MV Capacitor Bank Sizing

\[ Q_c = P \times \left(\tan\varphi_1 - \tan\varphi_2\right) \]

Where \(P\) is maximum active power demand (kW), \(\varphi_1 = \cos^{-1}(\text{pf}_{existing})\), and \(\varphi_2 = \cos^{-1}(\text{pf}_{target})\).

\[ S_{existing} = \frac{P}{\text{pf}_{existing}} \qquad S_{new} = \frac{P}{\text{pf}_{target}} \qquad \Delta S = S_{existing} - S_{new} \]

\[ Q_{c,inst} = \left\lceil \frac{Q_c}{50} \right\rceil \times 50 \]

The bank is specified by its output at system voltage, in 50 kVAr steps — also for a detuned bank, whose steps are sold by effective output. Every result below is computed from \(Q_{c,inst}\).

\[ f_r = f_n \times \sqrt{\frac{S_{sc}}{Q_{c,inst}}} \]

Where \(S_{sc}\) is the short-circuit capacity at the PCC (kVA) and \(Q_{c,inst}\) is the installed bank output (kVAr). This is the resonance of the bank without a reactor; the harmonic order is \(n_r = f_r / f_n\).

A harmonic above 3% is significant. With reactor_selection = auto, detuning is required when \(f_r\) lies within ±20% of a significant harmonic (3rd, 5th, 7th, 11th or 13th). The reactor is then chosen by the lowest significant harmonic, so that the bank is tuned below every significant harmonic with margin for capacitance tolerance and ageing (both raise the tuning frequency):

Lowest significant harmonic	Reactor \(p = X_L/X_C\)	Tuning order \(1/\sqrt{p}\)	At 50 Hz	At 60 Hz
3rd	14%	2.67	134 Hz	160 Hz
5th	7%	3.78	189 Hz	227 Hz
7th or above	5.67%	4.20	210 Hz	252 Hz

Practice per Schneider Electric Electrical Installation Guide ch. L (detuned reactors) and ABB Power factor correction and harmonic filtering in electrical plants. Any other reactor_selection forces the choice. The tuning frequency of the capacitor + reactor branch is:

\[ f_t = \frac{f_n}{\sqrt{p}} \]

Undetuned bank: \(f_r\) must not lie within ±20% of any significant harmonic. Detuned bank: \(f_t\) must lie below the lowest significant harmonic, so the bank is inductive at every significant harmonic (IEEE 1036). With auto the selection rule satisfies the check; a forced choice (e.g. no reactor near a significant harmonic, or 5.67% with a significant 3rd) fails it.

Per phase, the capacitor (\(X_C\)) and reactor (\(X_L = pX_C\)) are in series:

\[ Q_{eff} = \frac{V^2}{X_C - X_L} = \frac{V^2}{X_C(1-p)} \qquad V_C = \frac{V}{1-p} \qquad Q_C = \frac{V_C^2}{X_C} = \frac{Q_{eff}}{1-p} \]

The bank output at system voltage is \(Q_{eff} = Q_{c,inst}\). The capacitors run at \(V_C\) and generate \(Q_C\); the reactor absorbs \(pQ_C\). With a rated voltage \(V_{rated} = 1.1\,V_C\) the capacitor nameplate rating is

\[ Q_{N} = Q_C\left(\frac{V_{rated}}{V_C}\right)^2 = \frac{1.21\,Q_{c,inst}}{1-p} \]

(IEEE 1036; equivalently \(Q_N = Q_{eff}(1-p)(V_N/V)^2\) as used in manufacturers' detuned-filter tables). Without a reactor \(p = 0\): \(V_C = V\), \(Q_C = Q_{c,inst}\), \(Q_N = 1.21\,Q_{c,inst}\). The standard voltage is the next value at or above \(V_{rated}\) in 3.3, 6.6, 7.2, 11, 12, 13.2, 13.8, 22, 24, 33 kV, continuing above 33 kV with the IEC 60038 \(U_m\) values 36, 40.5, 52, 72.5 kV.

\[ Q_{after} = P\tan\varphi_1 - Q_{c,inst} \qquad \text{pf} = \frac{P}{\sqrt{P^2 + Q_{after}^2}} \]

The target is met when the power factor is lagging (\(Q_{after} \ge 0\)) and at or above the target. A negative \(Q_{after}\) means the bank over-corrects to a leading power factor: this fails the target whatever its magnitude, because it raises the bus voltage, gets worse at lighter load and risks motor self-excitation. Use smaller steps or an automatic stepped bank.

\[ I_{rated} = \frac{Q_{c,actual}}{\sqrt{3} \times V_{LL}} \]

The harmonic current multiplier for a detuned bank is \(k_h = 1/(1 - p)\). Maximum expected current with a 30% margin:

\[ I_{max} = I_{rated} \times k_h \times 1.3 \]

\[ X_C = \frac{V_C^2}{Q_C} \qquad C_{phase} = \frac{1}{2\pi f_n X_C} \]

Where \(X_C\) is the capacitive reactance from the capacitor voltage and the capacitors' own reactive power (for an undetuned bank \(V_C = V_{LL}\), \(Q_C = Q_{c,inst}\)), and \(C_{phase}\) is the capacitance per phase for a Wye-connected bank.

\[ X_L = p \times X_c \qquad L_{phase} = \frac{X_L}{2\pi f_n} \]

Reactor resistance is estimated assuming \(X/R = 50\) (typical for MV detuning reactors):

\[ R_L = \frac{X_L}{50} \]

\[ I_{CB} = 1.35 \times I_{rated} \]

The calculated value is rounded up to the next standard breaker size (16 … 2000 A, then the IEC 60059 R10 series 2500, 3150, 4000, 5000, 6300 A …). Overload setting: \(1.25 \times I_{rated}\). Unbalance alarm at 5% and trip at 10% of \(I_{rated}\).

\[ R_{discharge} = \frac{-t} {C_{phase} \times \ln\!\left(\dfrac{V_{target}}{V_{peak}}\right)} \]

Where \(V_{peak} = V_{cap} \times \sqrt{2}\) (converted to volts), \(t = 300\) s (5 minutes per IEEE 18), and \(V_{target} = 50\) V.

\[ I_{inrush} \approx 15 \times I_{rated} \]

Typical range is 10–20 × rated current. A pre-insertion resistor to limit inrush to 10× rated:

\[ R_{pre} = \frac{V_{cap} \times 1000}{10 \times I_{rated}} \]

\[ E_{phase} = \tfrac{1}{2}\,C \,V^2 \qquad E_{total} = 3 \times E_{phase} \]

Where \(C\) is in farads and \(V\) is in volts.

\[ \text{Demand savings} = 12 \times \Delta S \times r_{demand} \] \[ \text{Energy savings} = \Delta S \times 0.02 \times h_{year} \times r_{energy} \] \[ \text{Payback} = \frac{\text{Investment}} {\text{Demand savings} + \text{Energy savings}} \]

Where \(\Delta S\) is the kVA demand reduction, \(r_{demand}\) is the demand rate ($/kVA/month), \(r_{energy}\) is the energy rate ($/kWh), and \(h_{year}\) is annual operating hours. The 2% factor approximates reduced system losses.

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