The Inverter Power Factor Penalty: Reactive Power Displacements in High-Density Residential Solar Feeds

When Every Roof Exports at Once

The low-voltage network was designed to carry power in one direction, from the substation to the customer, and its voltage profile was engineered accordingly. The distribution transformer's tap is set so that, with the network lightly loaded, the voltage at the first customer is high enough to leave margin for the drop along the feeder when it is heavily loaded, and the customer at the far end still receives a voltage within the statutory range of 230 V plus 10% and minus 6%, that is, from 216.2 V to 253 V. Distributed solar generation reverses the logic. On a sunny afternoon, when household demand is low and generation is at its peak, the power flows from the customers towards the substation, the voltage along the feeder rises instead of falling, and it is the customers at the end of the line that see the highest voltage. The upper limit of 253 V, which was formerly a remote consideration, becomes the binding constraint on how much solar a street can host.

The inverter is the interface between the array and the network, and it is a controllable device whose behaviour determines how the street responds. Beyond the active power it exports, it can exchange reactive power with the network and, within limits, reduce its active output. This audit sets out the power-system mathematics of apparent power and displacement, derives the voltage rise at the point of connection and shows quantitatively how much of it reactive power can offset on a low-voltage network, describes the control loops by which inverters respond to voltage under the ENA Engineering Recommendations, and quantifies the thermal and yield penalties that arise when the inverter's apparent-power capability becomes the limit. It is intended for power-quality consultants, solar specifiers and network automation planners, and it corrects a number of points that are often overstated in summary descriptions of the topic.

The Vector Mathematics of Apparent Power and Displacement

An alternating-current circuit exchanges two kinds of power with its source. Active power, P, measured in watts, is the component that does useful work and is averaged over a cycle. Reactive power, Q, measured in volt-amperes reactive (var), is the component that oscillates between the source and the energy stored in the circuit's electric and magnetic fields, with zero average value. The two combine as a complex quantity, the complex power S = P + jQ, whose magnitude is the apparent power, |S| = (P² + Q²)^1/2, measured in volt-amperes (VA). The three form the sides of the power triangle, with P and Q at right angles and S as the hypotenuse.

The phase angle φ between the fundamental voltage and the fundamental current determines the division of the apparent power: P = S × cos φ and Q = S × sin φ. The displacement power factor is cos φ, and it is equal to P / S for sinusoidal waveforms. At unity power factor the current is in phase with the voltage, Q is zero and S equals P. As φ increases, Q grows and a larger apparent power, and hence a larger current, is required to deliver the same active power. The current is I = S / V, or for a single-phase 230 V supply I = P / (V × cos φ).

Sign Conventions and Distortion

The sign of Q depends on the convention, and the confusion it causes is a frequent source of error in specifying inverter settings. In the generator convention commonly used for inverters, a unit that injects reactive power into the network behaves like a capacitor, with the current leading the voltage, and is said to be over-excited or leading. A unit that absorbs reactive power behaves like an inductor, with the current lagging the voltage, and is under-excited or lagging. For the purposes of voltage control, injection tends to raise the local voltage and absorption to lower it.

The displacement power factor considers only the fundamental. The true power factor also takes account of harmonic distortion in the current, and is PF = cos φ / (1 + THD_I²)^1/2, where THD_I is the total harmonic distortion of the current. For a modern inverter with a current distortion of 3% to 5%, the distortion factor is between 0.9989 and 0.9997, so that it is the displacement, and not the harmonics, that dominates the power factor at normal loading. Distortion matters more at low output, where the fundamental current is small and the harmonic content, determined by the switching and the filter, is a larger share. The relevant limits on harmonic current emissions are set by product standards and by the connection requirements, and a specifier should confirm that the inverter's declared performance meets them.

Grid Impedance and the Voltage Rise Vector

The voltage change along a feeder with series impedance Z = R + jX, carrying active power P and reactive power Q at a nominal voltage V, is approximated for small angles by ΔV ≈ (R × P + X × Q) / V. The approximation is the classical line-regulation formula and holds well when the voltage drop is small compared with the nominal voltage. For a load, the P and Q are positive and the voltage falls along the feeder. For a generator, the power flows the other way and the voltage rises. In the generator convention, with Q positive when the unit injects reactive power, the voltage rise at the point of connection relative to the substation is ΔV ≈ (R × P + X × Q) / V, so that the active power always raises the voltage, injected reactive power raises it further, and absorbed reactive power, being negative Q, reduces it.

