Central Improvement of Voltage Sags in the IEEE 33-Bus Distribution System by a Number of D-STATCOMS

n the case of using two D-Statcoms (Fig. 1) 
assumed to connect to bus j and k (such as k>j), the 
matrix of additional injected bus current only has two 
elements at bus j and bus k that do not equal zero 
(∆𝐼𝐼𝑗𝑗 = 𝐼𝐼𝐷𝐷𝐷𝐷𝑗𝑗 ≠ 0 and ∆𝐼𝐼𝑘𝑘 = 𝐼𝐼𝐷𝐷𝐷𝐷𝑘𝑘 ≠ 0). Other elements 
equal zero (∆Ii = 0 for ∀i≠j,k). Therefore, (6) can be 
rewritten as follows 
 �
∆�̇�𝑈𝑗𝑗 = 𝑍𝑍𝑗𝑗𝑗𝑗 × 𝐼𝐼�̇�𝐷𝐷𝐷𝑗𝑗 + 𝑍𝑍𝑗𝑗𝑘𝑘 × 𝐼𝐼�̇�𝐷𝐷𝐷𝑘𝑘
∆�̇�𝑈𝑘𝑘 = 𝑍𝑍𝑘𝑘𝑗𝑗 × 𝐼𝐼�̇�𝐷𝐷𝐷𝑗𝑗 + 𝑍𝑍𝑘𝑘𝑘𝑘 × 𝐼𝐼�̇�𝐷𝐷𝐷𝑘𝑘 (7) 
The injected currents to bus j and bus k, their bus 
voltages can boost Uj and Uk from 𝑈𝑈𝑗𝑗0 = 𝑈𝑈𝑏𝑏𝑠𝑠𝑠𝑠.𝑗𝑗 and 
𝑈𝑈𝑘𝑘
0 = 𝑈𝑈𝑏𝑏𝑠𝑠𝑠𝑠.𝑘𝑘 up to desired value, say 1p.u. That means 
 �
∆�̇�𝑈𝑗𝑗 = 1 − �̇�𝑈𝑏𝑏𝑠𝑠𝑠𝑠.𝑗𝑗
∆�̇�𝑈𝑘𝑘 = 1 − �̇�𝑈𝑏𝑏𝑠𝑠𝑠𝑠.𝑘𝑘 (8) 
replace (8) to (7) and solve this system of two 
equations, we get 
 �
𝐼𝐼�̇�𝐷𝐷𝐷.𝑘𝑘 = 𝑍𝑍𝑘𝑘𝑘𝑘×�1−�̇�𝑈𝑠𝑠𝑠𝑠𝑠𝑠.𝑘𝑘�−𝑍𝑍𝑘𝑘𝑘𝑘×�1−�̇�𝑈𝑠𝑠𝑠𝑠𝑠𝑠.𝑘𝑘��𝑍𝑍𝑘𝑘𝑘𝑘×𝑍𝑍𝑘𝑘𝑘𝑘−𝑍𝑍𝑘𝑘𝑘𝑘×𝑍𝑍𝑘𝑘𝑘𝑘� 
𝐼𝐼�̇�𝐷𝐷𝐷.𝑗𝑗 = 𝑍𝑍𝑘𝑘𝑘𝑘×�1−�̇�𝑈𝑠𝑠𝑠𝑠𝑠𝑠.𝑘𝑘�−𝑍𝑍𝑘𝑘𝑘𝑘×�1−�̇�𝑈𝑠𝑠𝑠𝑠𝑠𝑠.𝑘𝑘��𝑍𝑍𝑘𝑘𝑘𝑘×𝑍𝑍𝑘𝑘𝑘𝑘−𝑍𝑍𝑘𝑘𝑘𝑘×𝑍𝑍𝑘𝑘𝑘𝑘� (9) 
The power of corresponding D-Statcoms placed 
at buses j and k ��̇�𝑆𝐷𝐷𝐷𝐷.𝑗𝑗 = �̇�𝑈𝑗𝑗 × 𝐼𝐼𝐷𝐷𝐷𝐷.𝑗𝑗 
�̇�𝑆𝐷𝐷𝐷𝐷.𝑘𝑘 = �̇�𝑈𝑘𝑘 × 𝐼𝐼𝐷𝐷𝐷𝐷.𝑘𝑘 (10) 
The voltage upgrade at other buses i (i≠j,k) can 
also be calculated 
 ∆�̇�𝑈𝑖𝑖 = 𝑍𝑍𝑖𝑖𝑗𝑗 × 𝐼𝐼�̇�𝐷𝐷𝐷.𝑗𝑗 + 𝑍𝑍𝑖𝑖𝑘𝑘 × 𝐼𝐼�̇�𝐷𝐷𝐷.𝑘𝑘 (11) 
Finally, the voltage at other buses i (i≠j,k) after 
placing two D-Statcoms at buses j and k 
 �̇�𝑈𝑖𝑖 = ∆�̇�𝑈𝑖𝑖 + �̇�𝑈𝑖𝑖0 = ∆�̇�𝑈𝑖𝑖 + �̇�𝑈𝑏𝑏𝑠𝑠𝑠𝑠.𝑖𝑖 (12) 
2.2.3. Placing m D-Statcoms in the test system 
Assume that M is the set of m buses to connect 
to D-Statcom (Fig. 2), so the column matrix of bus 
injected current [∆I] in (6) has m non-zero elements 
and n-m zero elements. From (6), we have 
