H2M robot: a new prototype robot for insulation of high voltage transmission

Abstract

There are numerous uncovered long-distance overhead cables in the distribution network system which can be inappropriate in densely populated urban. Therefore, in most countries, the Electric Power Corporations (EPCO) are attempting to find a solution to replace the uncovered overhead lines with covered one. Since, these replacements are not economic, some corporations constructing a covering apparatus for insulating electrical distribution lines. This paper proposes a first and novel prototype robot for on-line insulating overhead robot. The suggested tele-operated mobile wrapped insulator cover on overhead cables without any experience of a skilled technician. The experiment results demonstrated that the proposed mechanism is an appropriate methodology for insulating of the bare wire. In future, by developing the robot, EPCOs will utilize it on live high voltage line to (1) reduce the power outage, as well as operation and maintenance costs, (2) save resources and time, (3) prevent the electrical interruptions caused by contacting trees and animals with overhead lines and (4) increase the power network reliability, significantly.

Appendices

Appendix A

There is a simple relation between the travelling and wrapping velocity to cover a cable as follows

$$ \dot{\theta } = \frac{{2 \times \pi \times d_{2} d_{3} }}{{OL \times W \times d_{1} }}\dot{\varphi } $$

(1)

where OL is the overlap length coefficient and W is width of tape (Fig. 13). The parameters $\dot{\varphi }$ and $\dot{\theta }$ are the angular velocity of the DC motor of the travelling unit and wrapping unit, respectively. See Fig. 13. The $d_{1}$, $d_{2}$ and $d_{3}$ are 60, 150 and 70 mm, respectively. Moreover, the W and OL are investigated 20 mm and 10%, respectively.

Fig. 13

The key physical parameters on wrapping velocity

Appendix B

2.1 Control system design

Since a cable is a flexible object with uniform weight per unit length, it in a single span of a transmission line can be described by a set of hyperbolic functions which describe catenary curves (Thayer 1924). Therefore, the constant speed DC motor controller is designed for controlling two wrapping and travelling speed on the cable. In this robot a cascade controller is employed. Figure 14 shows the structure of the proposed cascade controller. The overall control scheme is composed of two main blocks: (1) a serial PI-based velocity control loop that provides a reference command current and (2) a serial PI-based current control loop generating the reference voltage.

Fig. 14

The block diagram of the cascade controller loop

Figure 15 indicates the overall structure of the current loop. The open-loop transfer function of the current controller is obtained as

Fig. 15

The current control loop

$$ G_{{{\text{loop}}}} \left( s \right) = {\text{PI}}\left( s \right) \times \frac{{I_{{\text{a}}} \left( s \right)}}{{V_{{\text{a}}} \left( s \right)}} = \left( {\frac{{K_{{\text{p}}}^{{{\text{series}}}} K_{{\text{i}}}^{{{\text{series}}}} \left( {1 + \frac{s}{{K_{{\text{i}}}^{{{\text{series}}}} }}} \right)}}{s}} \right) \times \left( {\frac{{\frac{1}{{R_{{\text{a}}} }}}}{{1 + \frac{{L_{{\text{a}}} }}{{R_{{\text{a}}} }}s}}} \right) $$

(2)

The back electromotive force (back EMF), e_b, is depending on rotor speed which is a mechanical parameter. Therefore, since the mechanical time constant is much slower than the electrical time constant, the e_b is ignored in the transfer function. The ${\text{K}}_{\text{i}}^{\text{series}}$ is defined as

$$ {\text{K}}_{\text{i}}^{\text{series}} = \frac{{R_{a} }}{{L_{a} }} $$

(3)

In other words, the value of the ${\text{K}}_{\text{i}}^{\text{series}}$ is selected as the zero of a PI controller to neutralize the pole of the transfer function of the current to voltage. This satisfies the stability of the inner current loop and we have a first order transfer function. The ${\text{K}}_{\text{p}}^{\text{series}}$ is obtained as

$$ {\text{G}}_{current} \left( {\text{s}} \right) = \frac{1}{{1 + \frac{{L_{a} }}{{{\text{K}}_{\text{p}}^{\text{series}} }}{\text{s}}}} = > {\text{K}}_{\text{p}}^{\text{series}} = L_{a} \times {\text{Bandwidth}} $$

