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Research article

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Journal of the Korean Institute of Illuminating and Electrical Installation Engineers

KIIEE Vol. 36, No. 10, p.36-41

ISSN (print) :

1229-4691

ISSN (online) :

2287-5034

Received : 22 August 2022Revised : 14 September 2022Accepted : 19 September 2022

DOI :

http://doi.org/10.5207/JIEIE.2022.36.10.036

Derivation of Minimum Sending-end Voltage for Constant Load P+jQ in AC Two-bus System

(Gene-Que Lee) 1iD (SangJoong Lee) 2iD (JuChul Kim) †iD
  1. (Research Member, Elmin Electric Co, Seoul, Korea)
  2. (PE, Research Director, Gang Dong E&C, Korea)

†Corresponding Author：Professor, Chuncheon Campus of Korea PolytechnicⅢ, Korea Tel：033-260-7681 E-mail : cjfwnxkq@kopo.ac.kr

Copyright © The Korean Institute of Illuminating and Electrical Engineers(KIIEE)

License :

This work was supported by the Korean Federation of Science and Technology Societies(KOFST) Grant funded by the Korean Government.(www.kiiee.or.kr).

Abstract

For a series circuit with a voltage source and a constant power load connected by an impedance, the magnitude of voltage source must be equal to or greater than a specific threshold value in order to feed its load. This implies that the minimum magnitude of source voltage depends on the magnitude of load power. This paper shows a two-bus power system which is composed of a voltage source, a transmission line impedance and a constant power load. And the authors find out the minimum sending-end voltage that enables the voltage source to successfully feed its load in example calculation. A formula for obtaining the minimum sending-end voltage in general two-bus system with load P+jQ and line impedance r+jx, is also proposed. We have been more accustomed to a load represented by constant impedance, however, in power systems, the loads are represented by kVA or P+jQ in general. The two-bus system can be interpreted as a Thevenin’s equivalent that has a constant power load instead of the external impedance load. In this case, the constant power load can be interpreted as the total consumers’ demand in a power system, and the minimum sending-end voltage can be interpreted as the least system voltage which the power provider must maintain to keep the entire system from voltage collapse. The proposed formula can be a useful tool for power system analysis.

Key words

AC two-bus system, Constant power load, Discriminant, Minimum sending-end voltage

1. Introduction

Fig. 1 is a two-bus system with a complex load $S=12+j4\sqrt{3}$. $\dot{E}$ is the sending-end voltage of bus 1 and $\dot{V}$ is the load voltage of bus 2. Bus 1 and bus 2 are connected thru a transmission line with impedance $z=1+j\sqrt{3}$ [1-4].

Fig. 1. Two-bus system with a complex load $S=12+j4\sqrt{3}$

../../Resources/kiiee/JIEIE.2022.36.10.036/fig1.png

$Question 1 :$ If $\dot{E}=1\angle 0$ in Fig. 1, can this voltage supply the load with $S=12+j4\sqrt{3}$?

To say the answer in advance, it is ‘Impossible’. With $\dot{E}=1\angle 0$, there exists no solution for V. If we have no transmission line impedance, i.e., when $z=0$, every $E>0$ can feed the load $S=12+j4\sqrt{3}$. However, Fig. 1 has a transmission line with impedance $z=1+j\sqrt{3}$, hence, $\dot{E}=1\angle 0$ is too low a voltage to transfer the power $S=12+j4\sqrt{3}$ into the load. In other words, Fig. 1 is a physically impossible circuit if the sending-end voltage E is given ‘1 volt’ [1].

Constant power load, by definition, is the load that consumes constant power[5-7]. For a circuit including a constant power load as shown in Fig. 1, however, the minimum magnitude of source voltage depends on the magnitude of load power. We should investigate that the magnitude of voltage source is high enough to feed constant power $S=12+j4\sqrt{3}$ into load bus 2. Question 1 can be revised to the following Question 2.

$Question 2 :$ We want to extract complex power $S=12+j4\sqrt{3}$ from load bus 2 via transmission line $z=1+j\sqrt{3}$ as shown in Fig. 2 Then, what is the least magnitude of the sending-end voltage E for feeding the load?

