“This work has been submitted to IEEE for possible publication. Copyright may be transferred without notice, after which

this version may no longer be accessible.”

A Sensitivity-Based Three-Phase Weather-

Dependent Power Flow Approach for Networks

with Local Controllers—PART II: Case Studies

Evangelos E. Pompodakis

*

, Arif Ahmed

†

, and Minas C. Alexiadis

‡

*‡

Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Greece

†

TUMCREATE, 1 CREATE Way, #10-02 CREATE Tower, Singapore 138602

Email:

*

bobodakis@hotmail.com,

†

arif.ahmed@tum-create.edu.sg,

‡

minalex@auth.com

Abstract—Power flow is an integral part of distribution

system planning, monitoring, operation, and analysis. This two-

part paper proposes a sensitivity-based three-phase weather-

dependent power flow approach for accurately simulating

distribution networks with local voltage controllers (LVC). This

part II, firstly, presents simulation results of the proposed

approach in an 8-Bus and 7-Bus network, which are validated

using dynamic simulation. Secondly, simulation results for the

IEEE 8500-node network are also presented. An extensive

comparison is conducted between the proposed sensitivity-based

approach and the other existing power flow approaches with

respect to result accuracy and convergence speed. Moreover, the

influence of weather and magnetic effects on the power flow

results and the LVC states is also investigated. Simulation

results confirm that the proposed sensitivity-based approach

produces more accurate results than the existing approaches

since it considers the actual switching sequence of LVCs as well

as the weather and magnetic effects on the network. Moreover,

the proposed algorithm exhibits accelerated convergence due to

the usage of the sensitivity parameters, which makes it an

important tool for distribution system analysis.

Index Terms—Distributed power generation, Magnetic effects,

Power flow, Time delays, Local voltage controllers, Weather

effects.

I. INTRODUCTION

HE first part of this paper presents the theoretical

development of the proposed sensitivity-based three-

phase weather-dependent power flow approach for

distribution networks with LVCs. The proposed approach has

four distinct characteristics: a) it considers the three-phase

unbalanced nature of distribution systems, b) the operating

state of LVCs is calculated using sensitivity parameters to

accelerate the convergence speed of the algorithm, c) it

considers the exact switching sequence of LVCs based on

their reaction time delays, and d) the influence of weather

variations on the power flow is also taken into consideration.

This second part complements the theoretical development

of part I by presenting simulation results in a balanced 8-Bus

and an unbalanced 7-Bus network as well as in the large IEEE

8500-node network.

The rest of this Part II is structured as follows: Section II

validates the proposed algorithm against dynamic simulation

This research is co-financed by Greece and the European Union (European

Social Fund- ESF) through the Operational Programme «Human Resources

Development, Education and Lifelong Learning» in the context of the project

“Strengthening Human Resources Research Potential via Doctorate

Research – 2nd Cycle” (MIS-5000432), implemented by the State

Scholarships Foundation (ΙΚΥ).

via MATLAB

®

Simulink. Moreover, a comparison between

the proposed algorithm and two existing approaches is

presented in the same section. In Section III, a simulation case

study of the proposed algorithm is presented for the large-

scale IEEE 8500-node network under two distinct weather

conditions. The simulation results presented highlight the

importance of considering the weather and magnetic effects

in power flow analysis since they can significantly affect the

power flow results and LVC state estimation. Finally,

Section IV concludes the paper.

II. VALIDATION AND PERFORMANCE OF THE PROPOSED

SENSITIVITY-BASED ALGORITHM

First and foremost, the proposed algorithm is validated

against dynamic simulation in MATLAB

®

Simulink using an

8-bus balanced and a 7-bus unbalanced network. In addition,

the proposed approach is compared against the power flow

methods of [1]

*

and [3], with respect to result accuracy and

convergence speed. All algorithms were coded and

implemented in MATLAB

®

. It should be noted that for the

purpose of validation against the dynamic simulation, the

weather-dependent impacts were neglected in this section.

