There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

(1990). ENERGY FUNCTION ANALYSIS FOR POWER SYSTEM STABILITY. Electric Machines & Power Systems: Vol. 18, No. 2, pp. 209-210.

Energy function analysis for power system stability

Historically, centrally computed algorithms have been the primary means of power system optimization and control. With increasing penetrations of distributed energy resources requiring optimization and control of power systems with many controllable devices, distributed algorithms have been the subject of significant research interest. This paper surveys the literature of distributed algorithms with applications to optimization and control of power systems. In particular, this paper reviews distributed algorithms for offline solution of optimal power flow (OPF) problems as well as online algorithms for real-time solution of OPF, optimal frequency control, optimal voltage control, and optimal wide-area control problems.

https://ieeexplore.ieee.org/ielaam/5165411/8075182/7990560-aam.pdf

A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems

Modeled after the hierarchical control architecture of power transmission systems, a layering of primary, secondary, and tertiary control has become the standard operation paradigm for islanded microgrids. Despite this superficial similarity, the control objectives in microgrids across these three layers are varied and ambitious, and they must be achieved while allowing for robust plug-and-play operation and maximal flexibility, without hierarchical decision making and time-scale separations. In this paper, we explore control strategies for these three layers and illuminate some possibly unexpected connections and dependencies among them. Building from a first-principle analysis of decentralized primary droop control, we study centralized, decentralized, and distributed architectures for secondary frequency regulation. We find that averaging-based distributed controllers using communication among the generation units offer the best combination of flexibility and performance. We further leverage these results to study constrained ac economic dispatch in a tertiary control layer. Surprisingly, we show that the minimizers of the economic dispatch problem are in one-to-one correspondence with the set of steady states reachable by droop control. In other words, the adoption of droop control is necessary and sufficient to achieve economic optimization. This equivalence results in simple guidelines to select the droop coefficients, which include the known criteria for power sharing. We illustrate the performance and robustness of our designs through simulations.

Breaking the Hierarchy: Distributed Control and Economic Optimality in Microgrids

Spot Pricing of Electricity

Real-time control of power systems using nodal prices

This technical note presents a solution to the problem of regulating a general nonlinear dynamical system to an economically optimal operating point. The system is characterized by a set of exogenous inputs as an abstraction of time-varying loads and disturbances. The economically optimal operating point is implicitly defined as a solution to a given constrained convex optimization problem, which is related to steady-state operation. The system outputs and the exogenous inputs represent respectively the decision variables and the parameters in the optimization problem. The proposed solution is based on a specific dynamic extension of the Karush-Kuhn-Tucker optimality conditions for the steady-state related optimization problem, which is conceptually related to the continuous-time Arrow-Hurwicz-Uzawa algorithm. Furthermore, it can be interpreted as a generalization of the standard output regulation problem with respect to a constant reference signal.