Fig. 2. Two-bus system with unknown sending-end voltage E

../../Resources/kiiee/JIEIE.2022.36.10.036/fig2.png

We have been more accustomed to a load represented by constant impedance, however, in power systems, the loads are represented by kVA or P+jQ in general.

The answer for Question 2 is:

If the load $S=12+j4\sqrt{3}$ is given in Fig. 2, the magnitude of source voltage must be equal to or greater than a specific threshold value. On the contrary, if the sending-end voltage E is given, the magnitude of load power must be smaller than a specific threshold value in order to successfully feed its load. This implies that the magnitude of source voltage and the magnitude of load power depend on each other, and no such researches were found by the authors in publications.

In this paper, the authors present a calculation methodology for the minimum sending-end voltage of a two-bus power system with a constant power load as described in the next sections.

2.Calculation of minimum sending-end voltage

Let $\dot{V}=V\angle 0$ in Fig. 2, then, $I_{12}$ - the line current from bus 1 to bus 2 - is represented by [1-3]:

(1)
$\dot{I}_{12}=\left(\dfrac{12+j4\sqrt{3}}{V\angle 0^{\circ}}\right)^{*}=\dfrac{12-j4\sqrt{3}}{V}$

Thus, the vector of sending-end voltage is: [8-9]

(2)
$\dot{E}=\dot{V}+\dot{z}∙\dot{I}_{12}=V+(1+j\sqrt{3})∙\dfrac{12- j 4\sqrt{3}}{V} \\ =\left(V+\dfrac{24}{V}\right)+ j\dfrac{8\sqrt{3}}{V}$

We can get from (2):

(3)
$|\dot{E}| =\sqrt{\left(V+\dfrac{24}{V}\right)^{2}+\left(\dfrac{8\sqrt{3}}{V}\right)^{2}}$

or

(4)
$E^{2}= V^{2}+\dfrac{768}{V^{2}}+48$

The graph for E vs V of (4) is illustrated in Fig. 3. We see that solutions do not exist around E=1 but start to show up in the vicinity of E=10.

Fig. 3. Graph of E vs V

../../Resources/kiiee/JIEIE.2022.36.10.036/fig3.png

Let us find the exact point of E from which the power supply to load is possible.

Rewriting (4):

(5)
$V^{4}-(E^{2}-48)V^{2}+768=0$

Let $v=V^{2}$, then, we have the following quadratic equation with respect to v:

(6)
$v^{2}-(E^{2}-48)v +768=0$

In order for equation (6) to have a solution (or solutions), discriminant D of (6) must be equal to or greater than 0, i.e.,

(7)
$D=\left(E^{2}-48\right)^{2}-4×768≥0$

Let Emin be the sending-end voltage when D=0. Then we have:

(8)
$D=(E_{\min}^{2}-48)^{2}- 3,\: 072 =0$

And we get:

(9)
$E_{\min}^{2}-48 =\sqrt{3,\: 072}$

We finally obtain the minimum sending-end voltage Emin:

(10)
$Emin ≈ 10.17 volt$

Equation (10) implies that the magnitude of sending-end voltage must be equal to or greater than 10.17 volt in order to supply the load with $S=12+j4\sqrt{3}$ thru transmission line $z=1+j\sqrt{3}$Ω. In Fig. 3, we see that no solution for V is found until E reaches 10.17 but solutions begin to show up in the vicinity of E=10.17.

3.Derivation of a formula for minimum sending-end voltage of general AC two-bus system with constant power load

Let $\dot{V}=V\angle 0$ in Fig. 4, then, $I_{12}$ - the line current from bus 1 to bus 2 - is represented by [1-3]:

(11)
$\dot{I}_{12}=\left(\dfrac{P+j Q}{\dot{V}}\right)^{*}=\dfrac{P - j Q}{V}$

Fig. 4. General representation of AC two-bus system with load P+jQ

../../Resources/kiiee/JIEIE.2022.36.10.036/fig4.png

Thus, the vector of sending-end voltage is:

