Modeling and simulation of transients in electric power systems using hybrid system theory

Transients in electric power systems are of great interest to power engineers when designing a new or maintaining an existing system. The paper deals with using hybrid system theory for modeling and simulation of an electric power system with controllers. The presented technique is rather convenient and recommended as mathematical models of transients in electric power systems with controllers in general contain both continuous and discrete components. The modeling and simulation were carried out in the modeling and simulation environment ISMA, which is briefly presented in the paper. 1 Transients in electric power systems An electric power system (EPS) [1] is a network of elements generating, transforming (transformers, voltage stabilizers, power invertors), transmitting and distributing (power lines, feeders), and consuming (loads) electrical energy along with different controllers and protective devices. Transients might occur in an EPS due to faults or during its normal functioning. The examples are a threephase ground fault caused by a fallen tree and change of the loads as a result of, for example, a start of an asynchronous motor at a factory, which are rather common. In a transient state, the currents, voltages and other system characteristics change over time. States of this kind are not preferred by the users since they expect the system properties to be rather constant, i.e. the power line frequency and voltage magnitude in their electrical outlets to be about, for example, 50 Hz and 220V respectively. Components of the second type, controllers and protective devices, are intended for preventing the damage or malfunction of the electrical elements by making transients less destructive, bringing the system to a quasi-steady state.


Transients in electric power systems
An electric power system (EPS) [1] is a network of elements generating, transforming (transformers, voltage stabilizers, power invertors), transmitting and distributing (power lines, feeders), and consuming (loads) electrical energy along with different controllers and protective devices.
Transients might occur in an EPS due to faults or during its normal functioning. The examples are a threephase ground fault caused by a fallen tree and change of the loads as a result of, for example, a start of an asynchronous motor at a factory, which are rather common. In a transient state, the currents, voltages and other system characteristics change over time. States of this kind are not preferred by the users since they expect the system properties to be rather constant, i.e. the power line frequency and voltage magnitude in their electrical outlets to be about, for example, 50 Hz and 220V respectively. Components of the second type, controllers and protective devices, are intended for preventing the damage or malfunction of the electrical elements by making transients less destructive, bringing the system to a quasi-steady state.

Hybrid systems
Although electrical components are described by differential-algebraic systems of equations (DAEs), that is their models are continuous, elements of the second type are characterized as having discontinuities, more precisely the corresponding DAEs' right-hand sides have discontinuities. Models of this sort can be conveniently described as hybrid systems (HS) [2,3] exhibiting both discrete and continuous behaviors. An HS is a generalization of dynamical system and can be regarded as a sequence of classical dynamical systems (modes) activated one after another so the final conditions of one dynamical system are the initial conditions for the next one [4].
A tuple consequences. An event can appear instantly as a result of changing various hybrid system characteristics: parameters, initial conditions, continuous states and their derivatives (the right-hand sides of systems of first order ordinary differential equations), physical properties of the hybrid system.
A very elementary and classic example of a hybrid system, the bounce of a perfectly elastic ball off a perfectly elastic rigid surface, can be described with the following initial value problem: An HS mode can be constrained by one or more conditions ( 1 ≥ s ). In our case, the mode of falling down to the bounce surface corresponds to the simultaneous fulfillment of the following two conditions: the vertical velocity is negative 0 ) , ( . As events (transitions of an HS from one mode to another) may dramatically change the behavior of the hybrid system, their detection during the simulation is rather a crucial task. Even small errors can accumulate causing the visible change of the trajectory behavior. Failures to detect events may lead to even much worse simulation results having nothing to do with the system model.