(4)

Based on the Eqs. 3 and 4 the parameters of the PIs of current loops are determined by the resistance and inductance of the armature coils. To identify these characteristics, a cosine voltage, $V = 100 \cos (100 \times t)$, is applied to DC motors. In this case, the DC motor does not move ($e_{b} = 0$). Now, the current and voltage of motor terminal are measured. The measured voltage is defined as

$$ v_{m} = V_{m0} \cos \left( {\omega_{0} t + \theta } \right) $$

(5)

The transfer function of current to voltage can be defined as

$$ G\left( {j\omega } \right) = \frac{{I_{a} (j\omega )}}{{V_{a} (j\omega )}} = \frac{{\frac{1}{{R_{a} }}}}{{1 + j\frac{{L_{a} }}{{R_{a} }}\omega }} $$

(6)

Therefore, the measured current can be obtained as

$$ i_{m} = V_{m0} \left| {G\left( {j\omega_{0} } \right)} \right|\cos \left( {\omega_{0} t + \theta + \measuredangle G(j\omega_{0} )} \right) $$

(7)

where the magnitude an phase of the transfer function is defined as

$$ \left| {G\left( {j\omega_{0} } \right)} \right|^{2} = \frac{1}{{R_{a}^{2} (1 + (\frac{{L_{a} \omega_{0} }}{{R_{a} }})^{2} )}} $$

(8)

$$ \measuredangle G\left( {j\omega_{0} } \right) = - \tan^{ - 1} \left(\frac{{L_{a} \omega_{0} }}{{R_{a} }}\right) $$

(9)

By rewriting Eq. (5), we have

$$ v_{m} = V_{m0} \cos \left( {\omega_{0} t + \theta } \right) = V_{m0} \cos \left( \theta \right)\cos \left( {\omega_{0} t} \right) - V_{m0} \sin \left( \theta \right)\sin \left( {\omega_{0} t} \right) $$

(10)

Now we define

$$ \begin{aligned} V_{m0} \cos \left( \theta \right) = a \hfill \\ - V_{m0} \sin \left( \theta \right) = b \hfill \\ \end{aligned} $$

(11)

We have

$$ v_{m} = a\cos \left( {\omega_{0} t} \right) + b\sin \left( {\omega_{0} t} \right) $$

(12)

By multiplying both sides of Eq. (12) by $\cos \left( {\omega_{0} t} \right)$ and integrate from 0 to T, we have

$$ \begin{gathered} \int\limits_{{t = 0}}^{T} {v_{m} } \cos \left( {\omega _{0} t} \right)dt \hfill \\ \quad = \int\limits_{{t = 0}}^{T} {\left( {a\cos \left( {\omega _{0} t} \right) + b\sin \left( {\omega _{0} t} \right)} \right)} \cos \left( {\omega _{0} t} \right)dt \hfill \\ \quad = \int\limits_{{t = 0}}^{T} {\left( {a\cos \left( {\omega _{0} t} \right)^{2} + b\sin \left( {\omega _{0} t} \right)\cos \left( {\omega _{0} t} \right)} \right)} dt \hfill \\ \quad = \int\limits_{{t = 0}}^{T} {\left( {a\frac{{(1 + \cos \left( {2\omega _{0} t} \right))}}{2} + \frac{1}{2}b\sin \left( {2\omega _{0} t} \right)} \right)} dt \hfill \\ \quad = \frac{1}{2}\int\limits_{{t = 0}}^{T} a dt + \frac{1}{2}\int\limits_{{t = 0}}^{T} a \cos \left( {2\omega _{0} t} \right)dt + \frac{1}{2}\int\limits_{{t = 0}}^{T} b \sin \left( {2\omega _{0} t} \right)dt = \frac{T}{2}a \hfill \\ \end{gathered} $$

(13)

Moreover, by multiplying both sides of Eq. (12) by $\sin \left( {\omega_{0} t} \right)$ and integrate from 0 to T, we have

$$ \mathop \int \limits_{t = 0}^{T} v_{m} \sin \left( {\omega_{0} t} \right)dt = \frac{T}{2}b $$

(14)