(12)
$\dot{E}=\dot{V}+z\bullet\dot{I}_{12}=V+(r+jx)\bullet\dfrac{P-j Q}{V}\\ =V+\dfrac{1}{V}[(r P+ x Q)+j(x P - r Q)]\\ =\left(V+\dfrac{r P+x Q}{V}\right)+j\dfrac{x P-r Q}{V}$

We have from (12):

(13)
$|\dot{E}| =\sqrt{\left(V+\dfrac{r P+ x Q}{V}\right)^{2}+\left(\dfrac{x P - r Q}{V}\right)^{2}}$

or

(14)
$E^{2}=V^{2}+\dfrac{(r P+x Q)^{2}}{V^{2}}+2(r P+x Q)+\dfrac{(x P-r Q)^{2}}{V^{2}} \\ =V^{2}+\dfrac{(r P+x Q)^{2}+(x P-r Q)^{2}}{V^{2}}+2(r P+x Q)$

We get from (14):

(15)
\begin{align*} V^{4}+\left[2(r P+x Q)-E^{2}\right]\bullet V^{2}\\ \\ +(r P+x Q)^{2}+(x P-r Q)^{2}=0 \end{align*}

Let

(16)
$v=V^{2}$.

Then, we have the following quadratic equation with respect to v:

(17)
$ v^{2}+\left[2(r P+x Q)-E^{2}\right]\bullet v\\ \\ +(r P+x Q)^{2}+(x P-r Q)^{2}=0 $

In order for equation (17) to have a solution (or solutions), discriminant D must be 0 or positive, i.e.,

(18)
$ D=\left[2(r P+x Q)-E^{2}\right]^{2}\\ \\ -4\bullet\left[(r P+x Q)^{2}+(x P-r Q)^{2}\right]\ge 0 $

Let Emin be the minimum sending-end voltage when D=0, then we have:

(19)
$D=\left[2(r P+x Q)-E_{\min}^{2}\right]^{2}-4\left[(r P+x Q)^{2}+(x P-r Q)^{2}\right]=0$

and we get:

(20)

$\left[E_{\min}^{2}-2(r P+x Q)\right]^{2}=4\left[(r P+x Q)^{2}+(x P-r Q)^{2}\right]$

⇒

$E_{\min}^{2}-2(r P+x Q)=2\sqrt{(r P+x Q)^{2}+(x P-r Q)^{2}}$

We finally obtain:

(21)

$E_{\min}=\sqrt{2(r P+x Q)+2\sqrt{(r P+x Q)^{2}+(x P-r Q)^{2}}}$

$=\sqrt{2}\sqrt{r P+x Q+\sqrt{(r P+x Q)^{2}+(x P-r Q)^{2}}}$

Formula (21) gives the minimum sending-end voltage Emin for general AC two-bus system with constant power load P+jQ and transmission line impedance $r+jx$.

$Application of formula (21) to Fig. 2 :$

Let us apply formula (21) to Fig. 2 and compare the result with previous solution (10).

Substituting $r=1,\: x=\sqrt{3},\: P=12,\: Q=4\sqrt{3}$ into formula (21), we obtain:

(22)
$ E_{\min}=\sqrt{2}\sqrt{1\bullet 12+\sqrt{3}\bullet 4\sqrt{3}+(1\bullet 12+\sqrt{3}\bullet 4\sqrt{3})^{2}}\\ \\ \qquad\qquad\qquad\qquad\qquad\qquad\overline{+(\sqrt{3}\bullet 12-1\bullet 4\sqrt{3})^{2}}\\ \\ \qquad\approx 10.17 volt $

We see that the solution (22) by formula (21) is the same as (10).

The two-bus system can be interpreted as a Thevenin’s equivalent that has a constant power load instead of the external impedance load[10]. In this case, the constant power load can be interpreted as the total consumers’ demand in a power system, and the minimum sending-end voltage can be interpreted as the least system voltage which the power provider must maintain to keep the entire system from voltage collapse.

4. Conclusion

This paper has shown that, when a voltage source is connected to a constant power load via a transmission line in a power system, the magnitude of minimum source voltage depends on the magnitude of load power.

The authors have proposed a formula for obtaining the minimum sending-end voltage Emin for a general two-bus system with a constant power load P+jQ and transmission line impedance r+jx. Discriminant of quadratic equation with respect to the load voltage is used to derive Emin.