The Effect of the R/X Ratio

The effectiveness of reactive power control depends on the ratio of resistance to reactance. In high-voltage transmission networks, X far exceeds R and voltage is controlled almost entirely by reactive power. In low-voltage distribution networks, particularly those of underground cables and thin overhead conductors, R is comparable with or greater than X, and the ratio R/X is typically between 1 and 5. To cancel the resistive rise completely, the inverter would have to absorb reactive power Q = P × R / X, which is a multiple of its active power for any value of R/X above 1, and is far beyond the apparent power capability of the unit. Reactive power can therefore only reduce part of the rise, and the fraction depends on the impedance.

A numerical example illustrates the scale. Take a single-phase inverter at the G98 maximum of 3.68 kW (16 A at 230 V) connected at a point with a loop impedance, phase plus neutral, of R = 0.35 ohms and X = 0.12 ohms, a ratio of R/X of 2.9. At unity power factor the voltage rise is 0.35 × 3,680 / 230 = 5.6 V. With the inverter absorbing reactive power at a power factor of 0.95, Q = 3,680 × tan(acos 0.95) = 1,210 var and ΔV = (0.35 × 3,680 − 0.12 × 1,210) / 230 = 4.97 V, a reduction of 0.6 V or 11%. At a power factor of 0.90, Q = 1,782 var and ΔV = 4.67 V, a reduction of 0.9 V or 17%. The reduction per unit of reactive power is X / V = 0.52 millivolts per var, compared with R / V = 1.5 millivolts per watt for active power, a ratio of R/X of 2.9 in favour of active power.

For comparison, achieving the same voltage reduction by cutting active power requires ΔP = ΔV × V / R. To reduce the voltage by 0.9 V requires a cut of 0.9 × 230 / 0.35 = approximately 590 W, which is 16% of the rated output, compared with an apparent power capacity cost of nothing, so long as the inverter has spare apparent power. This asymmetry is the central point: reactive power is the cheaper first resort, because it costs no energy yield until the apparent-power limit is reached, but it is the weaker tool, and when the voltage must be reduced by more than it can deliver, active power reduction is the only remaining option.

The effect on a street is cumulative. The voltage at the end of a feeder carrying many inverters is the sum of the rises due to the currents of all the upstream inverters flowing through the shared impedance, so that the individual contributions of a few tenths of a volt accumulate to several volts at the point of greatest rise. A DNO that sets the substation voltage towards the top of the range, for example at 1.04 or 1.05 per unit (about 239 to 242 V) to compensate for load-driven drops at peak demand, leaves only about 11 to 14 V of headroom to the 253 V limit before any generation is added, and a street with a few kilowatts of solar on each of several dozen roofs can consume it.

G98 and G99 Voltage Control Loops

The requirements for connecting generation to the GB distribution networks are set out in the ENA Engineering Recommendations G98, for small-scale generation up to 16 A per phase, which is 3.68 kW on a single phase, and G99 for larger installations. Both build on the European standard EN 50549-1 and specify protection settings and the functional capabilities of the generating unit. Among the protection settings are over-voltage trip thresholds, which as I understand them include a stage at 262.2 V (230 V plus 14%) with a trip time of the order of a second, a further stage at about 273.7 V with a shorter trip time, and a requirement that the unit also disconnects if the rolling ten-minute average of the voltage exceeds 258.2 V. The ten-minute average limit is a frequent source of nuisance disconnections on high-voltage streets, since a sustained voltage in the region of 255 to 258 V at the terminals, which is within the protection limits for brief intervals but not as an average, will cause the inverter to trip, losing generation until the voltage falls and the reconnection delay expires.

It should be stated carefully what the documents require of reactive power. The summary that G98 and G99 mandate Volt-VAR control is stronger than the position as I understand it. The standards require the capability to operate in several modes: a fixed power factor, a fixed reactive power, a reactive power that varies with voltage, Q(U), and a reduction of active power at high voltage, P(U), often called Volt-Watt. Which mode is enabled by default, and to what settings, depends on the issue of the document, the size of the unit and the requirements of the DNO, and the common default for small units is operation at unity power factor with the protection thresholds applied. The DNO may request a different mode or setting for a particular connection, particularly where network studies show a voltage constraint. The specifier should check the current issue of G98 or G99, the manufacturer's declaration and the DNO's connection offer for the mode and settings that apply. In other markets, notably Australia under AS/NZS 4777.2, Volt-VAR and Volt-Watt modes are enabled by default with prescribed curves, and some of the commonly circulated descriptions derive from that experience.