A) 8-Bus Balanced Network

Fig. 1: 8-Bus network consisting of 3 SVRs and 1 DG.

Fig. 1 depicts the 8-Bus network consisting of 3 SVRs and

1 DG operating in droop control mode [4]. The

droop equation is given by (1) [4]:

(1)

* It is noted that the method proposed in [2] presents almost identical

results with the method of [1]. Therefore, the method of [2] is not

simulated in this paper due to its strong similarity with [1] with

respect to the produced results.

T

“This work has been submitted to IEEE for possible publication. Copyright may be transferred without notice, after which

this version may no longer be accessible.”

where

,

,

,

are the positive sequence reactive

power, the reference voltage, the droop gain, and the positive

sequence voltage of DG i, respectively. The topology of the

network is similar to the one investigated in [5]. Data about

the network, the controllers of SVRs, and the DG are

provided in Table I. The delays of the controllers are set based

on their distance from the substation.

The slack bus in Fig 1 is assumed to be a substation. The

SVR near the substation has the fastest reaction time and as

the distance from the substation increases, the reaction time

increases [6]. All SVRs are in wye configuration and

initialized to the 0 position for the purposes of simulation.

Each phase of an SVR is modelled with its own local

controller, which is independently controlled.

The DG at bus 8 operates in mode and generates

balanced phase-to-neutral voltages [7]. The network supplies

four balanced three-phase loads. All the loads are modelled

as constant impedance loads, as shown in Table II, due to the

inaccuracy that Simulink presents in the modeling of constant

power loads.

TABLE I

PARAMETERS OF 8-BUS NETWORK

Distance of the lines

10 km

Voltage of slack bus

7200 V

Frequency of the network

50 Hz

Resistance of lines

0.4 Ω/km

Self-reactance of the lines

0.3 Ω/km

Mutual-reactance of the lines

0.1 Ω/km

Reference voltage of SVRs

7500 V

Bandwidth of SVRs

70 V

Intentional delay of SVR1

10 s

Mechanical delay of SVR1

2 s

Intentional delay of SVR2

20 s

Mechanical delay of SVR2

3 s

Intentional delay of SVR3

30 s

Mechanical delay of SVR3

4 s

Active power of DG

1 MW

Reference voltage of DG (

)

7500 V

Droop gain of DG (

)

0.5∙

V/Var

TABLE II

LOADS OF 8-BUS NETWORK

Load

Bus 3

Load

Bus 5

Load

Bus 7

Load

Bus 8

The tap positions and switching sequence for all SVRs of

the 8-bus network obtained via the dynamic simulation of

Simulink are presented in Fig. 2. It should be noted that since

all the loads are balanced, the three phases of the SVRs

undergo similar tap changes in this case. SVR 1 executes the

first switching action after an intentional time delay (10 sec),

followed by subsequent switching actions with a mechanical

time delay (2 sec), until the voltage lies within the bandwidth.

Similarly, for SVR 2 and 3. Since SVR 1 has the lowest

intentional and mechanical delay, it undergoes the highest

number of tap changes. The final tap positions obtained via

dynamic simulation for SVR 1, SVR 2 and SVR 3 is 11, 3,

and 1, respectively.

Fig. 2: Switching sequence of SVRs in the 8-Bus network versus time as

calculated by MATLAB

®

Simulink.

In Figs. 3-6, the results of the SVR tap change versus

iteration number are presented for the proposed algorithm as

well as for the algorithms of [1] and [3]. Fig. 3 presents the

tap change profile of all the SVRs versus the iteration number

for the proposed algorithm without consideration of

sensitivity parameters (refer Section IV of part I). In Fig. 4,

the tap change profile versus iteration number is presented for

the proposed algorithm with the consideration of the

sensitivity parameters (refer Section VI of part I). The

proposed algorithm, for both with and without sensitivity

parameter, considers the actual reaction delays of LVCs.

However, as observed in Fig. 4, the consideration of

sensitivity parameters yields accelerated convergence. It is

reminded that the proposed algorithm, both with and without

sensitivity parameters, considers the same switching

algorithm, thus, it presents identical results in both cases.

