In the first part of this thesis, we introduce a specific class of Linear Matrix Inequalities (LMI) whose optimal solution can be characterized exactly. This family corresponds to the case where the associated linear operator maps the cone of positive semidefinite matrices onto itself. In this case, the optimal value equals the spectral radius of the operator. It is shown that some rank minimization problems, as well as generalizations of the structured singular value ($mu$) LMIs, have exactly this property.

In the same spirit of exploiting structure to achieve computational efficiency, an algorithm for the numerical solution of a special class of frequency-dependent LMIs is presented. These optimization problems arise from robustness analysis questions, via the Kalman-Yakubovich-Popov lemma. The procedure is an outer approximation method based on the algorithms used in the computation of hinf norms for linear, time invariant systems. The result is especially useful for systems with large state dimension.

The other main contribution in this thesis is the formulation of a convex optimization framework for semialgebraic problems, i.e., those that can be expressed by polynomial equalities and inequalities. The key element is the interaction of concepts in real algebraic geometry (Positivstellensatz) and semidefinite programming.

To this end, an LMI formulation for the sums of squares decomposition for multivariable polynomials is presented. Based on this, it is shown how to construct sufficient Positivstellensatz-based convex tests to prove that certain sets are empty. Among other applications, this leads to a nonlinear extension of many LMI based results in uncertain linear system analysis.

Within the same framework, we develop stronger criteria for matrix copositivity, and generalizations of the well-known standard semidefinite relaxations for quadratic programming.

Some applications to new and previously studied problems are presented. A few examples are Lyapunov function computation, robust bifurcation analysis, structured singular values, etc. It is shown that the proposed methods allow for improved solutions for very diverse questions in continuous and combinatorial optimization.

Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization

A power management architecture for an electrical power distribution system, or portion thereof, is disclosed. The architecture includes multiple intelligent electronic devices (“IED's”) distributed throughout the power distribution system to manage the flow and consumption of power from the system using real time communications. Power management application software and/or hardware components operate on the IED's and the back-end servers and inter-operate via the network to implement a power management application. The architecture provides a scalable and cost effective framework of hardware and software upon which such power management applications can operate to manage the distribution and consumption of electrical power by one or more utilities/suppliers and/or customers which provide and utilize the power distribution system. Autonomous communication on the network between IED's, back-end servers and other entities coupled with secure networks, themselves interconnected, via firewalls, by one or more unsecure networks, is facilitated by the use of a back-channel protocol. The back-channel protocol allows a device coupled with a secure network to solicit communications from a device on the unsecure network, thereby opening a back-channel through the firewall through which the unsecure network device may send unsolicited messages to the secure network device. Communications between multiple secure networks is accomplished using a unsecure device on an intermediary unsecure network to relay communications between the secure network devices using the protocol described above.

Push communications architecture for intelligent electronic devices

Power system loading margin is a fundamental measure of a system's proximity to voltage collapse. Linear and quadratic estimates to the variation of the loading margin with respect to any power system parameter or control are derived. Tests with a 118-bus system indicate that the estimates accurately predict the quantitative effect on the loading margin of altering the system loading, reactive power support, wheeling, load model parameters, line susceptance and generator dispatch. The accuracy of the estimates over a useful range and the ease of obtaining the linear estimate suggest that this method will be of practical value in avoiding power system voltage collapse.

/pdf/sensitivity-of-the-loading-margin-to-voltage-collapse-with-3u8e1smi2k.pdf

Sensitivity of the loading margin to voltage collapse with respect to arbitrary parameters

The work described in this paper was motivated by a perceived increase in the frequency at which power system operators are encountering high stress in bulk transmission systems and the corresponding need to improve security monitoring of these networks. Online risk-based security assessment provides rapid online quantification of a security level associated with an existing or forecasted operating condition. One major advantage of this approach over deterministic online security assessment is that it condenses contingency likelihood and severity into indices that reflect probabilistic risk. Use of these indices in control room decision making leads to increased understanding of potential network problems, including overload, cascading overload, low voltages, and voltage instability, resulting in improved security-related decision making. Test results on large-scale transmission models retrieved from the energy-management system of a U.S. utility company are described.

http://home.engineering.iastate.edu/~jdm/WebJournalPapers/OnlineRBSA.pdf

Online risk-based security assessment

Fault location is an important application among intelligent monitoring and outage management tasks used for realization of self healing networks, one of the most attractive features of smart grids. The data gathered from various intelligent electronic devices (IEDs) installed throughout the power system could be utilized for smart approaches to locating faults in both transmission and distribution systems. This paper discusses issues associated with improving accuracy of fault location methods in smart grids using an abundance of IED data. Two examples of how the gathered data from different IEDs is used to improve fault location accuracy in transmission and distribution systems are discussed in detail.