Modeling transients in electric power systems
Let us consider the following electric power system, a six-machine test system of Institute "Energosetproekt" (Russia) [6]. Its one line diagram is shown in Figure 1. The system consists of six synchronous generators (G 1 -G 6 ), the corresponding generator transformers (T 1 -T 6 ), three autotransformers (AT 1 -AT 3 ), four loads modeling the power consumers, five power transmission lines denoted by bold dashed lines, and a current limiting reactor (R).
After applying Park's transformation [7], in the corresponding d-q rotating reference frame, generators without damper windings are described as , All the generators are equipped with automatic voltage regulators, which stabilize the output voltage as the loads change or a fault occurs. The generators G 1 -G 5 have turbine speed controllers maintaining the angular speeds of the rotors as close to the rated one as possible since even pretty small variations can lead to the damage of the synchronous machines and other components of the network.
The block diagram for the turbine speed controllers of the generators G 2 and G 3 is shown in Fig. 2. Given the angular speed, the controller computes the displacement of the centrifugal pendulum clutch having a dead zone. The output signal, turbine torque, is bounded to an upper and a lower limits by the saturation block.
The controller can be represented in the LISMA language [8] as shown in Fig. 3. Transformers and autotransformers can be represented as active-inductive elements as shown in Fig. 4 a.
The power transmission lines are modeled as activeinductive elements with two active-capacitive connections to ground. The equivalent circuit is depicted in Fig. 4 b.
One of the simplest equivalent circuits for loads is an active-inductive element connected to ground (Fig. 5 a).
Reactors can be represented in the same way but with resistances of zero (Fig. 5 b).

Fig. 4. Equivalent circuits for a) transformers and autotransformers, b) power transmission lines.
In order to obtain the full model, we also need to include a system of equations of current conservation according to Kirchhoff's first law. The full system of equations contains 144 differential and 65 algebraic equations. The given models allow us to study both electromechanical and electromagnetic transient phenomena.

Modeling and simulation environment ISMA
ISMA [8] (the acronym for "Computer-aided analysis tools" in Russian) is a modeling and simulation environment developed by the Automated Control Systems department of Novosibirsk State Technical University (Russia).
ISMA supports several input modeling languages, including the textual general-purpose modeling language LISMA, a graphical language of block diagrams, and a graphical language for modeling electric power systems (Fig. 6).
It allows one to model EPSs using the aforementioned component models, carry out simulations with different scenarios including changes in the system parameters, short circuits, line breaks.
In ISMA's EPS editor, controllers of generators as well as protective devices can be conveniently modeled using block diagrams. The model in Fig. 2 was created in ISMA's block diagram editor.
Events in HSs can be classified into three groups: unilateral, bilateral, and accuracy critical events [4]. The division is necessary because it is impossible to accurately detect a switching point as phase space trajectories are computed using finite precision arithmetic. The problem of detection of the time instant * t t = , when an event function ( ( ), ) 0 g y t t = , is actually a complex problem of accurate event detection [9,10,11].
If there are singularities or the physical meaning of a problem imposes the condition that the phase space trajectories must not ever cross the event surface, then such cases fall into the group of unilateral events. What is exactly the case in the presented model since the controllers have dead zones, the turbine powers are bounded as well as the rate at which the controllers can change them. For event detection ISMA employs an original event detection algorithm based on the ideas of J. Esposito [8,9], which allows correct detection of events. The algorithm is based on the following theorem.
Theorem. Computing the step size for an explicit numerical method of the form guarantees the event dynamics to behave as a stable linear system, which approaches the surface g(y, t) = 0. Besides, if g(y 0 , t 0 ) < 0, then g(y n , t n ) < 0 for all n.
Also, a distinctive feature of ISMA is that it allows one to use one-step explicit numerical methods with improved stability regions and stability control [12] for simulating moderately stiff [13] systems of equations, whereas it is a common practice to employ implicit methods for this purpose [14,15].

Simulation
Let the load at Node 8 decrease by 20% at 0 t s = . The reaction of the electric torque of the nearest generator G 1 is shown in Fig. 7. All the values are given in a per-unit system [7]. Apart from the electromagnetic transient, the event results in a much longer lasting electromechanical transient (Fig. 8).