∆�̇�𝑈𝑘𝑘 = 𝑍𝑍𝑘𝑘𝑘𝑘 × 𝐼𝐼�̇�𝐷𝐷𝐷.𝑘𝑘 + ∑ 𝑍𝑍𝑗𝑗𝑘𝑘 × 𝐼𝐼�̇�𝐷𝐷𝐷.𝑗𝑗𝑗𝑗∈𝑀𝑀,𝑖𝑖≠𝑘𝑘 (13) 
Journal of Science & Technology 139 (2019) 012-017 
14 
For bus k, k∈M, the rule of voltage compensation 
is as follows 
 ∆�̇�𝑈𝑘𝑘 = �̇�𝑈𝑘𝑘 − �̇�𝑈𝑏𝑏𝑠𝑠𝑠𝑠.𝑘𝑘 = 1 − �̇�𝑈𝑏𝑏𝑠𝑠𝑠𝑠.𝑘𝑘 (14) 
Replace (14) to (13) we have m equations to 
calculate m variables IDS.k of m D-Statcoms. Solve this 
system of m equations, we get m values of IDS.k. 
Replace m values of IDS.k in (6), we can calculate 
the voltage upgrade of n-m buses without connecting 
to D-Statcoms 
 ∆�̇�𝑈𝑖𝑖 = ∑ 𝑍𝑍𝑖𝑖𝑘𝑘 × 𝐼𝐼�̇�𝐷𝐷𝐷𝑘𝑘𝑛𝑛𝑖𝑖=1 (15) 
Finally, we calculate voltages of all buses in the system 
after placing m D-Statcoms similar to (12). 
Fig. 2. Test system short-circuit modeling using [Zbus] 
with the presence of m D-Statcoms (m<n) 
3. Problem Definition 
3.1. IEEE 33-Bus Distribution System 
This paper uses the IEEE 33-bus distribution 
feeder (Fig. 3) as the test system for the research. It 
features a balanced three-phase distribution system, 
with three-phase lines and loads. This research 
assumes: base values are 11kV; 100MVA. The system 
voltage is 1pu. System impedance is 0.1pu. 
Fig.3. IEEE 33-bus distribution feeder 
3.2. Short-circuit calculation 
According to point 2.2a, Section 2, we assume 
the initial status of the test system is a short-circuit in 
the system. The paper considers a number of short-
circuit positions with different fault impedance Zf. 
Three-phase short-circuit calculations are performed 
in Matlab using the method of bus impedance matrix 
and resulting bus voltage sags can be calculated. 
With the calculation of system bus voltage in the 
short-circuit event with the presence of D-Statcom, we 
can define the problem of optimization as follows. 
