1. Introduction

Situation 1 : Shortest connection of two transmission lines with intersection angle $\theta$

In Fig. 1, we see an existing transmission line X (x axis) and another transmission line Y’ (y’ axis) that intersect at point O(0,0) with intersection angle $\theta$. A hydro power plant is under construction at point Q between transmission line X and Y’.

A new transmission line is planned to connect two lines X and Y’ via the power plant. If the coordinate of the power plant Q is (p,q), what is the shortest length and what are the coordinates of x and y’ intercept of the new transmission line? Assume that all transmission lines are straight.

Situation 2 : Minimal connection of two straight roads with intersection angle $\theta$

In Fig. 2, we see along the lakeside a road X (x axis) and another road Y’ (y’ axis) that meet at point O(0,0) with intersection angle $\theta$. A building lies at point Q between road X and Y’.

A military road is planned to connect two roads X and Y’ via this building. We want this new road to be as short as possible, but it must be straight without bends because it may be used for landing of airplanes in case of emergency. And then, the building is to be used as the control tower of the airport. If the coordinate of the building is (p,q), what is the shortest length and what are the coordinates of x and y‘ intercept of new road?

This paper presents a derivation to determine the length of the straight line that minimally connects two axes with intersection angle $\theta$ via a specific point (p,q). A formula, represented by a cube equation w.r.t. the coordinate of specific point (p,q) and intersection angle $\theta$, is derived to obtain the minimal length and x-y’ intercepts of the connection line using optimization technique ⁽¹⁾.

2. Shortest connection of perpendicular x-y axes via specific point (p, q)

Suppose that two axes x and y’ are perpendicular to each other for Situation 1 of Section 1. Let point O(0,0) be the origin where axis x and axis y(=y’) intersect with $\theta =90^{\circ}$ as shown in Fig. 3. Let the length of new transmission line be L, and x intercept of L be point A(a, 0) and y intercept be point H(0, h). Then, we have,

If (p, q) is the coordinate of the power plant Q, L becomes the length of straight line $\overline{A H}$ that passes through specific point Q(p, q), forming the hypotenuse of a right triangle HOA. What is the shortest length of the new transmission line when $\theta =90^{\circ}$? This problem can be formulated as an optimization problem minimizing the length L that passes through specific point Q(p, q). Now we take the length L of new transmission line as the objective function to be minimized^(1-4) such that:

Constraint (2) implies that $\overline{A H}$ is a straight line connecting points A(a,0) and H(0,h) via specific point Q(p, q). To solve above optimization problem in a simpler way, let us rewrite the objective function (1) and constraint (2) as follows:

Defining Lagrange dual function Λ with (1-1) and (2-1):

we obtain the optimality conditions:

where λ is the Lagrangian multiplier.

From (4),

From (5),

Thus, we have from (6) and (7):

from which we finally obtain:

From (9), we obtain h/a - the optimal slope of $\overline{A H}$ via specific point (p, q) - as follows:

Equation (10) implies L, the length of $\overline{A H}$, is minimized when the slope of the new transmission line is given by cube root of the coordinates of specific point Q(p, q).

Substituting (10) into (2) yields:

Thus, we obtain,

and from (10) and (12),

Now we can obtain the following formula for minimal length of the straight line passing through specific point Q(p, q) ⁽⁴⁾.

or

3. Shortest connection of two axes with intersection angle $\theta$ via specific point (p,q)

In Section 2, we derived the minimal length of the straight line forming the hypotenuse of a right triangle via specific point Q(p, q). What if the intersection angle of the two axes is not 90° but acute or obtuse?

For Situation 2 of Section 1, let the road X be axis x and road Y’ be axis y’. Let point O(0,0) be the origin where axis x and axis y’ intersect with intersection angle $\theta$ as shown in Fig. 5.

Let the length of new road be L and the x-axis intercept of L be point A(a, 0). And let the y’ axis intercept of L be point H and the length $\overline{OH}$ be h. Then, the coordinate of point H becomes $(h\cos\theta ,\: h\sin\theta)$ as shown in Fig 5.

If the coordinate of the building is (p,q), how can we connect axis x and axis y’ via specific point Q(p,q) with minimal length? And what are the coordinates of x and y‘ intercept of new road?

