1 Overview of the power system module

The THYME model is typical of models used for electro-mechanical transients.

A brief overview of the model is given here. For a more detailed treatment,
see the references listed at the end of this document. The model has two parts:
differential equations for the generators and algebraic equations for the loads and
transmission system. This note describes first the equations for the generators,
second the equations for the loads and transmission systems and the inclusion
of the generators within this model, and third the loads. Specific information on
the models and simulation algorithms can be found in [2] and [5] respectively.

2 Generators
Each generator has the following state variables.

1. The frequency ω in the ”per unit” system. Frequency is zero during normal
operations: positive and negative values indicate excursions away from
normal.

2. The mechanical power Pm that drives the generator’s turbine. This also
is in the “per unit” system.

3. The excitation voltage amplitude E, also in the “per unit” system. This
value is adjusted dynamically to maintain the voltage at the terminals of
the generator.
4. The phase angle θ of the voltage at the terminals of the generator.

5. A control signal c that regulates the mechanical power Pm for the purposes
of i) keeping ω near zero and ii) keeping Pe at its reference level (i.e., at
the desired output for the generator).

Each generator has the following parameters.

1. The “inertia” M of the turbine in the “per unit” system.

2. The synchronous reactance X of the turbine motor.

3. The droop setting R of the frequency regulator.

4. The reference point Ps for the mechanical power.

5. The area control gain A.

6. The reference voltage Vs for the terminals of the generator.

7. The time constants T1 , T2 , and Te of the control circuits.

8. A resistance D to operation away from nominal frequency.

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Each generator acts on the voltage phasor V ∠φ at terminals. The quantity V ∠φ
is an input that originates with the loads, transmission network, and interaction
(via the transmission network) with other generators. The voltage controller in
the generator acts on V . The speed of the generator reacts to changes in Pe ,
which is ½µ ¶µ ¶∗ ¾
E∠θ − V ∠φ
Pe = Re V ∠φ (1)
X
The generator output is a current phasor I that it injects into the transmission
network. This current is
E∠θ
I= (2)
X
The behavior of each generator is dictated by the following equations:
Pm − Pe − Dω
ω̇ = (3)
M
θ̇ = ω (4)
ċ = T1 (Ps − ω/R − c) (5)
Ṗm = T2 (ck − Pm ) (6)
Ė = Te (Vs − V ) (7)

The power set point Ps is adjusted whenever a signal arrives from the area
generation controller. At these times Aω is subtracted from Ps . Over and
under frequency protection devices are built into the generator models. These
open when ω reaches its upper or lower threshold.

3 Transmission, loads, and generators in the net-

work
Every load and generator is modeled by a Norton equivalent circuit: a current
source and complex impedance in parallel and connected to ground at one end
and its network terminal at the other. For generators, the impedance is its syn-
chronous reactance X and the current source is I. For loads the current source
is usually negative (i.e., it draws current from the network) or zero (this case
creating a constant impedance load). Transmission lines are modeled by com-
plex impedances that connect network nodes. The resulting network equations
are in the form
I = YV (8)
The current vector I contain the current sources provided as output from the
generators and any currents injected (or drawn) by the loads. The voltages Vt
are the voltages seen at the network’s terminals and contribute the V input to
each generator. An example of a transmission network is shown in Fig. 1. Using
X = j, yline = 1/Zline = 100.0, yload = 1/Zload = 1, and Iload = 0 we obtain

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the admittance matrix
· ¸ · ¸
1/X + yline −yline j + 100 −100
Y= = (9)
−yline yload + yline −100 101
and voltage and current vectors
· ¸ · ¸
I V1
I = genr V= (10)
0 V2

Zline

V1 V2

Iinj Xd I load Z load

(a) One line drawing (b) Equivalent circuit

Figure 1: An example of the transmission model.

4 Loads
Loads are modeled as recommended in [1]. A reduced form of this model and
parameters for it can be found in [3]. The power consumed by the load is
modulated by changing the current injected by that load into the network. The
load admittance remains constant. The equation for calculating injected load
is derived from the load’s Norton equivalent circuit with injected current Iinj ,
load admittance y, terminal voltage V , and base demand S0 = P0 + jQ0 for the
circuit. These give the injected current as
µ ¶∗
αr P0 + αi jQ0
Iinj = V y −
V
where S0 = V0 y and V0 is the initial voltage at the load and the model is always
parameterized so that Iinj = 0 at V = V0 . The terms αr and αi modulate Iinj .
These are
µ ¶2 µ ¶
|V | |V |
αr = Kpz + Kpi + Kpc
|V0 | |V0 |
µ ¶n µ ¶n
|V | pv1 |V | pv2
+ Kp1 (1 + npf 1 f ) + Kp2 (1 + npf 2 f )
|V0 | |V0 |
µ ¶2 µ ¶
|V | |V |
αi = Kqz + Kqi + Kqc
|V0 | |V0 |
µ ¶nqv1 µ ¶n
|V | |V | qv2
+ Kq1 (1 + nqf 1 f ) + Kq2 (1 + nqf 2 f )
|V0 | |V0 |

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A description of the parameters can be found in [1]. The voltages V are calcu-
lated from the network equations and frequency f at the load using the method
described in [4]. Note that this requires the solution to a fixed point problem
in the form I¯inj = f (x, I¯inj ) where x is a vector with the state variables of the
generators and I¯inj is the vector of currents injected by the loads.