Their main difference is that the usage of sensitivity

parameters makes the execution of a complete power flow

after each switching action unnecessary, reducing the total

iteration number required for the final power flow solution.

More details are provided in Part I.

Fig. 3: Switching sequence of the three SVRs for the 8-Bus network versus

iteration number, as calculated by the proposed method without sensitivity.

“This work has been submitted to IEEE for possible publication. Copyright may be transferred without notice, after which

this version may no longer be accessible.”

Fig. 4: Switching sequence of the three SVRs for the 8-Bus network versus

iteration number, as calculated by the proposed method with sensitivity.

It is observed that the proposed power flow algorithm

yields correct tap positions when compared to the dynamic

simulation, both with and without sensitivities. This is due to

the consideration of the actual switching sequence of LVCs,

as presented in Part I Section IV. On the other hand, the

LVC’s states estimated by the power flow algorithms of [1]

and [3] do not conform to the dynamic simulation, as shown

in Figs. 5 and 6.

Fig. 5: Switching sequence of the three SVRs for the 8-Bus network versus

iteration number, as calculated by the algorithm of reference [1].

Fig. 6: Switching sequence of the three SVRs for the 8-Bus network versus

iteration number, as calculated by the algorithm of reference [3].

Algorithm [1] is only able to correctly estimate the tap

position of SVR 2 as shown in Fig 5. It is reminded that the

authors in [1] divide the LVCs in delay groups based on their

reaction delays. In this example, SVR 1, SVR 2 and SVR 3

belong to the first, second and third delay group, respectively.

Initially, the SVR of the first delay group reacts, by varying

its taps until its voltage lies inside the bandwidth.

Subsequently, the SVR of the second delay group undertakes

switching actions, and so on until all SVR voltages lie inside

their bandwidths.

The algorithm of [3] reaches final tap position in only a few

iterations (Fig. 6), because the algorithm completely neglects

the LVC’s reaction delays in its formulation, and thus, it

updates all the LVC states simultaneously.

A detailed depiction of the convergence characteristic of

the investigated algorithms is presented in Fig. 7. Figure 7

presents the maximum mismatch voltage error between two

consecutive iterations in per-unit versus the iteration number.

As observed, the proposed algorithm converges quickly to the

correct solution when sensitivity parameters are considered.

Although algorithm of [3] appears to have the fastest

converge, the produced power flow results are not accurate

compared to the dynamic simulation, as explained above. It

is again highlighted here that although a difference in

convergence of the proposed algorithm exists with and

without the usage of the sensitivity parameters, the final

results in both cases match those of the dynamic simulation.

Fig. 7: Maximum mismatch voltage error (pu) between two consecutive

iterations versus iteration number for all investigated algorithms.

In Fig. 8, the evolution of the voltage of bus 2 and 5 versus

iteration is presented for the investigated methods. The

proposed algorithm with and without the sensitivity yields the

same final voltages but with different convergence speed as

observed in Fig. 8. On the other hand, the algorithms of [1]

and [3] yield different voltages, which are not in agreement

with the proposed algorithm. The final voltages of the

network for the investigated methods and Simulink are

provided in Table III at the end of the paper. The proposed

approach with and without sensitivities present almost

identical results with those of Simulink, in contrast to the

methods of [1] and [3] that show significant deviations.

Fig. 8: Bus voltage evolution versus iteration for bus 2 and 5.

In Fig. 9, the total three-phase reactive power of the DG

connected at bus 8 is presented. It operates in inductive mode

according to droop equation (1), to mitigate the voltage rise

caused by the high amount of generated active power [8]. It

is observed that the proposed algorithm indicates a

consumption of 542 kVar, which is exactly the same as in the

dynamic simulation. Algorithms [1] and [3] yield different

results. The deviation of the reactive power in algorithms [1]

this version may no longer be accessible.”

and [3] is caused due to the inaccurate state estimation of

SVRs, which inevitably leads to an imprecise reactive power

calculation.

