FrB06.4
43rd IEEE Conference on Decision and Control December 14-17, 2004 Atlantis, Paradise Island, Bahamas
A Theoretical Diagnosis Approach Applied to a Power Transmission System G. K. Fourlas, K. J. Kyriakopoulos, and N. J. Krikelis
Abstract—Fault detection and isolation is a challenging task in the control of large scale complex systems. In this work we proceed to the fault diagnosis of an electric power transmission system based on a fault diagnosis of hybrid systems method presented in our previous work. Power systems often exhibit complex behavior in response to large disturbances. Such behavior is characterized by interactions between continuous dynamics and discrete events. Components such as loads drive the continuous dynamics, while other components such as protection devices exhibit event-driven discrete dynamics. Therefore, power systems constitute an important case of hybrid systems for fault detection.
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I. INTRODUCTION
HE increasing requirements to achieve more reliable performance on complex systems such as power systems [3, 7, 11, 12, 13], air traffic management systems [15, 21], automated highway systems [16, 22], manufacturing systems [2], have necessitated the development of fault diagnosis schemes for accurate diagnosis of system failures. Such systems can be viewed as hybrid systems and therefore fault diagnosis is a challenging task in the control of hybrid systems. Hybrid systems are systems including both continuous and discrete dynamics influencing each other [1], and therefore the global dynamics. The issues of safe operation for such systems are of major importance and require their supervision in order to timely handle the occurrence of faults or failures [19]. Power systems often exhibit complex behavior in response to large disturbances. Such behavior is characterized by interactions between continuous dynamics and discrete events. Components such as loads drive the continuous dynamics, while other components such as protection devices exhibit event-driven discrete dynamics. Therefore, power systems constitute an important case for fault detection in hybrid systems.
Manuscript received February 29, 2004. G. K. Fourlas is with the Department of Informatics and Computer Technology, Technological Educational Institute (T. E. I.) of Lamia, Lamia 35100, GREECE, (e-mail:
[email protected]). K. J. Kyriakopoulos and N. J. Krikelis are with the Control Systems Laboratory, Mechanical Eng. Dept., National Technical University of Athens (NTUA), Athens 15700, GREECE (tel:++30-210-7723595, e-mail: {kkyria,nkrik}@central.ntua.gr).
0-7803-8682-5/04/$20.00 ©2004 IEEE
In this paper we focus on fault diagnosis of power systems. We use the theoretical framework proposed in our previous work [5, 6 7, 8, 9] which include the notion of diagnosability as well as the necessary and sufficient conditions for fault diagnosis of hybrid systems. More specifically in section II we describe the method for fault diagnosis of hybrid systems, in section III we present the modeling of an electrical power transmission system and in section IV we discus the application of fault diagnosis to the power system. II. DESCRIPTION OF FAULT DIAGNOSIS METHOD At this section we describe briefly the fault diagnosis procedure using a diagnoser. The whole system is modelled by a hybrid input output automaton HIOA [17], which captures both continuous and discrete behavior. It is considered to consist of several distinct parts, each of them modelled by a (HIOA). The overall model is the composition of a number of automata containing the dynamics of all parts. The faults are modelled as discrete transitions from the normal to a faulted state, as well as deviation of trajectories describing the continuous evolution from a predefined set point. Each of the aforementioned parts can be affected directly by a fault and is considered as a component. Thus, a number of faults may occur for each of these components. According to [18] each of these faults can be classified into different fault modes. Therefore, for each component only one fault mode may occur at a time. The basic diagnosis procedure is based on a new framework called hybrid structure hypothesis testing which has no restrictions on the type of faults [8]. Its mathematical foundation lies on hypothesis testing and mathematical logic. Although we share some ideas with [18] and [20], our approach is more general in the sense that it addresses hybrid systems and not merely discrete events or continuous systems. The development of hypothesis tests requires the partition of the state space into different fault modes. The methodology of fault diagnosis proposed in the above framework is based on variables called faulty guards which are described below. When there is a fault, visible or not, the system enters to a certain partition of the faulty state space. Consequently, the faults could be detected from the transitions to the faulty state space. Generally, the
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discrete transitions take place whenever some conditions on the variables are satisfied, which determine whether a transition is enabled. These variables are called guards. Therefore, the problem of diagnosis consists of the measurement problem of faulty transition guards called faulty guards. A. Diagnosability First, we present the notion of diagnosability of hybrid systems. Generally speaking, a system is said to be diagnosable, if it is possible to detect the occurrence of a fault in a short period of time. Thus, Definition 1: A Hybrid System H is to be diagnosable if the following holds, Fi E , ki N , ai H , htrace(ai ) H | t htrace(ai ), t t ki , Zi VF , ai DF , (Zi , ai ) htrace( a )
The above definition means that for every fault Fi E , (E denotes the set of all faults modes) and for every faulty guard contained in a hybrid execution a (is an alternating infinite or finite sequence of trajectories and actions) there are ki alternations of trajectories and actions after the occurrence of the fault. Then, for the hybrid system H to be diagnosable, a certain hybrid trace with a sufficiently long duration t from alternations of trajectories and actions after the occurrence of the fault containing the same faulty guard, must exist. The hybrid trace of a hybrid execution Į of H, denoted by htrace(Į), records the visible behavior of the execution and is the sequence obtained by projecting Į onto the external variables of H and subsequently removing all inert internal and environment actions. It is noted that this definition does not lead to concluding which fault mode has occurred. To describe this situation we give the next definition. Definition 2: A hybrid faulty state is indistinct if it is not clear which fault mode has occurred.