/pdf/on-constrained-steady-state-regulation-dynamic-kkt-50jwx457w6.pdf

On Constrained Steady-State Regulation: Dynamic KKT Controllers

During the past decade, electrical power systems have been going through some major restructuring processes. From monopolistic, highly regulated and one utility controlled operation, a system is being restructured to include many parties competing for energy production and consumption, and for provision of many of the ancillary services necessary for system operation. With the emergence of competitive markets as central operational mechanisms, the prime operational objective has shifted from a centralized, utility cost minimization objective to decentralized, profit maximization objectives of competing parties. The market-based (price-based) operation is shown to be practically the only approach that is capable to simultaneously provide incentives to hold the prices at marginal costs and to minimize the costs. As a result, such an operational structure inherently tends to maximize the social welfare of the system during its operation, and to accelerate developments and applications of new technologies. Another major change that is taking place in today’s power systems is an increasing integration of small-scale distributed generation (DG) units. Since in future power systems, a large amounts of DG will be based on renewable, intermittent energy sources, e.g. wind and sun, these systems will be characterized by significantly larger uncertainties than those of the present power systems. Power markets significantly deviate from standard economics since the demand side is largely disconnected from the market, i.e. it is not price responsive, and it exhibits uncertain, stochastic behavior. Furthermore, since electrical energy cannot be efficiently stored in large quantities, production has to meet these rapidly changing demands in real-time. In future power systems, efficient real-time power balancing schemes will become crucial and even more challenging due to the significant increase of uncertainties by large-scale integration of renewable sources. Physical and security limits on the maximal power flows in the lines of power transmission networks represent crucial system constraints, which must be satisfied to protect the integrity of the system. Creating an efficient congestion management scheme for dealing with these constraints is one of the toughest problems in the electricity market design, as the line power flows are characterized by complex dependencies on nodal power injections. Efficient congestion control has to account for those limits by adequately transforming them into market signals, i.e. into electricity prices. One of the main contributions of this thesis is the development of a novel dynamic, distributed feedback control scheme for optimal real-time update of electricity prices. The developed controller (which is called the KKT controller in the thesis) reacts on the network frequency deviation as a measure of power imbalance in the system and on measured violations of line flow limits in a transmission network. The output of the controller is a vector of nodal prices. Each producer/consumer in the system is allowed to autonomously react on the announced price by adjusting its production/consumption level to maximize its own benefit. Under the hypothesis of global asymptotic stability of the closed-loop system, the developed control scheme is proven to continuously balance the system by driving it towards the equilibrium where the transmission power flow constraints are satisfied, and where the total social welfare of the system is maximized. One of the advantageous features of the developed control scheme is that, to achieve this goal, it requires no knowledge of marginal cost/benefit functions of producers/consumers in the system (neither is it based on the estimates of those functions). The only system parameters that are explicitly included in the control law are the transmission network parameters, i.e. network topology and line impedances. Furthermore, the developed control law can be implemented in a distributed fashion. More precisely, it can be implemented through a set of nodal controllers, where one nodal controller (NC) is assigned to each node in the network. Each NC acts only on locally available information, i.e. on the measurements from the corresponding node and on the information obtained from NC’s of the adjacent nodes. The communication network graph among NC’s is therefore the same as the graph of the underlying physical network. Any change is the network topology requires only simple adjustments in NC’s that are local to the location of the change. To impose the hard constraints on the level to which the transmission network lines are overloaded during the transient periods following relatively large power imbalances in the system, a novel price-based hybrid model predictive control (MPC) scheme has been developed. The MPC control action adds corrective signals to the output of the KKT controller, i.e. to the nodal prices, and acts only when the predictions indicate that the imposed hard constraint will be violated. In any other case, output of the MPC controller is zero and only the KKT controller is active. Under certain hypothesis, recursive feasibility and asymptotic stability of the closed-loop system with the hybrid MPC controller are proven. Next contribution of this thesis is formulation of the autonomous power networks concept as a multilayered operational structure of future power systems, which allows for efficient large-scale integration of DG and smallscale consumers into power and ancillary service markets, i.e. markets for different classes of reserve capacities. An autonomous power network (AN) is an aggregation of networked producers and consumers, whose operation is coordinated/controlled with one central unit (AN market agent). By performing optimal dispatching and unit commitment services, the main goals of an AN market agent is to efficiently deploy the AN’s internal resources by its active involvement in power and ancillary service markets, and to optimally account for the local reliability needs. An autonomous power network is further defined as a major building block of power system operation, which is capable of keeping track of its contribution to the uncertainty in the overall system, and is capable of bearing the responsibility for it. With the introduction of such entities, the conditions are created that allow for the emergence of novel, competitive ancillary service market structures. More precisely, in ANs based power systems, each AN can be both producer and consumer of ancillary services, and ancillary service markets are characterized by double-sided competition, what is in contrast to today’s single-sided ancillary service markets. One of the main implications of this novel operational structure in that, by facilitating competition, it creates the strong incentive for ANs to reduce the uncertainties and to increase reliability of the system. On a more technical side, the AN concept is seen as decentralization and modularization approach for dealing with the future, large scale, complex power systems. As additional contribution of this thesis, motivated by the KKT controller for price-based real-time power balancing and congestion management, the general KKT control paradigm is presented in some detail. The developed control design procedure presents a solution to the problem of regulating a general linear time-invariant dynamical system to a time-varying economically optimal operating point. The system is characterized with a set of exogenous inputs as an abstraction of time-varying loads and disturbances. Economic optimality is defined through a constrained convex optimization problem with a set of system states as decision variables, and with the values of exogenous inputs as parameters in the optimization problem. A KKT controller belongs to a class of dynamic complementarity systems, which has been recently introduced and which has, due to its wide applicability and specific structural properties, gained a significant attention in systems and control community. The results of this thesis add to the list of applications of complementarity systems in control.

Price-based optimal control of electrical power systems

Model predictive control (MPC) is one of the few advanced control methodologies that have proven to be very successful in real-life applications. An attractive feature of MPC is its capability of explicitly taking state and input constraints into account. Recently, there has been an increasing interest in the usage of MPC schemes to control electrical power networks. The major obstacle for implementation lies in the large scale of these systems, which is prohibitive for a centralised approach. In this article, we therefore assess and compare the suitability of several non-centralised predictive control schemes for power balancing, to provide valuable insights that can contribute to the successful implementation of non-centralised MPC in the real-life electrical power system.

https://web.eecs.umich.edu/~hiskens/publications/00207179.2012.pdf

Assessment of non-centralised model predictive control techniques for electrical power networks

This paper deals with stabilization of interconnected discrete-time nonlinear systems under given, arbitrary information constraints on the controller structure. As a prominent example, the considered information constraints on the controller structure include decentralized and distributed control over a given communication network. The main contribution of this paper is twofold. Firstly, we introduce the notion of structured control Lyapunov functions (CLFs) as a suitable tool for stabilizing controller synthesis under information constraints. This includes the relaxation of Lyapunov conditions at the local level. Secondly, we present a method for constructing structured CLFs and we show that the controller synthesis problem using structured CLFs can be formulated as a convex optimization problem. Possible solutions for solving this problem efficiently under several different types of information constraints are also indicated.

/pdf/on-decentralized-stabilization-of-discrete-time-nonlinear-21loqsokp1.pdf

Andrej Jokić

Papers

Real-time control of power systems using nodal prices

On Constrained Steady-State Regulation: Dynamic KKT Controllers

Price-based optimal control of electrical power systems

Assessment of non-centralised model predictive control techniques for electrical power networks

On decentralized stabilization of discrete-time nonlinear systems