Fig. 9: Total reactive power consumption of the DG at bus 8.

B) 7-Bus Unbalanced Network

An unbalanced 7-bus network consisting of an OLTC, a

voltage-controlled capacitor, an SVR, and a DG is considered

for further simulation and validation, as shown in Fig. 10.

This network was selected for simulation since it includes all

kind of LVCs (OLTC, SVR, Capacitor, DG), it is unbalanced

and also its simple topology facilitates the comparison and

interpretation of the simulation results. Moreover, its small

size allows the execution of simulations in the time domain

environment of MATLAB

®

Simulink.

Fig. 10: 7-Bus unbalanced network consisting of one OLTC, one SVR, one

three-phase capacitor and one DG.

The DG operates in droop control mode (see Eq. (1)).

Data about the network, the controller of OLTC, SVR,

capacitors and DG are presented in Table IV. The SVR is

connected in wye configuration, while OLTC in Y

g

-Y

g

connection [11, Section II.C]. Each phase of OLTC, SVR,

and capacitor has its own local controller and is

independently controlled. The DG generates balanced phase-

to-neutral voltages [7]. The network supplies three balanced

and one unbalanced three-phase constant impedance load, as

shown in Table V.

Table VI summarizes the calculated LVC’s states of each

phase in the 7-Bus network using dynamic simulation, the

proposed algorithm without and with sensitivity parameters,

as well as the methods of [1] and [3]. It is observed that the

proposed algorithm presents identical results with those of

Simulink, confirming its accuracy. On the other hand, the

calculated states of the approaches in [1], [3] deviate from

those of Simulink. Moreover, Table VII at the end of the

paper depicts indicatively the voltages of each phase for the

last three buses of the network for the investigated

approaches. The remaining buses are not depicted due to

space limitation. As shown, the proposed method with and

without sensitivity parameter yields near identical results

with those of Simulink, while the other investigated power

flow methods present significant deviations.

TABLE IV

PARAMETERS OF 7-BUS NETWORK

Distance of the lines

10 km

Voltage of slack bus

7200 V

Frequency of the network

50 Hz

Resistance of lines

0.4 km

Self-reactance of the lines

0.3 /km

Mutual-reactance of the lines

0.1 /km

Reference voltage of SVR and OLTC

7500 V

Bandwidth of SVR and OLTC

70 V

Reference voltage of CAP

7500 V

Capacitance of each phase

F

Bandwidth of Capacitors

350V

Intentional delay of OLTC

10 s

Mechanical delay of OLTC

2 s

Intentional delay of CAP

20 s

Intentional delay of SVR

30 s

Mechanical delay of SVR

4 s

Active power of DG

1 MW

Reference voltage of DG

7500 V

Droop gain of DG (

)

V/Var

TABLE V

LOADS OF 7-BUS NETWORK

Load

Bus 3

Load

Bus 4

Load

Bus 6

Load

Bus 7

TABLE VI

LVC’S STATES CALCULATED BY THE INVESTIGATED APPROACHES FOR

THE 7-BUS NETWORK

OLTC 1 Taps

(phase a, b, c)

CAP 1

(phase a, b, c)

SVR 1 Taps

(phase a, b,

c)

DG Reactive

Power

Simulink

(10, 12, 10)

(ON, ON, OFF)

(2, 4, 3)

690 kVar

(inductive)

without

sensitivity

(10, 12, 10)

(ON, ON, OFF)

(2, 4, 3)

690 kVar

(inductive)

with

sensitivity

(10, 12, 10)

(ON, ON, OFF)

(2, 4, 3)

690 kVar

(inductive)

Algorithm

[1]

(11, 10, 10)

(OFF, ON, OFF)

(3, 2, 2)

666 kVar

(inductive)

Algorithm

[3]

(11, 11, 10)

(OFF, ON, OFF)

(3, 3, 2)

625 kVar

(inductive)

Finally, the number of iterations required by the algorithms

to converge with an accuracy of 10

-4

pu are presented in

Tables VIII and IX, for the 8-Bus and 7-Bus networks,

respectively. As shown, the algorithm of [3] presents the

fastest convergence since it calculates all LVC’s state

simultaneously ignoring their reaction delays. The proposed

algorithm without the sensitivity parameters has the slowest

convergence due to the successive power flow executions

after each switching action. The proposed method with

this version may no longer be accessible.”

sensitivity parameters combines the high accuracy with the

fast convergence, as no other power flow method so far.