Smart Fault Location for Smart Grids

Voltage collapse and blackout can occur in an electric power system when load powers vary so that the system loses stability in a saddle node bifurcation. This paper computes load powers at which bifurcation occurs and which are locally closest to given operating load powers. The distance in load power parameter space to this locally closest bifurcation is an index of voltage collapse and a minimum load power margin. The computations are illustrated for several power systems. Monte-Carlo optimization techniques are applied to obtain multiple minimum load power margins. The use of load power margin sensitivities to select system controls is discussed. >

/pdf/computation-of-closest-bifurcations-in-power-systems-wyy1rxnesq.pdf

Computation of closest bifurcations in power systems

Both fault location and fault resistance of a fault are calculated by the present method and system. The method and system take into account the effects of fault resistance and load flow, thereby more accurately calculating fault resistance by taking into consideration the current flowing through the distribution network as well as the effect of fault impedance. A direct method calculates fault location and fault resistance directly while an iterative fashion method utilizes simpler calculations in an iterative fashion which first assumes that the phase angle of the current distribution factor Ds is zero, calculates an estimate of fault location utilizing this assumption, and then iteratively calculates a new value of the phase angle βs of the current distribution factor Ds and fault location m until a sufficiently accurate determination of fault location is ascertained. Fault resistance is then calculated based upon the calculated fault location. The techniques are equally applicable to a three-phase system once fault type is identified.

System for locating faults and estimating fault resistance in distribution networks with tapped loads

Power system security is characterized by using distances to system operational limits (including voltage collapse limits, stability limits, line and transformer overloads, and generator limits). It uses the point of collapse method based on the singularity of the power flow Jacobian to define an operational limit boundary in load demand space. Distances to this boundary are then translated into probabilistic measures of likelihood of system failure: probability of normal status, expected demand not served, expected unserved energy, and, ultimately, expected outage cost. Sensitivities of these measures to load and generation changes (important for operational decisions and real-time pricing) are also described. A numerical example is presented. Computational issues are discussed. >

/pdf/engineering-foundations-for-the-determination-of-security-3u430kzrbo.pdf

Engineering foundations for the determination of security costs

A fault is located in a transmission line with a sending end, a receiving end, and a tapped load connected to the transmission line at a tap node. The tap node divides the transmission line into a sending side and a receiving side. The sending end and the receiving end each include a measuring device. The fault location is determined by obtaining measured circuit parameters including measured pre-fault and faulted current and voltage values at the sending end and at the receiving end of the transmission line. The phase angle difference due to unsynchronized measurement using the measured pre-fault current and the measured pre-fault voltage values may be calculated. The load impedance of the tapped load is calculated. A first fault location is calculated assuming that the fault is located on the sending side of the tap node. A second fault location is calculated assuming that the fault is located on the receiving side of the tap node. The fault location is selected from one of the first fault location and the second fault location, by selecting the fault location having a value within a predetermined range representing a full distance between two nodes.

Systems and methods for locating faults on a transmission line with a single tapped load

A system for implementing accurate V/Hz value measurement and trip time determination for generator/transformer overexcitation protection independent of the conventional frequency tracking and phasor estimation based on Discrete Fourier Transformation (DFT) techniques. The half-cycle summation technique of the invention is a non-recursive digital technique which measures the per unit V/Hz value by summing the sampled data points in every half cycle of a sinusoidal input signal and dividing the sum with the ideal base sum value. When the input voltage signal is sampled at a reasonable frequency, the technique of the invention approximates the accurate per unit V/Hz value of the input voltage signal and thus obtains an accurate V/Hz characteristic directly without computing voltage and frequency separately.

Yi Hu

Papers

Computation of closest bifurcations in power systems

System for locating faults and estimating fault resistance in distribution networks with tapped loads

Engineering foundations for the determination of security costs

Systems and methods for locating faults on a transmission line with a single tapped load

Half-cycle summation V/Hz relay for generator and transformer over-excitation protection