Based on Eqs. (11), (13) and (14) we have

$$ \left( {\frac{T}{2}a} \right)^{2} + \left( {\frac{T}{2}b} \right)^{2} = \left( {\frac{T}{2}} \right)^{2} \left( {\left( {V_{m0} \cos \left( \theta \right)} \right)^{2} + \left( { - V_{m0} \sin \left( \theta \right)} \right)^{2} } \right) = \left( {\frac{T}{2}} \right)^{2} V_{m0}^{2} $$

(15)

$$ \frac{{\left( {\frac{T}{2}b} \right)}}{{\left( {\frac{T}{2}a} \right)}} = - \tan \theta $$

(16)

where T is the overall time of the applying voltage to identify characteristics of the motor. Now, by expanding Eq. (7), we have

$$ \begin{aligned} i_{m} & = V_{{m0}} \left| {G\left( {j\omega _{0} } \right)} \right|\cos \left( {\omega _{0} t + \theta + \measuredangle G\left( {j\omega _{0} } \right)} \right) \\ & = V_{{m0}} \left| {G\left( {j\omega _{0} } \right)} \right|\cos \left( {\theta + \measuredangle G\left( {j\omega _{0} } \right)} \right)\cos \left( {\omega _{0} t} \right) \\ & \quad - V_{{m0}} \left| {G\left( {j\omega _{0} } \right)} \right|\sin (\theta + \measuredangle G\left( {j\omega _{0} } \right))\sin (\omega _{0} t) \\ \end{aligned} $$

(17)

we have

$$ \begin{array}{*{20}l} {V_{m0} \left| {G\left( {j\omega_{0} } \right)} \right|\cos \left( {\theta + \measuredangle G\left( {j\omega_{0} } \right)} \right) = c} \\ { - V_{m0} \left| {G\left( {j\omega_{0} } \right)} \right|\sin (\theta + \measuredangle G\left( {j\omega_{0} } \right)) = d} \\ \end{array} $$

(18)

Therefore,

$$ i_{m} = c\cos \left( {\omega_{0} t} \right) + d\sin (\omega_{0} t) $$

(19)

By multiplying both sides of Eq. (19) by $\cos \left( {\omega_{0} t} \right)$ and $\sin \left( {\omega_{0} t} \right)$ and integrate from 0 to $T$, we have

$$ \mathop \int \limits_{t = 0}^{T} i_{m} \cos \left( {\omega_{0} t} \right)dt = \frac{T}{2}c $$

(20)

$$ \mathop \int \limits_{t = 0}^{T} i_{m} \sin (\omega_{0} t)dt = \frac{T}{2}d $$

(21)

According to Eqs. (18), (20) and (21) we have

$$ \left( {\frac{T}{2}c} \right)^{2} + \left( {\frac{T}{2}d} \right)^{2} = \left( {\frac{T}{2}} \right)^{2} \left( {\left( {V_{m0} \left| {G\left( {j\omega_{0} } \right)} \right|\cos \left( {\theta + \measuredangle G\left( {j\omega_{0} } \right)} \right)} \right)^{2} + \left( { - V_{m0} \left| {G\left( {j\omega_{0} } \right)} \right|\sin \left( {\theta + \measuredangle G\left( {j\omega_{0} } \right)} \right)} \right)^{2} } \right) = \left( {\frac{T}{2}} \right)^{2} V_{m0}^{2} \left| {G\left( {j\omega_{0} } \right)} \right|^{2} $$

(22)

$$ \frac{{\left( {\frac{T}{2}d} \right)}}{{\left( {\frac{T}{2}c} \right)}} = - \tan \left( {\theta + \measuredangle G\left( {j\omega_{0} } \right)} \right) $$

(23)

By Considering Eq. (8) and dividing Eq. (22) on Eq. (15), we have

$$ \frac{{\left( {\frac{T}{2}c} \right)^{2} + \left( {\frac{T}{2}d} \right)^{2} }}{{\left( {\frac{T}{2}a} \right)^{2} + \left( {\frac{T}{2}c} \right)^{2} }} = \left| {G\left( {j\omega_{0} } \right)} \right|^{2} = \frac{1}{{R_{a}^{2} \left( {1 + \left( {\frac{{L_{a} \omega_{0} }}{{R_{a} }}} \right)^{2} } \right)}} $$