information about the fault mode responsible for the behavior of the process. The diagnoser is considered to be passive, which implies that it does not affect the system to be diagnosed. Also it is assumed to be static, that is, the same observations always give the same diagnostic result (statement). The diagnoser consists of three parts: the discrete diagnoser, the continuous diagnoser and the mechanism of decision logic. The discrete component offers estimation for the discrete state of the hybrid system as well as the fault diagnosis at the level of discrete event. The continuous part provides the fault diagnosis of continuous behavior of the hybrid system. Then, the decision logic mechanism is used to produce the final diagnosis statement expressed as S i S Di j SC j i.e., it is a combination of discrete and continuous sub-statements. According to the aforementioned approach, a flow chart of the fault diagnosis procedure is depicted in figure 1 [9]. Identification of subsystems and components
Specification of faults
Definition of variables and guards
Diagnosability checking
Creation and correspondence of fault labels
B. Diagnosability Conditions We then determine the conditions under which a system is diagnosable. These conditions are necessary and sufficient for a hybrid system to be diagnosable. Theorem 1: A hybrid system without multiple failures of the same mode is diagnosable if and only if the following conditions are satisfied: C1: There is a measurable faulty guard. C2: No state in a htrace(a) is indistinct. C. Diagnoser To perform diagnosis we use a diagnoser which is a hybrid automaton that generates a signal whenever a fault occurs. Its role is to observe and check the behavior of the system automaton and to compare its evolution with the predefined acceptable behavior. Moreover, whenever it detects a fault it generates a diagnostic statement S, indicating the malfunctioning component, and providing
Residuals production and threshold checking
Diagnosis statement Actions of faults handling (e.g. Restoration) Fig. 1. Flow chart of diagnosis procedure
III. POWER SYSTEM MODELING The power system under investigation consist of a voltage source connected at the first bus, an infinite bus (bus 4) representing the connection of the system with a large system having no variations, six-transmission lines, equipped with inverse time relays for overcurrent
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protection and loads of variable conductivity connected at bus 2 and 3 (figure 2). The system is built and simulated with the SimulinkTM environment, and the StateflowTM, all running on top of MatlabTM.
f
V1 1
CB
CB
L1
4
CB
L4
V4
We investigate the following scenario. At time t 1s the load starts to change and its dynamics are represented by a ramp. It should be noted that we are not studying the case of system islanding. The simulation of the electric power transmission system behavior under the aforementioned disturbance is accomplished with analysis of load flow equations in Cartesian form [14]. This analysis generates at every simulation step the bus voltages of the network.
CB
CB
L3
L2
Iˆij
Vˆi
L6
Vˆj
CB
Fig. 4. Typical transmission line of power energy L5
CB
CB
CB
2
CB
G1,B1
We mention that for the construction of the admittance matrix the operation/interruption of a line controlled by a CB is taken into account according to the status of the corresponding CB. In figure 4, a typical transmission line is shown. The element Y i, j of the admittance matrix is
3
G2,B2
Fig. 2. Diagram of four bus and six transmissions lines
The system isn’t solved analytically but numerically using the MATLAB. The hybrid behavior of this system is due: x to the close/open position of the circuit breakers (CB) x to the continuous dynamics of the inverse time relays and the loads x the output w of an emergency control system which expresses the shedding of a percentage of the system load. The inverse time relay module of Simulink model is depicted in figure 3. The inverse time relay compares the line current with a limit value (1.5 pu for the first line in the example of figure 3) and is used as protection devise against overcurrent of the transmission line. The output (state zi ) drives the CBs at both ends of the line and when zi 0 gives a trip signal.