The Q(U) Droop Curve

In Q(U) mode the inverter's microcontroller measures the voltage at its terminals, typically as a filtered RMS value, and sets its reactive power according to a piecewise-linear droop curve. An illustrative curve, expressed as a fraction of the rated apparent power, has a dead band in which Q is zero up to a voltage V_a of 240 V, then ramps linearly to a maximum absorption at a voltage V_b, for example 253 V, where Q = −0.436 × S_rated, corresponding to a power factor of 0.9. Between those points Q(V) = −Q_max × (V − V_a) / (V_b − V_a), and above V_b the absorption is held at its maximum. A similar curve with a negative slope below a lower voltage injects reactive power to support low voltages. The response is smoothed by a first-order filter whose time constant, the open-loop response time, is typically several seconds, to avoid the inverter responding to transients.

The behaviour of an aggregation of such inverters on a shared feeder raises a stability question. Each unit senses a voltage that is influenced by its own reactive power and by that of the others. The sensitivity of voltage to reactive power at the shared point is X_c / V, in volts per var, and the droop slope is m = Q_max / (V_b − V_a) in vars per volt. The loop gain of the aggregate is approximately N × m × X_c / V, where N is the number of inverters and X_c the common reactance. With Q_max = 1,604 var and a voltage span of 13 V, m is 123 var per volt. With X_c = 0.06 ohms, the contribution of each inverter to the loop gain is 123 × 0.06 / 230 = 0.032, and for 20 inverters it is 0.64. If the aggregate gain approaches or exceeds one, in combination with the delays in the measurement and filtering, the voltage can oscillate, with the inverters hunting between absorbing and releasing reactive power. The open-loop response time and the slope are therefore design parameters of the whole network, not of individual units, and the permitted ranges in the standards reflect this concern.

Priority and the Volt-Watt Backstop

The inverter's apparent-power capability is the constraint that couples the two functions. If the unit is limited to S_max, then P_max = (S_max² − Q²)^1/2, and any reactive power commanded reduces the active power that can be delivered when the unit is at its limit. A well-designed controller gives priority to one or the other. In a combined strategy the Q(U) curve acts first, absorbing reactive power as the voltage begins to rise, and if the voltage continues to rise beyond the capability of reactive power, the Volt-Watt curve then reduces active power, typically ramping from full output at a threshold of the order of 250 V to a lower limit at about 253 V to 260 V, depending on the settings. The sequence ensures that yield is sacrificed only after the cheaper option is exhausted.

Inverter Bridge Thermal Stress, Ohmic Losses and Real Power Clipping

At the hardware level, an inverter is a power-electronic bridge, usually a full or half bridge of insulated-gate bipolar transistors (IGBTs) or, increasingly, silicon carbide MOSFETs, switching at tens of kilohertz and followed by an output filter that recovers the sinusoidal waveform. The ratings that matter are the maximum current of the switches, the thermal capability of the heat sink and the maximum junction temperature, which together set the apparent power S_max the unit can sustain. The switches do not distinguish active from reactive current: the current in the bridge is the total current, which is I = S / V.

The consequence is that, for a given active power, a lower power factor requires a higher current. At a power factor of 0.95 the current is 1 / 0.95 = 1.053 times the unity value, a rise of 5.3%, and at 0.90 it is 1.111 times, a rise of 11.1%. The conduction losses in the switches and in the copper of the filter inductor are proportional to the square of the current, so that they increase by 10.8% at 0.95 and by 23.5% at 0.90. If an inverter of 97% efficiency has total losses of 3% of 3.68 kW, or 110 W, and half of that is conduction loss, then the additional loss is approximately 6 W at 0.95 and 13 W at 0.90. These are modest in absolute terms, a change in efficiency of the order of 0.15 to 0.35 percentage points, but they appear at the time of peak output, when the heat sink is hottest, and so bear directly on the thermal margin.

