by Phil Kreveld

Australia is leading the way in the quest for a renewable electricity grid. Being at the forefront comes with its challenges — the most pressing of which is maintaining network stability. This in light of a reduction in baseload and a reduction in traditional generation that supplies a stable voltage and frequency. Cutting through the dense fog enveloping network stability problems, the incidence of low energy prices driving out investment in synchronous generation and solar and wind farms, gas instead of coal as frequency support, and the number of players, operationally and regulatory, all of whom are pushing their own barrows, might seem an impossible task.

Yet, there is a straightforward action plan that could almost immediately provide the foundation for network stability and provide an incentive for keeping baseload generation going beyond currently flagged retirements and taking the short-term pressure of deciding on new gas-fired plant proposals.

## Disruptive Form of Generation

In short, the action plan would put a stop on unregulated, uncontrolled solar PV installations, the most disruptive form of generation Australia has. What started off as ‘feel good’ installations representing minor incursions in distribution networks, are now responsible for well over 20% of daytime power demand. Distributed solar PV installations are continuing their growth, much of it encouraged by state governments.

Distribution networks are mainly worried about voltage and its impacts on energy consumption and operation of solar inverters (i.e., these being cut off when high voltage limits are exceeded). However, network stability is affected by rapid changes in substation power and reactive power due to generation by distributed energy sources (DER). This is increasingly playing a role in AEMO response mechanisms including islanding of large parts of the national grid. South Australia, a state that more than others in the NEM, facing islanding by virtue of its very large renewable generation, is attempting to limit power export of rooftop solar.

The major issue is that, the vast array of ** grid following grid tie inverters** in the NEM require the presence of voltage so that they can synchronise to supply frequency. That requires a minimum amount of power flow from the high voltage grid to substations to establish stable voltage and frequency for inverter phase-locked loop synchronisation.

The much talked-up self sufficiency and so-called independence from the grid is, when thought through, demanding that there be ‘grid islands’ with voltage forming inverters—a truly brave idea that, we can be sure, is not part of AEMO’s ISP.

It will come as no surprise that substation minimum power flow from the HV grid to zone substations is not seen as a matter of concern, yet phase jumps, through lines dropping out, or caused by switching in of capacitor banks for voltage correction purposes, can cause anti-islanding criteria to be invoked and to switch off inverters as has been shown by studies supported by AEMO. ** Voltage control of lightly loaded transmission lines is looming as a technical challenge**.

The complexity of mathematical analysis of the NEM grid is beyond the ken of this writer, requiring elaborate modelling. However, highly simplified analytical examples as presented here focus well on the issues of stability —the more since it is a reasonable assumption that, given the complexity of interactions between generators and loads in the physical, extensive NEM grid, any instabilities due to rapid changes in demand would be exacerbated.

The (above diagram – *Figure 1*) AC circuit is of a generator, **G**, whose voltage, **v _{1}** is described by the versor

**V**, and which, via a short transmission line of impedance

_{1}e^{j0}**Z**, equal to

**R+**, is supplying a load

*j*X**P+**. The voltage at the load

*j*Q**v**, is the versor,

_{2}**V**, (note: the assumption is made that

_{2}e^{-jδ}**v**lags

_{2}**v**by

_{1}**δ**radians).

The diagram is a highly simplified representation of an AC network connecting a variable load drawing real power **P**, and exchanging reactive power **Q** with the generator, **G**. Notwithstanding its simplicity, it draws attention to the above remarks on the need for substation stability. Frequency (phase change of **δ**) stability is affected by changes in load.

Of concern is the variability of **δ** under conditions of varying load, i.e.

*Note*: **P** and **Q** are assumed independent variables, i.e., the load is not a constant power factor load.

The variability of the load reflects the effects in power and reactive power fluctuation due the high level of distributed generation in the distribution network defined by **P+ jQ**.

Note: ^{dδ}/_{dt} is a change of frequency during the change of state of the load, assumed to be the distribution network. The change in frequency during the changes in load is assumed to be deleterious to the operation of solar inverters within the distribution network as these, by virtue of their phase locked loop (PLL) operation, require **δ** to be constant or to vary slowly in reaching a new stable state.

The circulating current,* i *is given by:

Apparent power, **S**

Where* *** i*** is the vector conjugate of

**i**Expanding in Euler form

Replacing * Z* with

**,**

*R*+*jX*

*Z**=*R*–*jX*Therefore:

Expanding

For a transmission line of **X** only,

If **V _{1}** and

**V**≅

_{2}**V,**the equation reduces to

Therefore, for a load,

Manipulating the terms, the phase angle, **δ** is defined by:

Writing the above as ** tanδ = a**, then

**.**

*δ=arctan a=1/(1+ a2)*Although a highly simplified example is being made use of, we can differentiate this for influence of **P** and **Q** on phase angle. This is relevant for networks with a high proportion of phase-locked loop, grid following inverters.

Therefore:

Having come this far with some mathematical manipulation, the question is ‘so what’?

Part of the answer could be in bespoke control schemes for grid supporting inverters.

It is also answered in part by considering the form of time variations experienced by power and reactive power at the load end of the highly simplified transmission line, with impedance** jX**.

Without having to resort to more maths, let’s imagine large step functions for** P **and** Q**, resulting in large phase changes in **δ**. The step functions could be the result of a system fault, a large cloud or a ’non-contingent’ event.