An example calculation of the minimum sending-end voltage Emin also has been presented for a two-bus system with constant load $S=12+j4\sqrt{3}$ and transmission line impedance $z=1+j\sqrt{3}$. Calculation result has shown Emin= 10.17 volt, which implies that the magnitude of sending-end voltage must be equal to or greater than 10.17 volt in order for the system to successfully feed its load.

The constant power load can be interpreted as the total consumers’ demand in a power system, and the minimum sending-end voltage can be interpreted as the least system voltage which the power provider must maintain to keep the entire system from voltage collapse.

The proposed formula and its derivation process can be a useful tool for power system analysis.

References

1 
Lee G. Q., Kim J. C., 2019, A Wrong Application Case of Equivalent Resistance Method for Voltage Calculation of Distribution System, in Proceedings of 2019 Spring Conference of the Korean Institute of Illuminating and Electrical Installation Engineer, pp. 108DOI
2 
Nilsson J., Riedel S., 2011, Electric Circuits, 9th ed. Pearson, pp. 440Google Search
3 
Lee S. J., Kim J. C., 2020, An Exact Voltage-Power Equation with no Angle Terms for an AC Two-bus System, Trans of KIIE(Korean Institute of Electrical Engineers), Vol. 69, No. 5, pp. 637-643DOI
4 
Song K. Y., 2016, Power Transmission and Distribution, Dong-Il Book, pp. 487Google Search
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Ajjarapu V., 2006, Computational Techniques for Voltage Stability Assessment and Control, Springer, pp. 5-10DOI
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Glover J., Overbye T., Sarma M., 2017, Power System Analysis & Design, 6th ed., Cengage Learning, pp. 309-412Google Search
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Bergen A., Vittal V., 2000, Power Systems Analysis, 2nd ed., Prentice Hall, pp. 325-326Google Search
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Kreyszig E., 2021, Advanced Engineering Mathematics, 10th ed., Wiley, pp. 608-618Google Search
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Alexander C. K., Sadiku M., 2017, Fundamentals of Electric Circuits, 6th ed., McGraw Hill, pp. 137-143Google Search

Biography

Gene-Que Lee
../../Resources/kiiee/JIEIE.2022.36.10.036/au1.png

He graduated from Sejong University in 2010 majoring in Mathematics. He also graduated from SeoulTech University in 2017 majoring in Electrical Engineering. He has worked for Elmin Electric Co. and Gangdong Engineering & Consultant in Seoul, Korea since 2018. He is interested in Mathematics in Power Systems and has published papers on Smart Grid and Steiner Tree during his Bachelor and Master course at Electrical Engineering Dept. of SeoulTech University.

SangJoong Lee
../../Resources/kiiee/JIEIE.2022.36.10.036/au2.png

He worked for Korea Electric Power Corporation(KEPCO) for 22 years since 1976, mostly at Power System Research Center. He also completed the Power System Engineering Course at GE Research Center in Schenectady, NY, USA, during 1987～1988. He has been a professor of Seoul National University of Science and Technology since 1998. His research interest includes power generation, large power system and engineering mathematics. He proposed ‘Angle reference transposition(ART) in power system computation’ on IEEE Power Engineering Review in 2002, which describes that the loss sensitivities for all generators including the slack bus can be derived by specific assignment of the angle reference on a bus where no generation exists, while the angle reference has been specified conventionally on the slack bus. He applied the loss sensitivities derived by ART to ‘Penalty factor calculation in ELD computation’, ‘Optimal MW generation for system loss minimization’ and etc.

JuChul Kim
../../Resources/kiiee/JIEIE.2022.36.10.036/au3.png

He has worked for SD E&GC Co., Ltd, for 12 years since 2002 as Chief Executive of R&D Center. He has been a Professor of Chuncheon Campus of Korea Polytechnic University since 2014. His research interest includes Power system optimization, Quiescent power cut-off and Human electric shock. He published many papers on ELCB(Earth Leakage Circuit Breakers), Human body protection against electric shock, Improvement of SPD, Quiescent power cut-off, and etc. He received Ph.D. at Seoul National University of Science and Technology.