TABLE VIII

REQUIRED ITERATIONS FOR THE INVESTIGATED APPROACHES IN THE 8-BUS

NETWORK

without

sensitivity

with

sensitivity

Algorithm

[1]

Algorithm

[3]

Required

iterations

74

27

61

9

TABLE IX

REQUIRED ITERATIONS FOR THE INVESTIGATED APPROACHES IN THE 7-BUS

NETWORK

without

sensitivity

with

sensitivity

Algorithm

[1]

Algorithm

[3]

Required

iterations

96

29

76

8

III. INFLUENCE OF WEATHER ON THE LVC STATES AND

POWER FLOW

We conduct a case study on the large IEEE 8500-Node

network to highlight the important impacts of weather on the

LVC’s state and power flow results, which cannot be

investigated via conventional approaches. Moreover, the

accuracy of the proposed sensitivity-based algorithm in a

large-scale network is investigated.

A) Network description

Fig. 11 depicts the IEEE 8500-Node Network, which has

been modified by including 4 DGs operating in several

modes. The network originally includes 4 SVRs and 4

capacitors [9], as shown in Fig. 11.

Fig. 11: IEEE 8500-Node network consisting of 4 SVRs and 4 three-phase

capacitors [9].

To get a sense on the position of the buses in the network,

the distance of each bus from the substation is depicted in Fig.

12.

Fig. 12: Distance of the buses from the substation for the IEEE 8500-Node

network.

The 4 additional DGs are connected to the buses 100, 350,

835, 1600. Data about the power profile and voltage/current

profile of DGs are provided in Table X, while data about the

reference voltages, the droop gains, the active powers, etc. are

given in Table XI. For DG 1, which is a synchronous

generator (SG), the negative- and zero-sequence impedance

are

. DG 1 and DG 4 operate in constant

voltage mode (treated as PV buses) and generate constant

active power and positive-sequence voltage. DG 2 and DG 3

operate in droop control mode generating balanced currents

(refer to Section III.D of Part I for more details about the

operational modes of DGs).

Moreover, data of the LVCs are provided in Table XI. Each

phase of the LVCs is independently controlled. The

capacitors are voltage controlled and the SVRs are in wye

configuration. The time delays of SVRs and capacitors are set

based on their distance from the substation [6], while DGs

react instantaneously. LVCs near to substation have faster

reaction times as shown in Table XI. Each controller

regulates the voltage of the connection point of LVC and no

line drop compensator or any other remote voltage control are

applied.

Finally, all the lines of the network were replaced with the

Penguin ACSR [10]. It is a single-layer conductor with a

cross-sectional area 125.1

composed of a 6 Aluminum

and 1 steel wire. Penguin is the largest single-layer ACSR

conductor and can successfully withstand the full load of the

network in both investigated environmental conditions,

without thermal violation. This modification was necessary

in order to simulate the influence of weather and magnetic

effects into the power flow results. With the original lines of

IEEE 8500-node network, this would not be possible since all

lines consist of constant impedances with unknown

conductor specific details.

In Figs 13 and 14, the resistance and self-reactance of

Penguin ACSR conductor are presented, respectively, as a

function of conductor temperature and current. The

methodology to calculate the resistance and self-reactance of

Figs. 13 and 14 is presented in [7].

It is observed in Fig. 13 that the resistance of Penguin

ACSR is strongly related to both current and conductor

temperature (conductor temperature is a function of both

current and weather conditions [7]). On the other hand, the

self-reactance is more related to the current than the