It matters ought, as a rapid phase change, however occasioned, will trip the anti-islanding functions of connected, grid-following inverters resulting in large power swings as the inverters resynchronise.

## Simplified Model

The NEM looming problems certainly don’t end with a review of what the consequences at a particular substation are as a result of large penetrations of solar PV.

Let’s go back to our very simple transmission line, which in truth could only be a short line on the basis of inductive reactance only. ** What characterises the NEM is long transmission lines with large voltage angles between generator buses and load centres**.

We’ll resort again to a simplified model but do so without diminishing the argument of straightforward, if possibly unpopular solution—controlling power and reactive power at zone substations.

The power network in *Figure 1* has the generator in idealised form, supplying voltage at a phase angle of 0°. We’ll assume we are dealing with a synchronous generator—unfortunately a species rapidly disappearing from our national electricity ecosystem.

Under steady load conditions, the armature reaction will see to it that the generator voltage angle ‘developed’ across its steady-state inductance is ** EjӨ**. Under steady state conditions, the total phase difference is equal to

**.**

*Ө*+*δ*=*α*In *Figure 2* below, the power versus total angle curve is shown and it is the familiar half sine wave with maximum power delivered at the angle ** π/2**. In ‘classical’ power system analysis, synchronous generators are often assumed to be connected to an infinite bus (one which behaves like voltage source with infinite inertia).

We cannot use this model in the NEM and even less so as synchronous generation is becoming a small proportion of total power demand. This means that in considering the stability of our electrical system we have to look at ‘first swing’ stability, realising that there is no contribution available from so-called ‘grid- firming’ battery banks.

Under steady state operation of a synchronousgenerator, mechanical torque **Tm** of the prime mover is countered by the electrical torque **Te**. Power output from the generator is given by the electrical torque multiplied by angular velocity, ω in radians per second, and torque in Newton-metres. The voltage angle, Ө is measured with reference to a pole of the generator. Therefore, under steady state conditions, with reference to some imaginary mark on the stator, is a constant, the synchronous speed of the generator, ω. A sudden increase in electrical power demand requires increase in mechanical shaft torque from the prime mover, causing an acceleration of the rotor to hopefully a new equilibrium condition.

The power change demand is reflected by the equation below:

Where **J** is the moment of inertia (kg-m^{2}) of the combination of prime mover and generator. Multiplying by the synchronous speed, provides the power response curve for the generator. However, the simple transmission line phase shift of **δ** can be included for overall stability considerations.

This gives us the following relationship for determining stability of the system in *Figure 1*.

Where **P _{m}** and

**P**are respectively instantaneous mechanical and electrical power, and

_{e}**is the angular momentum of the combined prime mover and generator, and this brings back to**

*M**Figure 3*relating to first swing stability for the system of generator and power line.

The diagram illustrates three important points on the power-angle (**α**). At power level **P _{1}** the system is considered stable by definition, i.e., point 1 on the curve and the system power angle is

**. Now a sudden increase in power demand occurs, requiring power**

*α*_{1}**P**at point 2. The additional torque required accelerates the generator rotor but rather then settling immediately at power angle

_{2}**, the rotor overshoots to point 3, on the other side of the maximum power point**

*α*_{2}**P**.

_{m}For ‘first swing’ stability to occur, the rotor should return, initially swinging back with decreasing angle amplitude to point 2. Obviously this swinging period is what rate of change of frequency, RoCoF is all about during this settling period of the generator which will have a general form of **α _{M}e **

**–**

^{𝑡}/_{𝑇}*where*

**sin 2π⨍**_{α}t**is the maximum power angle excursion following a power increment,**

*α*_{M}**T**is a damping constant, and fα adds to the synchronous frequency.

When it comes to the continually changing NEM with a decreasing proportion of synchronous generation, the bad dream of first swing instability becomes closer to a reality at some stage.

Coming back to point 3 on the power angle curve, this the critical point, if generator rotor swings past this point, the chances of restoring torque to take the rotor back to a damped oscillatory response diminish. In order for there to be sufficient restoring torque available, area **a _{2}** (between points 2 and 3) needs to exceed the area

**a**(between points 1 and 2).

_{1}Expressed mathematically:

The foregoing discussion is limited in the same way as those on, for example, economics which base elaborate theories on convenient models that leave out lots of difficult to predict detail.

The same observations are warranted here but—and it’s an important ‘but’ because even based on these ‘electrical engineering 101’ concepts, the perils to which NEM networks are subjected through continued diminishment of synchronous generation are clear. It is difficult— if not impossible to remain confident about network resilience and stability with the replacement of synchronous generation by inverter-based resources (IBR) that have no voltage forming or inertia associated with them.

The voltage forming properties in IBR are even more important in NEM networks that by virtue of long transmission lines are subject to large voltage angle differences—in other words, the simple explanations above relating to ‘first swing stability, really come into force.

## Resilience and Stability

The electricity sector needs to develop markets for spinning reserve/capacity because renewables are steadily destroying the income base for traditional generation. Yet we cannot do without this basis for resilience and stability. At the same time, substation control over distributed generation is now essential as well as narrowing the limits regarding allowable generation that can be connected.

First published in Transmission & Distribution.