Diagnosability of Hybrid Power Systems

In recent years great research effort has been directed toward fault diagnosis of Electrical Power Generation Systems. Many papers have been published on this subject [1, 2], and even if we restrict ourselves to the theoretical treatment of faults, there is a variety of problem formulations depending on fault models, measurement conditions, the final object of the fault diagnosis, and so forth. Key factors in the classification of diagnostic problems are the following.

1. Subsystems under diagnosis:

a. Elements contained in the subsystem: 1) “RLC or RC”, 2) “

1) DC Excitation. 2) Sinusoidal excitation – Single-frequency or multi-frequency.

c. Number of exciting sources (independent sources or inputs):

1) Single exciting source. 2) Multiple exciting sources. 2. Fault models:

a) Short circuits or open circuits (hard faults). b) Malfunction of an element or subsystem that occurs at intervals, usually irregular, in an element or subsystem that functions normally at other times (intermittent faults). c) Element or parameter-value deviations outside the tolerance bounds (soft faults) 3. Measurements: a) Voltage, b) Current, c) Power, d) Phase. 4. Final object of fault diagnosis: a) To determine voltages and currents, b) To identify the faulty elements, c) to locate and isolate the faults within a subsystem.

In addition to the problem formulation directly related to electrical and mechanical elements, attempts have been made to apply techniques of system diagnosis to electrical power (EP) diagnosis.

In diagnosis of large- scale system, use of the computers is inevitable. One special feature of EP diagnosis which is quite different from those of circuit analysis is that the information for diagnosis or knowledge about the internal part of the EP system is very restricted. Therefore diagnosability, or whether or not the diagnostic problem formulated is solvable under the specific requirements and the expected performance indexes are achievable, must be preliminarily researched in order to discriminate instability intrinsic to the problem from that due to the computational method and errors. Since there are several problem formulations, diagnosability must be described as structural as functional according to them.

In the following sections, the diagnosability of soft faults in a EP system which is in a “Fuzzy MLD Fault Model” is consider. An advantage of assuming soft faults only is that the EP model structural and functional description is known, and the equation derived for the Mixed Logical Dynamic (MLD) definition included simple, directed and co-directed element-value relationship can be inferred.

It is possible to assume that the system contains as active elements, controlled-transferring conversion subsystems and several element-values relationship in terms of passive and active energy conversion in the different steps of the model. Most elements and function conversion can be represented by their equivalent equation conversion model and at last, a complete dynamic and recursive equation. Therefore, this approach not suffers loss of generality.

For simplicity of discussion we further assume that the system contains no “VCVS” and CCCS”. Either of these controlled sources should be replaced, described and modeled by a cascade connection of a VCCS and a CCVS.

Computability of the parameters in the system from measured and known values is developed first, since it is the basis of fault diagnosis. Then the result is applied to fault location by the fault verification or assume-and-check method. It is shown that the diagnosability of an EP system, like the system diagnosability, depends on the connectivity of the system under test. Finally, a brief discussion is given on diagnosis by multiple taps of evaluation mode.

The system under consideration is denoted by n, and its structural and functional description by \(G\). In order to overcome the difficulty arising from mutual coupling in controlled sources we use two subsystems \(N_v\) and \(N_i\) and two graphs \(G_v\) and \(G_i\).

Then \(G_v\) and \(G_i\) are called the voltages and current graphs, respectively (Shifang 2007), and Kirchhoff’s voltage and current laws are applied to \(G_v\) and \(G_i\), respectively. These subsystems and graphs are derived from \(N\) and \(G\) in the following way. Nv is derived from N by contracting the current sensors and nullators and deleting dependent sources and noratos. \(G_v\) is the graph of \(N_v\). Next, \(N_i\) is the circuit which is derived from \(N\) by deleting voltage sensors and nullators and contracting dependent voltage sources and norators. \(G_i\) is the graph of \(N_i\). Note that either the sensor or the dependent source of a controlled source remains in \(N_v\) or \(N_i\), and a two-port element in \(N\) is represented by a two-terminal element in \(N_v\) and a two-terminal element in \(N_i\), in the same way as a resistor. Thus there is no need to distinguish a controlled source from a resistor in \(N_v\) and \(N_i\). The element in \(N_v\) and the element in \(N_i\) representing the same element in \(N\) are given the same label. Then \(N_v\) and \(N_i\) have a common element set, which is denoted by \(E\). \(G_v\) and \(G_i\) also have a common edge set. Strictly speaking, circuits \(N\), \(N_v\), and \(N_i\) are constituted by “elements” which correspond to “edges” of graphs \(G\), \(G_v\), and \(G_i\). For simplicity we will not distinguish edges from elements hereafter. Thus the common edge set of \(G_v\) and \(G_i\) is also denoted by \(E\).

Next let us formulate the measurement on \(N\) [4]. The element set \(E\) of \(N_v\) and \(N_i\) is partitioned into \(E_b\), \(E_e\), \(E_j\), \(E_k\) and \(E_u\), which are defined as follows.

\(E_b\) => the set of elements whose voltages and currents can be both measured and know.

\(E_e\) => the set of elements whose voltages only can be measured and know.

\(E_j\) => the set of elements whose currents only can be measured or know.

\(E_k\) => the set of elements whose voltages and currents are unknown but whose element values are known.

\(E_u\) =>the set of elements whose voltages and currents are unknown and whose element values are also unknown.

More specifically, Table 9.1 applies to a resistor, inductor, capacitor, or controlled source. Table 9.2 applies to an independent voltage or current source.

To insert the Tables 9.1 and 9.2...

The above classification of elements can be interpreted in terms of degrees of freedom as follows. Each element in the circuit originally has two degrees of freedom, because two variables, a voltage and a current, are associated with it. The degrees of freedom are decreased by the information obtained for the element or the restriction imposed on it. Thus an element in \(E_b\) has zero; one in \(E_e\), \(E_j\) or \(E_k\) has one; and one in \(E_u\) has two degrees of freedom. Note that in an ordinary circuit analysis each element in the circuit has one degree of freedom. The total number of KVL and KCL equations is equal to the number of elements, and thus the voltages and currents can be uniquely determined. The measurement condition regarding a node can be expressed by an element which is connected to the node and the reference node, if such an element exists, or by an imaginary element which is added between the node and the reference node.

For any graph \(G\) with edge set \(E\) and subset \(E_s\) of \(G \cdot E_s\) (respectively \(G x E_s\) ) is the graph obtained from \(G\) by deleting (respectively contracting) the edges of \(\overline E_s:= E - E_s\). The rank and nullity of \(G\) are denoted by \(r(G)\) and \(n(G)\), respectively. \(\parallel E_s \parallel\) is the cardinality of \(E_s\) and + denotes the union of disjoint sets.

Regarding trees and cotrees of \(G\) we have the following propositions. For simplicity we use the same notation for a tree or cotree and its edge set.

**Proposition 2.1** Let \(T_a\) and \(T_b\) be trees of \(G \cdot E_s\) and \(G x E_s\), respectively. Then the union of \(T_a\) and \(T_b\) constitutes a tree of G, and

\[r(G) = r(G \cdot E_s) + r(G x \overline E_s), n(G) =n(G \cdot E_s) + n(G \cdot E_s + n(G x \overline E_s)\]

Let \(T\) be a tree of \(G\) and \(K\) be the cotree of \(T\)

**Proposition 2.2** Given a subset

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