance of distributed frequency control. In particular, we will show that the convergence time increases as the time interval between two messages increases. We propose a new algorithm that uses the dynamics of the power grid, and show through simulation that it improves the convergence time of the control scheme significantly., United States. Defense Threat Reduction Agency (Grant HDTRA1-13-1-0021), United States. Defense Threat Reduction Agency (Grant HDTRA1-14-1-0058)

High levels of penetration of distributed generation and aggressive reactive power compensation may result in the reversal of power flows in future distribution grids. The voltage stability of these operating conditions may be very different from the more traditional power consumption regime. This paper focused on demonstration of multistability phenomenon in radial distribution systems with reversed power flow, where multiple stable equilibria co-exist at the given set of parameters. The system may experience transitions between different equilibria after being subjected to disturbances such as short-term losses of distributed generation or transient faults. Convergence to an undesirable equilibrium places the system in an emergency or \textit{in extremis} state. Traditional emergency control schemes are not capable of restoring the system if it gets entrapped in one of the low voltage equilibria. Moreover, undervoltage load shedding may have a reverse action on the system and can induce voltage collapse. We propose a novel pulse emergency control strategy that restores the system to the normal state without any interruption of power delivery. The results are validated with dynamic simulations of IEEE $13$-bus feeder performed with SystemModeler software. The dynamic models can be also used for characterization of the solution branches via a novel approach so-called the admittance homotopy power flow method., Comment: 13 pages, 22 figures. IEEE Transactions on Smart Grid 2015, in press

The power flow equations are at the core of most of the computations for designing and operating electric power systems. The power flow equations are a system of multivariate nonlinear equations which relate the power injections and voltages in a power system. A plethora of methods have been devised to solve these equations, starting from Newton-based methods to homotopy continuation and other optimization-based methods. While many of these methods often efficiently find a high-voltage, stable solution due to its large basin of attraction, most of the methods struggle to find low-voltage solutions which play significant role in certain stability-related computations. While we do not claim to have exhausted the existing literature on all related methods, this tutorial paper introduces some of the recent advances in methods for solving power flow equations to the wider power systems community as well as bringing attention from the computational mathematics and optimization communities to the power systems problems. After briefly reviewing some of the traditional computational methods used to solve the power flow equations, we focus on three emerging methods: the numerical polynomial homotopy continuation method, Groebner basis techniques, and moment/sum-of-squares relaxations using semidefinite programming. In passing, we also emphasize the importance of an upper bound on the number of solutions of the power flow equations and review the current status of research in this direction., 13 pages, 2 figures. Submitted to the Tutorial Session at IEEE 2016 American Control Conference

The increasing development of the electric power grid, the largest engineered system ever, to an even more complicated and larger system requires a new generation of stability assessment methods that are computationally tractable and feasible in real-time. In this paper we first extend the recently introduced Lyapunov Functions Family (LFF) transient stability assessment approach, that has potential to reduce the computational cost on large scale power grids, to structure-preserving power grids. Then, we introduce a new geometry-based method to construct the stability region estimate of power systems. Our conceptual demonstration shows that this new method can certify stability of a broader set of initial conditions compared to the minimization-based LFF method and the energy methods (closest UEP and controlling UEP methods)., Masdar Institute of Science and Technology (Masdar Inititiative), Skolkovo Foundation (MIT/Skoltech initiative)

We study the interaction between the power grid and the communication network used for its control. We design a centralized emergency control scheme under both full and partial communication support, to improve the performance of the power grid. We use our emergency control scheme to model the impact of communication loss on the grid. We show that unlike previous models used in the literature, the loss of communication does not necessarily lead to the failure of the correspondent power nodes; i.e. the "point-wise" failure model is not appropriate. In addition, we show that the impact of communication loss is a function of several parameters such as the size and structure of the power and communication failure, as well as the operating mode of power nodes disconnected from the communication network. Our model can be used to design the dependency between the power grid and the communication network used for its control, so as to maximize the benefit in terms of intelligent control, while minimizing the risks due to loss of communication., Defense Threat Reduction Agency (DTRA) (Grant HDTRA1-13-1-0021)

We consider stochastic electro-mechanical dynamics of an overdamped power system in the vicinity of the saddle-node bifurcation associated with the loss of global stability such as voltage collapse or phase angle instability. Fluctuations of the system state vector are driven by random variations of loads and intermittent renewable generation. In the vicinity of collapse the power system experiences so-called phenomenon of critical slowing-down characterized by slowing and simultaneous amplification of the system state vector fluctuations. In generic case of a co-dimension 1 bifurcation corresponding to the threshold of instability it is possible to extract a single mode of the system state vector responsible for this phenomenon. We characterize stochastic fluctuations of the system state vector using the formal perturbative expansion over the lowest (real) eigenvalue of the system power flow Jacobian and verify the resulting expressions for correlation functions of the state vector by direct numerical simulations. We conclude that the onset of critical slowing-down is a good marker of approach to the threshold of global instability. It can be straightforwardly detected from the analysis of single-node autostructure and autocorrelation functions of system state variables and thus does not require full observability of the grid., National Science Foundation (U.S.) (Award ECCS-1128437), MIT Skoltech Initiative (Seed Funding Grant)

Dynamic response of loads has a significant effect on system stability and directly determines the stability margin of the operating point. Inherent uncertainty and natural variability of load models make the stability assessment especially difficult and may compromise the security of the system. We propose a novel mathematical "robust stability" criterion for the assessment of small-signal stability of operating points. Whenever the criterion is satisfied for a given operating point, it provides mathematical guarantees that the operating point will be stable with respect to small disturbances for any dynamic response of the loads. The criterion can be naturally used for identification of operating regions secure from the occurrence of Hopf bifurcation. Several possible applications of the criterion are discussed, most importantly the concept of Robust Stability Assessment (RSA) that could be integrated in dynamic security assessment packages and used in contingency screening and other planning and operational studies., Comment: 16 pages, 20 figures, IEEE Transactions on Power Systems 2015, in press

For a power system operating in the vicinity of the power transfer limit of its transmission system, effect of stochastic fluctuations of power loads can become critical as a sufficiently strong such fluctuation may activate voltage instability and lead to a large scale collapse of the system. Considering the effect of these stochastic fluctuations near a codimension 1 saddle-node bifurcation, we explicitly calculate the autocorrelation function of the state vector and show how its behavior explains the phenomenon of critical slowing-down often observed for power systems on the threshold of blackout. We also estimate the collapse probability/mean clearing time for the power system and construct a new indicator function signaling the proximity to a large scale collapse. The new indicator function is easy to estimate in real time using data from PMU and SCADA information about power load fluctuations on the nodes of the grid. We discuss control strategies leading to the minimization of the collapse probability., National Science Foundation (U.S.) (Award ECCS-1128437), MIT Skoltech Initiative (Seed Funding Grant)

We introduce and analyze Markov Decision Process (MDP) machines to model individual devices which are expected to participate in future demand-response markets on distribution grids. We differentiate devices into the following four types: (a) optional loads that can be shed, e.g. light dimming; (b) deferrable loads that can be delayed, e.g. dishwashers; (c) controllable loads with inertia, e.g. thermostatically-controlled loads, whose task is to maintain an auxiliary characteristic (temperature) within pre-defined margins; and (d) storage devices that can alternate between charging and generating. Our analysis of the devices seeks to find their optimal price-taking control strategy under a given stochastic model of the distribution market., 8 pages, 8 figures, submitted IEEE CDC 2011