5 Initializing from power flow data

The electrical model can be initialized from solved power flow data in the IEEE
Common Data Format or the PTI format. These are described at http://www.
ee.washington.edu/research/pstca/ and several sample data files are also
provided at that location. Parameters assigned to the generators and loads are
described in the source code in these locations: see the class AutoInitilizingData
for generator parameters and ElectricalModelEqns for load parameters.

6 Full equations from a one load, one generator

system
To better illustrate the numerical issues posed by the above equations, a com-
plete model consisting of a generator and load connected by a transmission line
is constructed in this section. The following set of equations is derived from Fig.
1, the above discussion and the material presented in [4]. Note that voltages,
currents, and impedances shown in Fig. 1 are complex quantities.

1. Transmission equations.
· ¸
1/X + yline −yline
Y=
−yline yload + yline
· ¸ · ¸
Igenr V
=Y 1
Iload V2

The transmission equations are used to solve for the voltages V1 and V2
given Y, which is a determined by the structure of the electrical network,
and the currents Igenr and Iload , which are determined by the generator
and load equations.

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 2. Generator equations.
Pm − Pe − Dω
ω̇ =
M
θ̇ = ω
ċ = T1 (Ps − ω/R − c)
Ṗm = T2 (ck − Pm )
Ė = Te (Vs − |V1 |)
Pe = ℜ{V1 Iout
∗
}
Iout = Igenr − V1 /X
E cos θ + jE sin θ
Igenr =
X
Note that the generator equations are coupled to the load equations via
the transmission equations. Specifically, the voltage V1 is a function of the
load current Iload . Because the load equations (see below) comprise a set
of non-linear, algebraic equations, the entirety of the model constitutes a
semi-explicit differential-algebraic model.
3. Load equations.
µ ¶∗
αr P0 + αi jQ0
Iload = V2 yload −
V2
µ ¶2 µ ¶ µ ¶n µ ¶n
|V2 | |V2 | |V2 | pv1 |V2 | pv2
αr = Kpz + Kpi + Kpc + Kp1 (1 + npf 1 f ) + Kp2 (1 + npf 2 f )
|V0 | |V0 | |V0 | |V0 |
µ ¶2 µ ¶ µ ¶n µ ¶n
|V2 | |V2 | |V2 | qv1 |V2 | qv2
αi = Kqz + Kqi + Kqc + Kq1 (1 + nqf 1 f ) + Kq2 (1 + nqf 2 f )
|V0 | |V0 | |V0 | |V0 |
cos(arg V2 ) −|V2 | sin(arg V2 ) |V˙2 |
· ¸· ¸ ·1 ¸
Eω(sin θ − j cos θ)
= Y−1 X
sin(arg V2 ) |V2 | cos(arg V2 ) f 0

As before, the load equations are coupled to the generator equations.

Specifically, the voltage V2 is a function of the load generator Igenr and
the frequency f is a function of the generator state variables E, ω, and θ.
The entirety of the model therefore constitutes a semi-explicit differential-
algebraic model. Numerically, it is the solution to the load equations that
pose the greatest difficulty. There are two special case of practical inter-
est, however, in which the non-linearities vanish and the complete model
becomes a set of simpler, ordinary differential equations. The first case is
when constant impedance loads are modeled by setting Kpz = Kqz = 1
and all other of the K parameters to zero. This gives Iload = 0. The
second case is when V ≈ V0 and so we can take Iload ≈ 0.
The general form of the system can be seen by taking the vector x to be the
state variables in the generator model(s), and the vector I = [Iload Igenr ]T to be

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the currents injected by the generator(s) and load(s). The form of the equations
is then

ẋ = f (x, Y−1 I)
Iload = g(x, Y−1 I)

where I depends, of course, on both x and Iload . In the special cases described
for the load model, the term Iload becomes zero and this is reduced to the system
of ordinary differential equations

ẋ = f (x)

where the vector I appearing in the previous equations now depends solely on
x via the generator equations.

References
[1] Standard load models for power flow and dynamic performance simulation.
IEEE Transactions on Power Systems, 10(3):1302–1313, August 1995.

[2] J. Arrillaga and N. R. Watson. Computer Modelling of Electrical Power

Systems, Second Edition. Wiley, 2001.

[3] IEEE Task Force on Load Representation for Dynamic Performance. Load
representation for dynamic performance analysis (of power systems). IEEE
Transactions on Power Systems, 8(2):472–482, May 1993.

[4] J. Nutaro and V. Protopopescu. Calculating frequency at loads in simula-

tions of electro-mechanical transients. Authors are with the Computational
Sciences and Engineering Division at Oak Ridge National Laboratory, Oak
Ridge, TN (contact email: nutarojj@ornl.gov)., November 2010.

[5] James J. Nutaro. Building Software for Simulation: Theory and Algorithms
with Applications in C++. John Wiley and Sons, Inc, Hoboken, New Jersey,
2011.