Mid-line reactor for Vr = Vs at no load

Treat the line as lossless for the reactor calculation (as usual; $r$ has a negligible effect, checked below). Split it into two equal halves, each with electrical length $\theta = \beta l/2$, with the reactor $jX_L$ at the midpoint.

Line constants:

$$\begin{aligned} \beta &= \omega\sqrt{LC} = 314.16\sqrt{1\times10^{-3} \times 10\times10^{-9}} = 9.9346\times10^{-4}\ \text{rad/km} \\ \beta l &= 9.9346\times10^{-4} \times 500 = 0.4967\ \text{rad} = 28.46^\circ \\ \theta &= \beta l/2 = 0.2484\ \text{rad} = 14.23^\circ \\ Z_c &= \sqrt{L/C} = \sqrt{10^{-3}/10^{-8}} = 316.23\ \Omega \end{aligned}$$

Without the reactor: $V_r/V_s = 1/\cos 28.46^\circ = 1.137$ pu.

Derivation

Second half (midpoint M to the open receiving end, $I_r = 0$):

$$V_m = V_r\cos\theta, \qquad I_{m2} = j\frac{V_r}{Z_c}\sin\theta$$

At M: the current leaving the first half feeds the reactor and the second half:

$$I_{m1} = \frac{V_m}{jX_L} + I_{m2} = -j\frac{V_r\cos\theta}{X_L} + j\frac{V_r\sin\theta}{Z_c}$$

First half:

$$\begin{aligned} V_s &= V_m\cos\theta + jZ_c\sin\theta\,I_{m1} \\ &= V_r\left[\cos^2\theta - \sin^2\theta + \frac{Z_c}{X_L}\sin\theta\cos\theta\right] \\ &= V_r\left[\cos\beta l + \frac{Z_c}{2X_L}\sin\beta l\right] \end{aligned}$$

Condition $V_r = V_s$

$$\cos\beta l + \frac{Z_c}{2X_L}\sin\beta l = 1 \;\Rightarrow\; X_L = \frac{Z_c\sin\beta l}{2(1 - \cos\beta l)} = \frac{Z_c}{2}\cot\frac{\beta l}{2}$$ $$\begin{aligned} X_L &= \frac{316.23}{2} \times \cot(14.23^\circ) \\ &= 158.11 \times 3.943 = 623.5\ \Omega \end{aligned}$$

Check: $\cos 28.46^\circ + \dfrac{316.23}{2 \times 623.5}\sin 28.46^\circ = 0.8792 + 0.2536 \times 0.4766 = 1.000$ ✓. Including $r = 0.031\ \Omega$/km, the same reactor gives $|V_s/V_r| = 1.00002$, so neglecting $r$ is justified.

Reactor rating (3-phase, 400 kV): $Q = 400^2/623.5 = 256.6$ MVAr.

 Vs o------ 250 km ------+------ 250 km ------o Vr
                         |
                        jXL = 623.5 ohm
                         |
 ------------------------+--------------------- gnd

Answer: $X_L \approx$ 623.5 Ω per phase (about 257 MVAr) at the midpoint.