IEEE 9-bus system

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The Western System Coordinating Council (WSCC) three-machine, nine-bus system is a standard benchmark for power-flow and transient studies. This chapter develops a three-phase electromagnetic-transient (EMT) representation of the benchmark and verifies its 60 Hz steady state against the conventional phasor solution.

Three generators, three step-up transformers, six 230 kV lines and three loads on a 100 MVA, 60 Hz base. Generator dispatch and load are the standard values; the transformer reactances are on the same 100 MVA base. Bus numbering follows the usual convention in which the generator terminals are buses 1 to 3 and the transmission network is buses 4 to 9.

1. Study objective

The reference case is normally specified as a positive-sequence phasor dataset containing bus, branch, transformer, generator, and load data. An EMT implementation requires an explicit time-domain model for each entry. The principal objective is to determine whether those element models reproduce the published operating point when observed at the fundamental frequency.

Three result sets are compared:

1. the published load-flow solution;
2. a Newton–Raphson power-flow solution evaluated from the circuit file;
3. the settled RMS values obtained from a 200 ms EMT simulation.

This comparison tests the consistency of the network representation without introducing synchronous-machine dynamics.

2. EMT representation

2.1 Generator equivalents

Each generator is represented by an ideal balanced three-phase voltage source behind a 1 mΩ series resistance. Source magnitude and phase are set equal to the solved terminal-bus values. The three generator buses therefore act as prescribed electrical boundary conditions; deviations at the remaining buses can be attributed to the network and load models rather than to machine initialization or controller dynamics.

2.2 Loads

The three loads are represented by constant impedances selected to absorb their specified MW and MVAr at the corresponding solved bus voltage. For a balanced three-phase load expressed in terms of total complex power and line-to-line voltage, the equivalent per-phase impedance is

$$Z_{\varphi }=\frac{|V_{LL}{|}^2}{S_{3\varphi }^{\ast }}.$$

The constant-impedance formulation introduces no additional control dynamics and matches the specified power at the reference operating point.

2.3 Transformers

The three step-up transformers retain the benchmark leakage reactances on the 100 MVA base. Their winding resistances are zero, consistent with the source dataset. The EMT models also include 1% magnetizing-current branches; this distinction is relevant when transformer terminal currents are compared with the load-flow result.

2.4 Transmission lines

Each 230 kV branch is represented by a distributed-parameter Bergeron travelling-wave model rather than a lumped π section. The positive-sequence $R$, $L$, and $C$ values are obtained from the branch data. They determine the line surge impedance and one-way propagation delay:

$$Z_c=\sqrt{\frac{L}{C}},\tau =\sqrt{LC}.$$

The resulting line parameters are:

Line	$R$ (pu)	$X$ (pu)	$B$ (pu)	$Z_c$ ($\mathrm{\Omega }$)	$\tau$ ($\mu$s)
4–5	0.0100	0.0680	0.176	329	290
4–6	0.0170	0.0920	0.158	404	320
5–7	0.0320	0.1610	0.306	384	589
6–9	0.0390	0.1738	0.358	369	662
7–8	0.0085	0.0576	0.149	329	246
8–9	0.0119	0.1008	0.209	367	385

3. Time-step selection

A Bergeron line stores the incident wave from each terminal and returns it after one propagation delay. The integration step must therefore be shorter than the smallest line delay. Line 7–8 is the limiting branch, with ${\tau }_{min}=246\mu \text{s}$. The circuit uses $\mathrm{\Delta }t=50\mu \text{s}$, corresponding to approximately 4.9 integration steps per traversal of that line.

4. Steady-state validation

4.1 Measurement procedure

One RMS meter is connected to each bus. Every meter evaluates line-to-line voltage over a sliding one-cycle window. Once the window is full and the network has settled, its output can be compared directly with a 60 Hz phasor magnitude.

The following table reports the published solution, the Newton–Raphson result obtained with Power Flow, and the settled meter value from the 200 ms EMT simulation.

Bus	Published	Newton-Raphson	EMT meters	EMT − published
1	1.04000	1.0400	1.040001	$+1\times {10}^{-6}$
2	1.02500	1.0250	1.025000	$0$
3	1.02500	1.0250	1.024999	$-1\times {10}^{-6}$
4	1.02531	1.0253	1.025170	$-1.4\times {10}^{-4}$
5	0.99972	0.9997	0.999584	$-1.4\times {10}^{-4}$
6	1.01225	1.0123	1.012299	$+4.9\times {10}^{-5}$
7	1.02683	1.0268	1.026689	$-1.4\times {10}^{-4}$
8	1.01727	1.0173	1.017191	$-7.9\times {10}^{-5}$
9	1.03269	1.0327	1.032657	$-3.3\times {10}^{-5}$

4.2 Voltage error

The maximum bus-voltage difference between the EMT result and the published solution is $1.4\times {10}^{-4}$ pu, equivalent to approximately 33 V on a 230 kV base. Agreement at buses 1–3 is prescribed by the ideal source models and is therefore not an independent validation result. Buses 4–9 are solved by the network model; their voltages agree with the reference solution to approximately one part in ten thousand.

4.3 Relation between the phasor and EMT line models

The two calculations use different mathematical formulations. The load-flow calculation solves a set of complex algebraic equations at 60 Hz using lumped π branches. The EMT calculation advances a differential-algebraic system through 4,000 time steps and represents each transmission line by delayed travelling waves.

For a lossless distributed line, the equivalent π parameters at angular frequency $\omega$ are

$$X_{\mathrm{eq}}=Z_c\sin (\omega \tau ),B_{\mathrm{eq}}=\frac{2}{Z_c}\tan \left(\frac{\omega \tau }{2}\right).$$

For this network, $\omega \tau$ is at most 0.25 rad at 60 Hz. Consequently, $\sin (\omega \tau )\approx \omega \tau$ and $\tan (\omega \tau /2)\approx \omega \tau /2$, so the distributed-line equivalent closely approaches the reactance and charging susceptance of the lumped branch data. The voltage comparison quantifies the resulting agreement at the fundamental-frequency operating point.

4.4 Current cross-checks

The phase-$a$ current into the bus 5 load settles at 338.9 A RMS. The load-flow quantities, 134.6 MVA at 229.94 kV, imply

$$I=\frac{|S|}{\sqrt{3}{V}_{LL}}=338.1\text{A}.$$

At the slack source, the EMT current is 2,601 A, compared with 2,586 A from the load flow. The approximately 0.6% difference is primarily due to the transformer magnetizing branches, which are present in the EMT model but absent from the reference load-flow model. Reducing the magnetizing-current settings on all three transformers from 1% to 0.001% lowers the EMT current to 2,589 A.

5. Exercises

1. Transformer winding resistance. Set r_hv_lv_pu = 0.005 on each transformer. Compare the bus-voltage magnitudes and the decay of the source-current DC offsets with the original case. Explain why winding resistance has little effect on the load-flow operating point but a substantial effect on the EMT startup transient.

2. Time-step sensitivity. Repeat the simulation with $\mathrm{\Delta }t=10\mu \text{s}$ and compare the settled voltages with the 50 µs results. Then investigate what occurs when the time step exceeds the 246 µs propagation delay of line 7–8. Relate the observed behavior to the delay-line storage requirement.

3. Single-contingency study. Open either line 4–5 or line 4–6 and calculate the resulting redistribution of line power. The line open/short sample provides an example of the required switching arrangement.

4. Dynamic source substitution. Replace one ideal source with a synchronous machine and excitation system, following the generator controls sample. Use the load-flow solution as the dynamic model's initial condition, and distinguish network-model errors from electromechanical rotor response.