3.3. The problem of optimization 
3.3.1. Objective function and constraints 
In this research, the problem of optimizing the 
location and size of a multiple D-Statcoms in the test 
system where the objective function is to minimize the 
total system voltage deviation, is established. It’s seen 
as the index of system voltage sag energy [16]. 
 𝐹𝐹 = �∑ �𝑈𝑈𝑟𝑟𝑟𝑟𝑓𝑓 − 𝑈𝑈𝑖𝑖�2𝑛𝑛𝑖𝑖=1 ⇒ 𝑀𝑀𝑀𝑀𝑀𝑀 (16) 
where 
Uref: Reference system voltage, equals 1p.u. 
Ui: Bus i voltage calculated in (14). 
For this problem of optimization, the main 
variable is the scenario of positions (buses) where D-
Statcoms are connected. We can see each main 
variable as a string of m bus numbers with D-Statcom 
connection out of n buses of the test system. Therefore, 
the total scenarios of D-Statcom placement to be tested 
is the m-combination of set N (n=33): 
 𝑇𝑇𝑚𝑚 = 𝐶𝐶𝑛𝑛𝑚𝑚 = 33!𝑚𝑚!×(33−𝑚𝑚)! (17) 
For example, if we consider the placement of 2 
D-Statcoms in the test system, m=2, the total scenarios 
for placing these two D-Statcoms is as follows 
 𝑇𝑇2 = 𝐶𝐶332 = 33!2!×(33−2)! = 528. 
Each candidate scenario to be tested is a pair of 
buses number k and l out from 33 buses where the two 
D-Statcoms are connected (e.g. 1,2; 1,3;). 
The only constraint is that the size of D-Statcom 
is limited to a certain maximum value (SDS.max). In this 
research D-Statcom’s size is not greater than 0.1p.u. 
(or 10MVA). For each bus where D-Statcom can be 
connected, if SDS > SDS.max, this bus is not qualified for 
D-Statcom placement. 
3.3.2. Problem solving 
For such a problem of optimization, under the 
assumption of a fault event, the objective function and 
the constraint are always determined. So, we use the 
method of direct search and testing all candidate 
scenarios in the set of scenarios of Tm. The flowchart 
of solving this problem in Matlab is given in Fig. 4. 
Each candidate scenario k defines positions 
where D-Statcoms are connected. According to this 
method, we have to determine the whole set of 
candidate scenarios Tm (17). For a candidate scenario 
k, we can calculate the D-Statcom’s power (size) and 
objective function Fk. We can sweep all candidate 
scenarios in Tm for constraint verification and 
minimization of the objective function. 
Journal of Science & Technology 139 (2019) 012-017 
15 
Fig. 4. Flowchart of the problem of optimization 
In the flowchart, input data that can be seen as 
parameters are fault events. “postop” is the 
intermediate variable that fixes the optimal scenario of 
D-Statcom placement where the objective function is 
minimized. The initial solution of objective function 
Min equals 4 which is big value for starting the search 
process. The method sweeps all cadidate scenarios in 
the set of Tm to find the global optimal solution. 
4. Result Analysis 
4.1. Fault event scenarios 
The research considers the following fault event 
scenarios that have significant influence on the D-
Statcom’s size and objective function: 
Short-circuit type and fault impedance: Three-
phase short-circuit through different values of fault 
impedances Zf is considered. Three alternatives of fault 
impedances Zf = 1.6(p.u.), 0.8(p.u.) and 0(p.u.) are 
considered for analysing its influences in the problem 
solutions. The paper mainly discusses the D-Statcom’s 
effectiveness on voltage compensation in an event of 
short-circuit in general, thus, other short-circuit types 
are not considered. 
Short-circuit positions: Two fault positions at 
buses 10 and 30 are considered. 
4.2. Result analysis 
The proposed method of modeling the system 
voltage sag mitigation for the case of using multiples 
of D-Statcom in Section 2.2 can be illustrated for the 
case of using two D-Statcom. Followings are step-by-
step clarification and analysis of the results. 
For a better understanding, we consider the case 
of fault position at bus 10. The Fig.5 is 3D graphic of 
the objective function for all scenarios of placement of 
2 D-Statcoms in case of Zf = 1.6p.u. A scenario is a 
point with its ordinates equal to D-Statcom’s locations. 