(24)

where $\frac{T}{2}d$, $\frac{T}{2}c$, $\frac{T}{2}b$ and $\frac{T}{2}a$ are known, we can define

$$ \begin{array}{*{20}c} {\frac{{\left( {\frac{T}{2}c} \right)^{2} + \left( {\frac{T}{2}d} \right)^{2} }}{{\left( {\frac{T}{2}a} \right)^{2} + \left( {\frac{T}{2}c} \right)^{2} }} = M,} & {\frac{{L_{a} \omega_{0} }}{{R_{a} }} = X} \\ \end{array} $$

(25)

Therefore, we have

$$ R_{a} = \sqrt {\frac{1}{{M(1 + X^{2} )}}} $$

(26)

Based on Eqs. (9) and (16), the Eq. (23) can be rewritten as

$$ \frac{{\left( {\frac{T}{2}d} \right)}}{{\left( {\frac{T}{2}c} \right)}} = - \tan \left( {\theta + \measuredangle G\left( {j\omega_{0} } \right)} \right) = - \frac{{\tan \left( \theta \right) + \tan \left( {\measuredangle G\left( {j\omega_{0} } \right)} \right)}}{{1 - \tan \left( \theta \right)\tan \left( {\measuredangle G\left( {j\omega_{0} } \right)} \right)}} = - \frac{{ - \frac{{\left( {\frac{T}{2}b} \right)}}{{\left( {\frac{T}{2}a} \right)}} - \frac{{L_{a} \omega_{0} }}{{R_{a} }}}}{{1 - \left( { - \frac{{\left( {\frac{T}{2}b} \right)}}{{\left( {\frac{T}{2}a} \right)}}} \right)\left( { - \frac{{L_{a} \omega_{0} }}{{R_{a} }}} \right)}} $$

(27)

where $\frac{T}{2}d$, $\frac{T}{2}c$, $\frac{T}{2}b$ and $\frac{T}{2}a$ are known, we have

$$ \begin{array}{*{20}c} {\frac{{\left( {\frac{T}{2}d} \right)}}{{\left( {\frac{T}{2}c} \right)}} = A,} & {\frac{{\left( {\frac{T}{2}b} \right)}}{{\left( {\frac{T}{2}a} \right)}} = B,} & {\frac{{L_{a} \omega_{0} }}{{R_{a} }} = X} \\ \end{array} $$

(28)

We can rewrite Eq. (27) as

$$ X = \frac{A - B}{1 + AB} $$

(29)

By substituting Eq. (29) on Eq. (26), the resistance of rotor coil is obtained. Finally, the inductance of the coils is calculated as

$$ \frac{{L_{a} \omega_{0} }}{{R_{a} }} = X \Rightarrow L_{a} = \frac{{R_{a} X}}{{\omega_{0} }} $$

(30)

Therefore, by Eqs. (26) and (30), the resistance and inductance of the coil are identified, respectively. The sufficient bandwidth of the current-loop is assigned 1 kHz. Now, the Eqs. (3) and (4) are utilized to compute ${\text{K}}_{\text{i}}^{\text{series}}$ and ${\text{K}}_{\text{p}}^{\text{series}}$, respectively. Up to now, the resistance and inductance of the coil are identified. The parameters of the PIs for velocity loops were selected by trial and error.

To consider the ability of the proposed cascade controller, we supposed that the reference angular velocity of the DC motor in travelling unit, $\dot{\varphi }$, is 0.1 rpm. According to Eq. (1), reference angular velocity of the DC motor in wrapping unit, $\dot{\theta }$, is equal to 54 rpm. Figure 16 shows the actual and reference speeds of two DC motors. In the first test, the robot moves with the specified speed when there is not any obstacle on the cable (Fig. 16a, b) as well as moves with the specified speed in order to pass over the obstacles (Fig. 16c). The experimental results have shown the robust performance of the suggested cascade controller scheme and have and put into evidence the ability of the mentioned controller in the our proposed robot.

Fig. 16

The actual and reference velocities with the PI-based cascade control, a the $\dot{\varphi }$ without obstacles, b the $\dot{\theta }$ without obstacles and c the $\dot{\varphi }$ with one obstacles on the cable