The problem of Situation 2 can be formulated as an optimization problem minimizing the length of the straight line $L=\overline{AH}$ that connects x-y‘ axes via specific point Q(p, q).

Length L can be represented by cosine rule as follows:

Now in Fig 5 we take the length L as the objective function to be minimized^(5-6), such that:

Constraint (22) implies that $\overline{A H}$ is a straight line connecting points A(a,0) and H$(h\cos\theta ,\: h\sin\theta)$ via specific point Q(p, q).

To solve above optimization problem in a simpler way, let us rewrite the objective function (21) and constraint (22) as follows:

Defining Lagrange dual function D with (21-1) and (22-1):

we obtain the optimality conditions:

Rewriting (24) and (25) w.r.t. the Lagrangian multiplier μ, we have:

Equalizing (26) and (27) yields:

Let us rewrite Equation (28) so that variables a and h occur only on RHS such that

Dividing the numerator and denominator of RHS by $a^{3}$ and rearranging (29), we finally obtain the following formula: [See Appendix A]

or

Equation (30) is the formula that derives the optimal ratio h/a by which minimal connection of two axes with intersection angle $\theta$ via specific point (p,q) can be obtained. Note that we can obtain the optimal ratio h/a by solving only one Equation (30) which is composed of two variables a and h.

4. Case studies

5. Conclusion

This paper has presented a derivation to determine the length of a straight line that minimally connects two axes x and y’ with intersection angle $\theta$ via a specific point.

A formula, represented by a cube expression w.r.t. the coordinate of the specific point and intersection angle, has been derived to obtain the minimal length and x-y’ intercepts of the connection line using optimization technique.

Case studies have been discussed to confirm if the solutions derived by the proposed formula are optimal.

It is expected that the proposed formula can be a reference for optimal routing of roads, optimal routing of power transmission or communication lines, optimal routing of gas pipes, optimal design of IC circuits and etc. for minimizing the construction, production and/or operation cost.

The authors also hope this paper be used as an optimization lecture note for students majoring in electrical engineering.

Appendix A

Dividing the numerator and denominator of RHS by $a^{3}$ and rearranging (29), we get:

Equation (A1) also can be expressed in a general cubic expression as follows:

or

where

Appendix B

Case study for $\theta =120^{\circ}$and $(p,\: q)=(\sqrt{3},\: 1)$.

Step 1

From (A4), we have:

Substituting $\cos 120^{\circ}= -1/2$ into (A2), we obtain the following relation:

Step 2

From (34) and (B2), we have:

We see in (B3) that the coordinate of x axis intercept of line L is given(3.8639, 0).

Step 3

Step 4

We have;

and the coordinate of y’ intercept of line L:

Appendix C

Calculation of transmission line impedance for a nuclear power plant

In Fig C1, we see along the seashore a T/L(transmission line) X (x axis) and T/L Y’ (y’ axis) that meet at point O(0,0) with intersection angle $45^{\circ}$.

A nuclear power plant is to be constructed at point N between T/L X and T/L Y’. According to the code and standard, a nuclear power plant needs to be connected to at

least two independent substations of 154 kV or higher level against emergency situation of the nuclear plant. (E.g., substation A and substation H in Fig C1)

So, a new T/L is planned to minimally connect two lines X and Y’ via the power plant. If the coordinate of the power plant is N(3km, 1km), what is the location of substation A and what is the series impedance of the new T/L from site N to substation A? where the series impedance per circuit-km of 345 kV T/L constructed for the power plant is given 0.0015+j0.0255 %/C-km.⁽⁷⁾ Assume that the new T/L is straight from substation A to substation H.

Solution

Step 1

Fig C1 is the case for $\theta =45^{\circ}$and $(p,\: q)=$$(3 km,\:$ $1km)$.

From (A4), we have:

Substituting $\cos 45^{\circ}=\dfrac{1}{\sqrt{2}}$ into (A3), we obtain the following relation:

Step 2

From (34) and (C2), we have:

We see in (C3) that the location of substation A(Coordinate of x intercept of new T/L) is given (3.4780km, 0).

Distance L from site N and substation A is:

Thus, the total series impedance per circuit of new T/L from site N and substation A is given:

Optimal routing of the superconducting cable - which requires very high operation cost for maintaining cryogenic temperature of the superconductor in addition to very high cost for construction - can be one example for path optimization in the power sector.