Also, because we don’t consider the permutation for 
the pair of D-Statcom’s location (e.g. 1-2 is the same 
as 2-1), we only consider points on the triangle from 
the main diagonal of the matrix of scenarios of 
placement of 2 D-Statcoms. The points in the other 
triangle of the above said matrix are not considered and 
thus its objective function is given a high value (e.g. 
F=4p.u.). Besides, for the scenarios that result in the 
power of one or both two D-Statcoms greater than 
SDSmax, they are also not considered as candidate 
scenarios and their objective function is also equal to 
4p.u. Objective function gets its minimum of 
0.1611p.u. for D-Statcoms placed at buses 9 and 13. 
The resulting system bus voltages are all upgraded 
above 0.8p.u. (Fig. 6). 
Fig. 5. Objective function for the placement of two D-
Statcoms for fault position at bus 10, Zf = 1.6p.u. 
Fig.6. System bus voltage without and with D-
Statcoms for short-circuit at bus 10, Zf = 1.6p.u. 
The main results are summarized in the Table 2. 
The system bus voltage before and after placing two 
D-Statcoms are also depicted in Fig. 7. 
Journal of Science & Technology 139 (2019) 012-017 
16 
Table 1. Remarked results for placing two D-Statcoms 
Fault impedance Zf (p.u.) 1.6 0.8 0 
Short-circuit position at bus 10 
Objective function (p.u.) 0.1611 0.2825 0.3184 
Optimal placement of DS 1 Bus 9 Bus 8 Bus 8 
Size (p.u.) of DS 1 0.0988 0.0822 0.0925 
Optimal placement of DS 2 Bus 13 Bus 13 Bus 13 
Size (p.u.) of DS 2 0.0518 0.0858 0.0965 
Number of buses U > 0.8p.u. 33 33 33 
Number of scena. SDS > SDS.max 310 358 423 
Short-circuit position at bus 30 
Objective function (p.u.) 0.1096 0.1247 1.8066 
Optimal placement of DS 1 Bus 28 Bus 28 Bus 9 
Size (p.u.) of DS 1 0.0707 0.0793 0.0918 
Optimal placement of DS 2 Bus 31 Bus 31 Bus 23 
Size (p.u.) of DS 2 0.0839 0.094 0.0589 
Number of buses U > 0.8p.u. 0.1096 0.1247 1.8066 
Number of scena. SDS > SDS.max 366 381 404 
The research considers the voltage tolerance of 
0.8p.u. in Table 1 and 2 because we know that the 
voltage sag duration is basically defined by 
protection’s tripping time and for distribution system, 
it’s normally in the range of 0.1-10s. According to 
voltage ride through curve (e.g. ITIC [1]), the safe 
voltage magnitude is 0.8pu. That’s why for the size of 
distribution system as the IEEE 33-bus system, we can 
only consider to use up to 2 D-Statcoms for system 
voltage sag mitigation. 
Fig. 7. System bus voltage without and with two D-
Statcom placements for short-circuit at buses 10, 30 
5. Conclusion 
This paper introduces a new method for 
considering “central improvement” voltage sag 
mitigation by a multiple of D-Statcoms in distribution 
system. D-Statcom modeling for voltage sag 
mitigation in short-circuit calculation of power system 
is introduced basing on the application of Thevenin’s 
superposition theorem. The problem of optimization is 
solved on the minimization of objective function 
which is the total system voltage deviation as per 
“central improvement” approach with regard to D-
Statcom’s power constraint. This method allows us to 
consider using a multiple of D-Statcoms in the case of 
large distribution system that helps improve totally 
system bus voltage in voltage sag events in distribution 
system. Different scenarios of fault event including 
short-circuit positions and fault impedances are taken 
into account for assessing their influence to the 
outcomes of the problem of optimization. 
A cost model is not introduced for the problem of 
optimization because the benefice from system voltage 
sag mitigation is impossibly determined. Research can 
be developed with regard to different fault events in 
the same time for a better illustration for D-Statcom’s 
system voltage sag mitigation. 
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