I lim i
given by the following equation Y i, j yij zij
yij
(2)
Rij jX ij
0 or 1 , Rij jX ij is the line impedance.
where zij
Therefore, the admittance matrix depends on the line parameters as well as the breakers outputs zij . With the knowledge of the bus voltages the current Iˆij (figure 2, 4) is given by the equation Iˆ y Vˆ Vˆ ij
ij
i
j
(3)
Consequently, the appropriate simulation model is constructed providing the current values as well as the circuit breaker status. According to the line data described in table I, the system starts by functioning in its normal mode. Subsequently, the load G1 , B1 starts increasing linearly with time. TABLE I LINE DATA
x 0
Ii
(1)
+
1 s
³
x ti
zi
1 k1
Fig. 3. Inverse time relay Simulink model
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Line
Ampacity (pu)
Resistance (pu)
Reactance (pu)
1
1
0.10
0.40
2
1.0
0.05
0.20
3
1.0
0.08
0.20
4
1.0
0.10
0.75
5
1.5
0.15
0.40
6
1.0
0.20
0.80
Fig. 5. System response
Fig. 6. Diagnosis Simulink model
As a consequence, the current passing through the lines increases. Because of their technical characteristics (table I), the transmission lines present different overload behavior. As shown at figure 5 the second line is overloaded first. The inverse time relay integrates the overcurrent value and when the output of the integrator becomes positive, a trip signal (Z2=0) is given. The CB is considered as a perfect switch interrupting abruptly the current. As a consequence, the second line is blocked and therefore all the current passes through the other lines which are in turn overloaded. The next line which overloads and causes its CB to turn it off is the first.
Because of the continuing overload, also the third line is turned off. At this point in order to avoid the islanding scenario we stop any further load increase. IV. FAULT DIAGNOSIS OF THE ELECTRIC POWER SYSTEM We now proceed to the diagnosis of the Electric Power Transmission System. According to the definition of diagnosability and the correspondence theorem [6, 9] the first step concerns the checking of diagnosability of the system. The aforementioned modeling of the power system contains the normal behavior as well as the faulty states of six lines.
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Fig. 7. Diagnosis unit of Electric Power Transmission System
The variables used as guards [6, 9] in this application are the six currents of the transmission lines and the output signals of the relays. These variables are measurable via telemetry and ensure the diagnosability of the system. As it is shown in figure 6 the diagnostic scheme involves two systems. The first is the physical system for which telemetry measurements are available. These measurements are inputs to the diagnosis unit. The second is the mathematical model which provides computed values
of the variables that are also inputs to the diagnosis unit. The diagnosis unit is a Stateflow diagram. Inputs of the diagnosis unit are the outputs of the physical system and of the mathematical model. Outputs are the different ALARM signals concerning the status in each line. Figure 7 shows the structure of the diagnosis unit. It is a StateflowTM chart. The chart consists of parallel states (denoted by dash-dotted boundaries) that represent concurrent modes of operation. The parallel states 1, 2, 3, 7, 8, 9 at the top of the figure 7 correspond to the six lines
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obtained from the mathematical model. Each state has various sub-states representing the status of that particular line. These sub-states are mutually exclusive. The transitions determine how states can change and how they are guarded by conditions. The other parallel states 4, 5, 6, 10, 11, 12 correspond to the six inverse time relays. Each one contains the sub-states of each relay which are mutually exclusive. The thirteen parallel states correspond to the diagnoser. Starting the description from outside to inside, we observe three mutually exclusive states: The first has six parallel states one for each line, while each one of them contains three mutually exclusive sub-states (NORMAL, ALERT, EMERGENCY) for each line. At each discrete state the checking of the corresponding residual is made and the appropriate alarm signal is produced. Finally, the second and the third mutually exclusive states represent the RESTORATIVE and BLACKOUT operating mode of the system. As noted above, outputs of the diagnosis unit are a number of signals called ALARMS. These signals are the following: x N1 and N2 refer to normal mode of operation for each line respectively x A1 and A2 denote the overload state for each line respectively x E1 and E2 denote the turn off of each line respectively x R refers to restorative state and x B refers to blackout.
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[5]
[6] [7]
[8] [9] [10] [11] [12]
[13]
V. CONCLUSION The handling of faults for large-scale systems is one of the major problems faced by control engineers today. The most significant challenge arises from the complexity of the system, which forces designers to develop more sophisticated diagnosis schemes. The contribution of this paper is mainly focused on the fault diagnosis of power system based on hybrid system modeling. The behavior of the electric power transmission system is modeled by a HIOA (Hybrid Input/Output Automaton), since it offers the ability to describe the interaction between the continuous dynamics and discrete event dynamics. This framework formalizes the description of faults and the diagnosis problem is solved using a diagnoser via a method which is based on hybrid structure hypothesis testing. In this paper, only a brief description of the theoretical fault diagnosis approach is given, due to limited space reasons. For a thorough presentation the reader is referred to [8, 9]. A topic of future research is to apply this methodology of fault diagnosis to more complex power systems, where an important issue for investigation is the algorithmic complexity, since a large number of components and subsystems are present.
[14] [15]
[16] [17] [18] [19] [20] [21] [22]
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