Robert E. Kooij

Limit cycles in the Holling-Tanner model

This paper deals with the following question: does the asymptotic stability of the positive equilibrium of the Holling-Tanner model imply it is also globally stable? We will show that the answer to this question is negative. The main tool we use is the computation of Poincare-Lyapunov constants in case a weak focus occurs. In this way we are able to construct an example with two limit cycles.

Algebraic invariant curves and the integrability of polynomial systems

Abstract In this note, we study the relation between the existence of algebraic invariants and integrability for planar polynomial systems. It is proved, under certain genericity conditions, that if the sum of the degrees of the algebraic invariants exceeds the degree of the polynomial system by one, then the system is integrable.

The impact of the topology on cascading failures in a power grid model

Cascading failures are one of the main reasons for large scale blackouts in power transmission grids. Secure electrical power supply requires, together with careful operation, a robust design of the electrical power grid topology. Currently, the impact of the topology on grid robustness is mainly assessed by purely topological approaches, that fail to capture the essence of electric power flow. This paper proposes a metric, the effective graph resistance, to relate the topology of a power grid to its robustness against cascading failures by deliberate attacks, while also taking the fundamental characteristics of the electric power grid into account such as power flow allocation according to Kirchhoff laws. Experimental verification on synthetic power systems shows that the proposed metric reflects the grid robustness accurately. The proposed metric is used to optimize a grid topology for a higher level of robustness. To demonstrate its applicability, the metric is applied on the IEEE 118 bus power system to improve its robustness against cascading failures.

A robustness metric for cascading failures by targeted attacks in power networks

Cascading failures are the main reason blackouts occur in power networks. The economic cost of such failures is in the order of tens of billion dollars annually. In a power network, the cascading failure phenomenon is related to both topological properties (number and types of buses, density of transmission lines and interconnection of components) and flow dynamics (load distribution and loading level). Existing studies most often focus on network topology, and not on flow dynamics. This paper proposes a new metric to assess power network robustness with respect to cascading failures, in particular for cascading effects due to line overloads and caused by targeted attacks. The metric takes both the effect of topological features and the effect of flow dynamics on network robustness into account, using an entropy-based approach. Experimental verification shows that the proposed robustness metric quantifies a power grid robustness with respect to cascading failures.

Decentralized planning of energy demand for the management of robustness and discomfort

The robustness of smart grids is challenged by unpredictable power peaks or temporal demand oscillations that can cause blackouts and increase supply costs. Planning of demand can mitigate these effects and increase robustness. However, the impact on consumers in regards to the discomfort they experience as a result of improving robustness is usually neglected. This paper introduces a decentralized agent-based approach that quantifies and manages the tradeoff between robustness and discomfort under demand planning. Eight selection functions of plans are experimentally evaluated using real data from two operational smart grids. These functions can provide different quality of service levels for demand-side energy self-management that capture both robustness and discomfort criteria.

Estimation of perceived quality of service for applications on IPv6 networks

To provide high quality service to future Internet applications, IPv6 performance measurements are needed. However, to the best of our knowledge, IPv6 delay and loss performance evolution and their impact on applications have not been studied on a large scale. In this paper, we have analyzed more than 600 end-to-end IPv6 paths between about 26 testboxes of RIPE NCC over the past two years, and compared the delay and loss performance evolution in IPv6 with their IPv4 counterparts. We present and discuss the measurement methodology, and we provide evidence that IPv6 network has a higher delay and loss evolution than IPv4. Finally, based upon our measurements, we assess the perceived quality of three real-life applications: VoIP, Video-over-IP and data communication services based upon TCP

An integrated packet/flow model for TCP performance analysis

Processor sharing (PS) models for TCP behavior nicely capture the bandwidth sharing and statistical multiplexing effect of TCP flows on the flow level. However, these ‘rough’ models do not provide insight into the impact of packet-level parameters (such as round trip time and buffer size) on, e.g., throughput and flow transfer times. This paper proposes an integrated packet/flow-level model: it exploits the advantages of PS approach on the flow level and, at the same time, it incorporates the most significant packet-level effects.

Uniqueness of limit cycles in polynomial systems with algebraic invariants

The uniqueness of limit cycles is proved for quadratic systems with an invariant parabola and for cubic systems with four real line invariants. Also a new, simple proof is given of the uniqueness of limit cycles occurring in unfoldings of certain vector fields with codimension two singularities.

Measuring and controlling unfairness in decentralized planning of energy demand

Demand-side energy management improves robustness and efficiency in Smart Grids. Load-adjustment and load-shifting are performed to match demand to available supply. These operations come at a discomfort cost for consumers as their lifestyle is influenced when they adjust or shift in time their demand. Performance of demand-side energy management mainly concerns how robustness is maximized or discomfort is minimized. However, measuring and controlling the distribution of discomfort as perceived between different consumers provides an enriched notion of fairness in demand-side energy management that is missing in current approaches. This paper defines unfairness in demand-side energy management and shows how unfairness is measurable and controllable by software agents that plan energy demand in a decentralized fashion. Experimental evaluation using real demand and survey data from two operational Smart Grid projects confirms these findings.

Virus spread in complete bi-partite graphs

In this paper we study the spread of viruses on the complete bi-partite graph KM,N. Using mean field theory we first show that the epidemic threshold for this type of graph satifies tauc = 1/radic(MN), hence, confirming previous results from literature. Next, we find an expression for the average number of infected nodes in the steady state. In addition, our model is improved by the introduction of infection delay. We validate our models by means of simulations. Inspired by simulation results, we analyze the probability distribution of the number of infected nodes in the steady state for the case without infection delay. The mathematical model we obtain is able to predict the probability distribution very well, in particular, for large values of the effective spreading rate. It is also shown that the probabilistic analysis and the mean field theory predict the same average number of infected nodes in the steady state. Finally, we present a heuristic for the prediction of the extinction probability in the first phase of the infection. Simulations show that, for the case without infection delay, this time dependent heuristic is quite accurate.