Alexey B. Iskakov

Gramians Method of Steady-State Stability Analysis for Large Electrical Power Systems

Abstract This paper provides a new approach to computing infinite and finite time Gramians and cross-Gramians relying on the usage of the Laplacian transformation of matrix exponential time functions and expansion of the product of these functions. The expansions are bilinear and quadratic forms of sequences of the Faddeev matrices generated by resolvents of original matrices. Matrix identities are obtained for bilinear and quadratic forms of these Faddeev sequences. Asymptotic expansions of controllability and observability Gramians of dynamic systems at the proximity of stability limit are found. Illustrative examples are given to demonstrate the perspective of using Gramians for the small-signal stability analysis of the electric power systems.

Characterization of power systems near their stability boundary by Lyapunov direct method

Abstract We propose a new method for the small-signal stability analysis of power systems based on the spectral decomposition of a square H 2 norm of the transfer function. Compared with the dynamics of H 2 and H ∞ norms of the transfer functions, the analysis of the behavior of individual eigen-components allows the earlier identification of the pre-fault condition occurrence. Since each eigen-component is associated with a particular eigenvector, the potential sources of instability can easily be localized and tracked in real time. An important class of systems operating under the pre-fault conditions near the boundary of stability is considered. We demonstrate that in such cases several ill-stable modes can increase the system energy up to a critical level much earlier due to their synergetic effect. In particular, an ill-stable low-frequency mode can act as a catalyst increasing the energy in the system. An illustrative test for the stability analysis of a real small power grid at Russky Island is provided.

Equilibrium contained by counter-threats and complex equilibrium in secure strategies

We present two generalizations of the concept of equilibrium in secure strategies. In equilibrium contained by counter-threats (ECCT), no player can increase its payoff by a unilateral deviation without creating a threat to lose more than it wins. This condition must be satisfied for any pseudo-equilibrium in the generalized sense and, therefore, any such equilibrium must belong to the set of ECCT. The second generalization is the complex equilibrium in secure strategies. The proposed concept allows identifying a hierarchical structure of mutual threats between players and will be useful for the analysis of problems with asymmetric behavior of players. Search algorithms for the proposed equilibria and their examples in matrix games are provided.

Equilibria in secure strategies in the Bertrand---Edgeworth duopoly

This paper analyzes the Bertrand–Edgeworth duopoly model using a solution concept of Equilibrium in Secure Strategies (EinSS), which describes cautious behavior in noncooperative games. The concept is suitable for studying games where the threats of other players represent an important factor in the decision-making process. We demonstrate that, in some cases where the Bertrand–Edgeworth price duopoly admits no Nash–Cournot equilibria, there exists a unique EinSS with both players choosing an identical equilibrium price lower than the monopoly price. The difference between these prices can be interpreted as an additional reduction in price that allows the players to secure themselves against the mutual threats of undercutting. We formulate and prove a criterion for the EinSS existence.

Energy approach to stability analysis of the linear stationary dynamic systems

A new approach was proposed to analyze the stability of the linear continuous stationary dynamic systems. It is based on the decomposition of a square H2 norm of the transfer function of the dynamic system into parts corresponding either to particular eigenvalues of the system matrix, or to pairwise combinations of these eigenvalues. The spectral decompositions of a square H2 norm of the transfer function with multiple poles were obtained using the residues of the transfer function and their derivatives. Exact analytical expressions for calculation of the quadratic forms of the corresponding expansions were derived for an arbitrary location of the eigenvalues in the left half-plane. The obtained decompositions allow one to characterize the contribution of individual eigen-components or their pairwise combinations into the asymptotic variation of the system energy. We propose the energy criterion for estimation of the system stability margins that uses an evaluation of energy accumulated in a group of weakly stable system modes. This approach is illustrated by calculating the energy of a band-pass filter.

Games for cautious players: the equilibrium in secure strategies

A non-cooperative solution, the Equilibrium in Secure Strategies (EinSS), is defined as an extension of the Nash equilibrium in pure strategies, and is meant to solve games where players are “cautious,” i.e., looking for secure positions and avoiding threats. This concept abstracts and unifies ad hoc solutions already formulated in various applied economic games that have been discussed extensively in the literature. A general existence theorem is provided and then applied to the price-setting game in the Hotelling location model, to Tullocks rent-seeking contests, and to Bertrand–Edgeworth duopoly. Finally, competition in the insurance market game is re-examined and the Rothschild–Stiglitz–Wilson contract is shown to be an EinSS even when the Nash equilibrium breaks down.

Chain equilibria in secure strategies

In this paper we introduce a modification of the concept of Equilibrium in Secure Strategies (EinSS), which takes into account the non-uniform attitudes of players to security in non-cooperative games. In particular, we examine an asymmetric attitude of players to mutual threats in the simplest case, when all players are strictly ordered by their relation to security. Namely, we assume that the players can be reindexed so that each player i in his behavior takes into account the threats posed by players j > i but ignores the threats of players j < i provided that these threats are effectively contained by some counterthreats. A corresponding equilibrium will be called a Chain EinSS. The conceptual meaning of this equilibrium is illustrated by two continuous games that have no pure Nash equilibrium or (conventional) EinSS. The Colonel Blotto two-player game (Borel 1953; Owen 1968) for two battlefields with different price always admits a Chain EinSS with intuitive interpretation. The product competition of many players on a segment (Eaton, Lipsey 1975; Shaked 1975) with the linear distribution of consumer preferences always admits a unique Chain EinSS solution (up to a permutation of players). Finally, we compare Chain EinSS with Stackelberg equilibrium.