THE ELECTROMAGNETIC TRANSIENT PROGRAM

The electromagnetic transient program (EMTP) is the creation of H. W.

Dommel, who started to work on the program at the Munich Institute of Technology in the early 1960s. He continued his work at BPA (Bonneville Power Administration) in the United States. The EMTP became popular for the calculation of power system transients when Dommel and Scott-Meyer, his collaborator in those days, made the source code public domain. This became both the strength and weakness of EMTP; many people spent time on program development but their actions were not always as concerted as they should have been. This resulted in a large amount of computer code for every conceivable power system component but very often without much documentation. This problem has been over-come in the commercial version of the program, the so-called EPRI/EMTP version. Electric Power Research Institute (EPRI) has recoded, tested, and extended most parts of the program in a concerted effort and this has improved the reliability and functionality of the transient program. Circuit breaker models are an example of the extended functionality added to the program in 1987 and improved in 1997, but are not available in the still-existing public domain version of the program; the alternative transient program (ATP). Presently, the EMTP and other programs that are built on a kernel (such as electromagnetic transients for DC (EMTDC) and power system computer-aided design (PSCAD)) based on the same principles are a widely used and accepted program for the computation of electrical transients in power systems.

The EMTP is based on the application of the trapezoidal rule to convert the differential equations of the network components to algebraic equations. This approach is demonstrated in the following text for the inductance, capacitance, and lossless line.

For the inductance L of a branch between the nodes k and m, it holds i_k,m(t)= ik,m(t− t) + 1

L

_t

t−t(ν_k− νm)dt (8.1)

Integration by means of the trapezoidal rule gives the following equations i_k,m(t)= t

2L(ν_k(t)− νm(t))+ Ik,m(t− t) I_k,m(t− t) = ik,m(t− t) + t

2L(ν_k(t− t) − νm(t− t)) (8.2) For the capacitance C of a branch between the nodes k and m, it holds

ν_k(t)− νm(t)= νk(t− t) − νm(t− t) + 1 C

_t

t−ti_k,m(t) dt (8.3) Integration by means of the trapezoidal rule gives the following equations

i_k,m(t)= 2C

t(ν_k(t)− νm(t))+ Ik,m(t− t) I_k,m(t− t) = −ik,m(t− t) −2C

t(ν_k(t− t) − νm(t− t)) (8.4) For a single-phase lossless line between the terminals k and m, the following equation must be true.

u_m(t− τ) + Zim,k(t− τ) = uk(t)− Zik,m(t) (8.5) τ the travel time (s)

Z the characteristic impedance ()

In words, the expression u+ Zi encountered by an observer when he leaves the terminal m at time t− τ must still be the same when he arrives at terminal k at time t. From Equation (8.5), the following two-port equations can be deduced.

i_k,m(t)= u_k(t)

Z + Ik(t− τ) I_k(t− τ) = −u_m(t− τ)

Z − im,k(t− τ) i_m,k(t)= u_m(t)

Z + Im(t− τ) I_m(t− τ) = −u_k(t− τ)

Z − ik,m(t− τ)

(8.6)

The resulting models for the inductance, capacitance, and the lossless line are shown in Figure 8.1.

They consist of current sources, which are determined by current values from previous time steps, and resistances in parallel. Thus a network can

THE ELECTROMAGNETIC TRANSIENT PROGRAM 139

Figure 8.1 EMTP representation of an inductance, capacitance, and a lossless line by current sources and parallel resistances. t is the step size of the computation. τ is the travel time of the line. Z is the characteristic impedance of the line

e(t )

Figure 8.2 Sample RLC circuit

be built up of current sources and resistances by using the equivalent circuits as shown in Figure 8.1. This approach will be demonstrated on the sample RLC network that is shown in Figure 8.2.

By means of the equivalent models for the inductance and the capaci-tance, as depicted in Figure 8.1, and the replacement of the voltage source

R = 2L /∆t

Figure 8.3 Equivalent EMTP circuit

and the series impedance by a current source with a parallel resistance, the RLC circuit can be converted into the equivalent circuit as shown in Figure 8.3. To compute the unknown node voltages, a set of equations is formulated by using the nodal analysis (NA) method.



In general, the following equations hold

Yu= i − I (8.8)

Y the nodal admittance matrix

u the vector with unknown node voltages i the vector with current sources

I the vector with current sources, that are determined by current values from previous time steps

The actual computation works as follows:

• The building up and inversion of the Y-matrix. This step has to be taken only once. However, when switching occurs, this step has to be repeated while the topology of the network changes;

• The time-step loop is entered and the vector of the right-hand side of Equation (8.8) is computed after a time step t;

THE ELECTROMAGNETIC TRANSIENT PROGRAM 141

• The set of linear equations is solved by means of the inverted matrix Y and the vector with the nodal voltages u is known; and

• The vector in the right-hand side of Equation (8.8) is computed for a time step t further in time and we continue this procedure till we reach the final time.

The advantages of the ‘Dommel–EMTP’ method, as described in the previous section, among others are

• simplicity (the network is reduced to a number of current sources and resistances of which the Y-matrix is easy to construct) and

• robustness (the EMTP makes use of the trapezoidal rule, which is a numerically stable and robust integration routine).

However, the method has some disadvantages too:

• A voltage source poses a problem. This becomes clear from the sample RLC circuit; a small series resistance will result in an ill-conditioned Y-matrix (see Equation (8.7));

• It is difficult to change the computational step size dynamically during the calculation, the resistance values and the current sources should be recomputed at each change [see Equation (8.7)] that entails Y-matrix re-inversion. This is time-consuming for larger networks;

and

• The Y-matrix is ill-conditioned. The resistances in the represen-tations of the capacitance and the inductance (see Figure 8.1 and Equation (8.7)) are treated oppositely with regard to the computa-tional step size t. Thus if the computacomputa-tional step size is decreased, it has a contrary effect on the inductances and capacitances. This can lead to numerical instabilities.

The arc models within the EMTP are implemented by means of the compensation method. The nonlinear elements are essentially simulated as current injections, which are superimposed on the linear network, after a solution without the nonlinear elements has been computed first.

The procedure is as follows: the nonlinear element is open-circuited and the Thevenin voltage and Thevenin impedance are computed. Now, the two following equations have to be satisfied. Firstly, the equation of the linear part of the network (the instantaneous Thevenin equivalent circuit

as seen from the arc model).

V_th− iRth= iR (8.9)

V_ththe Thevenin (open-circuit) voltage (V) R_ththe Thevenin impedance ()

i the arc current (A) R the arc resistance ()

Secondly, the relationship of the nonlinear element itself. Application of the trapezoidal method of integration yields for the arc resistance at the simulation time t:

R(t)= R(t − t) +t 2

dR dt

t

+ dR dt

t−t



(8.10)

t the time step (s)

The dR/dt is described by the differential equation of the arc model.

To find a simultaneous solution of Equation (8.9) and Equation (8.10), the equations have to be solved by means of an iterative process (e.g.

Newton–Raphson).

Therefore the solution process is as follows:

• The node voltages are computed without the nonlinear branch,

• Equation (8.9) and Equation (8.10) are solved iteratively, and

• The final solution is found by superimposing the response to the current injection i.

In the EMTP96, three arc models have been implemented; Avdonin–

Schwarz, Urbanek, and Kopplin.