Number of lamps an alternator can add

Lamps are resistive (unity pf), so they add only active power; the alternator limit is its rating, 850 kVA.

Present load (induction motor)

$$\begin{aligned} P_1 &= 400\ \text{kW} \\ Q_1 &= P_1\tan\phi = 400\times\frac{0.6}{0.8} = 300\ \text{kVAR} \\ S_1 &= \frac{400}{0.8} = 500\ \text{kVA} \end{aligned}$$

Maximum active power at 850 kVA with the same kVAR

$$\begin{aligned} P_{max} &= \sqrt{S^2 - Q_1^2} = \sqrt{850^2 - 300^2} = 795.30\ \text{kW} \\ \text{Lamp load} &= 795.30 - 400 = 395.30\ \text{kW} \\ \text{Number of lamps} &= \frac{395.30\times10^3}{100} = 3953 \end{aligned}$$

            S = 850 kVA
          .'|
        .'  | Q = 300 kVAR
      .'    |
    O-------+--------+
     400 kW   395.3 kW lamps

New pf of the alternator $= 795.3/850 = 0.936$ lagging.

Answer: about 3953 lamps of 100 W can be added (395.3 kW).