A further effect applies to single-phase inverters. The instantaneous power delivered to a single-phase grid pulsates at twice the grid frequency, with an amplitude equal to the apparent power S. The DC-link capacitor must absorb this 100 Hz power pulsation, and its ripple current and temperature rise are determined by S and not by P. A reactive power set-point therefore increases the ripple stress on the DC-link capacitors, which are among the life-limiting components of the unit, by an amount proportional to the increase in S. Electrolytic capacitors in particular have a life that falls exponentially with temperature, roughly halving for every 10 K rise, so a sustained increase in ripple heating at the hottest part of the day matters for long-term reliability.

Clipping and the Yield Penalty

The yield consequence follows from the power triangle. If the inverter is limited to S_max = 3.68 kVA, the maximum active power at a power factor of 0.95 is 3.68 × 0.95 = 3.50 kW, and at 0.90 it is 3.31 kW, reductions of 5% and 10% of the peak, but only at times when the array is producing at or beyond that power. Many arrays are installed with a DC capacity larger than the inverter's AC rating, commonly by a ratio of 1.2 to 1.4, so that the inverter already clips at the peak of a clear day. The additional loss from the lower P_max is the extra energy that falls between the new limit and the old one. Because the fraction of annual energy produced in the narrow band of highest power is small in the UK, where the climate is cloudy and the sun is low for much of the year, the annual yield penalty of a power factor setting of 0.95 is generally a small fraction of a percent to perhaps a couple of percent, depending on the DC to AC ratio, the orientation and the site. The figure should be estimated for the specific array from simulation, but it is seldom large, and it falls at times of the year when the power is lower.

The second point is that the penalty does not apply at all while there is spare apparent power. At an output of 2 kW from a 3.68 kVA inverter, the inverter can absorb up to (3.68² − 2²)^1/2 = 3.09 kvar without any loss of active power. Reactive power capability is therefore free for much of the day, and the energy cost arises only in the narrow window at the top of the output curve. This asymmetry argues for settings which apply reactive power responsively, as in Q(U) mode, so that the apparent power is used only when the voltage demands it, rather than a fixed power factor that applies throughout the day and raises current and losses when the network does not need support.

The reductions in voltage rise and the power figures in the matrix come directly from the worked example and the stated impedances, which are illustrative. At a different point on the network, with a different R and X, the numbers change in proportion, but the structure is general: reactive power is a cheap but partial remedy on a network with a high R/X ratio, and active power reduction is the guaranteed backstop.

Specification and Commissioning Guidance

For installers and specifiers, several practical conclusions follow. First, confirm the applicable connection process, G98 for units up to 16 A per phase or G99 above, the issue of the document and the DNO's offer, and record the mode and settings that are required, since they differ between networks. Second, understand the voltage environment before commissioning. A measurement of the terminal voltage over several days, including midday, will show whether the site is close to the 253 V limit or the 258.2 V ten-minute average, and is more informative than a theoretical calculation. Third, prefer responsive modes, Q(U) with appropriate settings or a combined Q(U) and P(U) strategy, to a fixed power factor, where the DNO and the standard permit, because they use apparent power only when voltage requires it and so incur the current, loss and capacitor-stress penalties only at the times that matter.

Fourth, size the inverter with the apparent-power budget in mind. A unit with a larger S_max than the array's peak active power has headroom for reactive power at no cost to yield, which is an argument for avoiding an aggressive DC to AC ratio on constrained feeders. Fifth, check the thermal design of the installation: an inverter in a hot loft or an enclosed cupboard has less thermal margin, and the extra losses at peak apparent power may trigger temperature derating, which curtails active power just as surely as a voltage limit. Sixth, where a street is persistently over-voltage, report it to the DNO, since the remedy may lie with the network, for example through a change of the substation tap setting or reinforcement of the feeder, and the inverter's control modes are a mitigation and not a substitute.

The underlying lesson is that the inverter's power factor setting is a trade between three scarce resources: the voltage headroom of the street, the apparent-power and thermal capability of the hardware, and the yield of the array. A setting that is chosen with an understanding of the impedance, the loop dynamics and the apparent-power budget can protect all three, and one that is chosen by default or by a misreading of the standard can cost yield, shorten the life of the unit, and fail to prevent the over-voltage trips it was intended to avoid.