Kundur two-area system

Authors: Martin Franke (Fraunhofer IEE)

Reviewers: Joy El Feghali (RTE)

Use case purpose​ and context

The Kundur two area system is a test system for studies on dynamic stability, oscillations, damping, power exchange and other use cases. It is a widely used standard test case defined in Kundur’s Power system stability and control [1] (Fig. E12.8).

Table of references

[1] P. Kundur, Power System Stability and Control. New York: McGraw-Hill Education Ltd, 1994.

[2] R. Ramos and I. Hiskens, “PES-TR18: Benchmark Systems for Small-Signal Stability Analysis and Control,” Aug. 2015.

[3] Shackshaft and Henser, “Model of generator saturation for use in power system studies.” Apr. 1979.

Network ​description

In Figure 1 the test system is shown. As [1] describes it, it consists of two similar areas connected by a weak tie.

The parameters are listed in Section 5. The system consists of 11 buses connected by lines or transformers. There are 4 generators G1…G4, two loads L7 and L9 as well as two shunt capacitors C7 and C9.

Static and Dynamic models​ description

This test case includes the following components:

Generators

Governors

The generator model does not include a governor.

AVRs

The following AVRs are used in this test system, depending on the used scenario:

• no AVR: Manual excitation, i.e. constant field voltage
• AVR DC1A (similar to DC1C, see colib page here)
  • \(K_\mathrm{a}=\) 20 pu
  • \(K_\mathrm{a}=\) 200 pu
• AVR ST1A (similar to ST1C, see colib page here)
  • (with/without transient gain reduction (TGR) and/or power system stabilizer (PSS))

For details on the scenarios, see Section 6. The parameters are given in the tables of Section 5.

Loads (static ZIP model)

Equation 1 and Equation 2 define ZIP load model. Depending on the coefficients in Table 1 it models a constant impedance, current or power load.

\(P = P_\mathrm{Ref} \cdot (1 + \Delta P) \cdot \left(Z_\mathrm{p} \cdot \left(\frac{U}{U_\mathrm{0}}\right)^2 + I_\mathrm{p} \cdot \frac{U}{U_\mathrm{0}} + P_\mathrm{p}\right) \qquad(1)\) \(Q = Q_\mathrm{Ref} \cdot (1 + \Delta Q) \cdot \left(Z_\mathrm{q} \cdot \left(\frac{U}{U_\mathrm{0}}\right)^2 + I_\mathrm{q} \cdot \frac{U}{U_\mathrm{0}} + P_\mathrm{q}\right) \qquad(2)\)

In [1] it is stated that the active power should be modelled as constant current and the reactive power as constant impedance: All coefficients can be set to zero except \(I_\mathrm{p} = Z_\mathrm{q} = 1\).

[!IMPORTANT]

Loads L7, L9 and shunt capacitors C7, C9 can all be modeled as ZIP loads. To reproduce the load flow results from Kundur, the loads need to be modeled as constant power (P and Q), while the shunt capacitors need to be modeled as constant impedance.

Input Data

Table 3: AVR DC1A parameters, see [2]

Table 4: AVR ST1A parameters, see [2]

Table 5: PSS parameters, see [2]

Scenarios

The following sceniarios are taken from [1] and [2], which are also considered for reference results.

The shown results are obtained from the Dyna\(\omega\)o Modelica implementation of the test case.

Load flow

The load flow results are given in Table 9 and Table 10.

• The active power exchange from Area 1 to Area 2 (see Figure 1) is 400.431 MW in the Dyna\(\omega\)o implementation and 400 MW in the Kundur reference.

Example a

Example a examines small-signal performance of the system with constant excitation.

• All four generators are on manual excitation (field voltage \(E_\mathrm{fd} = \mathrm{const.}\))
• Constant current characteristics of active load components
• Constant impedance characteristics of reactive load components

Figure 2 shows the result of an eigenvalue analysis of this system, compared to the reference values from [1]. The remaining differences probalby originate from differing model implementations of the synchronous machines in Dyna\(\omega\) compared to the ones used by Kundur.

Example b

[!NOTE]

To obtain a steady state initial condition, the exact values for the generators active and reactive power and voltage angles from load flow Section 6.1 need to be used as initial values in the dynamic simulation.

[!NOTE]

In the Dyna\(\omega\)o implementation, if the voltage transducer time constant \(T_\mathrm{R}\) = 0.01 s, then the voltage measurement first order lag and the field flux linkage \(\lambda_\mathrm{f}\) oscillate against each other with high amplitude. This is not represented in the reference results of [2]. To avoid this behaviour, set \(T_\mathrm{R} = 10^{-6}\) s. This has been done for the Dyna\(\omega\)o implementation.

Part 1: Self-excieted dc exciter (DC1A)

Instead of constant excitation voltage as in Section 6.2, now the self-excited DC exciter DC1A is used. For parameters, see Table 3.

In [2] this case has not been taken into account for comparative analysis. Like there, it has not been investigated using the Dyna\(\omega\)o implementation.

Part 2: Thyristor exciter (ST1A) with high transient gain

Instead of constant excitation voltage as in Section 6.2, now the thyristor exciter ST1A is used. For parameters, see Table 3. It has high transient gain, i.e. no transient gain reduction.

The implementation in Dyna\(\omega\)o does not allow to set saturation parameters \(A_\mathrm{sat}\) and \(B_\mathrm{sat}\) as in Table 2 because its synchronous machine model uses a different saturation model. Instead of the model from Kundur [1], the Shackshaft model [3] is used. To get the unstable pole close to the one in the Kundur reference, the Shackshaft parameters have been set to \(m_\mathrm{d}=1\), \(m_\mathrm{q}=1\), \(n_\mathrm{d}=9\), \(n_\mathrm{q}=9\).

The voltage transducer time constant has been

As in [2], a steps to the synchronous machines’ voltege references have been applied according to Table 11. Those were selected to excite the inter-area oscillation mode and the other electromechanical modes in the system [2].

The step response of the generators can be seen in Figure 4.

Part 3: Thyristor exciter (ST1A) with transient gain reduction (TGR)

Simlilarly to Section 6.3.2, the ST1A exciter is used. In this case, transient gain reduction (TGR) is activated by setting \(T_\mathrm{A} = 1\).

Part 4: Thyristor exciter (ST1A) with high transient gain and PSS

Simlilarly to Section 6.3.2, the ST1A exciter is used. In this case, a power system stabilizer according to [1] (Figure 6-8) is activated and parameterized according to Table 5.

Open source implementations

This model has been successfully implemented in :