Abstract
When urban distribution systems are gradually modernized, the overhead lines are replaced by underground cables, whose shunt admittances can not be ignored. Traditional power flow (PF) model with equivalent circuit shows non-convexity and long computing time, and most recently proposed linear PF models assume zero shunt elements. All of them are not suitable for fast calculation and optimization problems of modern distribution systems with non-negligible line shunts. Therefore, this paper proposes a linearized branch flow model considering line shunt (LBFS). The strength of LBFS lies in maintaining the linear structure and the convex nature after appropriately modeling the equivalent circuit for network equipment like transformers. Simulation results show that the calculation accuracy in nodal voltage and branch current magnitudes is improved by considering shunt admittances. We show the application scope of LBFS by controlling the network voltages through a two-stage stochastic Volt/VAr control (VVC) problem with the uncertain active power output from renewable energy sources (RESs). Since LBFS results in a linear VVC program, the global solution is guaranteed. Case study exhibits that VVC framework can optimally dispatch the discrete control devices, viz. substation transformers and shunt capacitors, and also optimize the decision rules for real-time reactive power control of RES. Moreover, the computing efficiency is significantly improved compared with that of traditional VVC methods.
Set of all the buses
Sets of buses with capacitors and renewable energy sources (RESs)
i, j Indices of buses i and j
Index of the scenario
Set of branches with transformers
Forecasted error of active power produced by RES
Coefficient of constant series impedance and shunt admittance of branch between buses i and j
Apparent load power at bus j
Shunt conductance and susceptance
The minimum and maximum voltage ratios of transformer
Number of scenarios
Active and reactive load power at bus j
Active power produced by RES
Forecasted value of active power produced by RES
The maximum reactive power available from shunt capacitor bank
The minimum and maximum reactive power
QRES,max produced by RES
Series resistance and reactance of branch between buses i and j
The minimum and maximum nodal voltage magnitudes
Line shunt admittance magnitudes at buses i and j
Shunt admittance magnitude of branch between buses i and j at bus i,
Equivalent shunt admittances of transformer at buses i and j
Series admittance magnitude of transformer located at branch between buses i and j at bus i
Shunt excitation admittance magnitude of transformer located at branch between buses i and j at bus i
Series impedance magnitude of branch between buses i and j
Reactive power adjustment by RES in response to its active power fluctuation
Voltage magnitude drop between buses i and j
Apparent load power at bus j for each scenario
Branch current magnitudes flowing through series impedance between buses i and j and other connecting buses
Sending- and receiving-end branch current magnitudes at bus i
Sending- and receiving-end branch current magnitudes at bus j
Sending- and receiving-end branch current magnitudes at bus j for each scenario
Transformer voltage ratio
Expected power loss of adjustable case
Expected power loss of deterministic case
Number of voltage violations
Number of nodes
Number of scenarios with at least one voltage violation
Reactive power from shunt capacitor bank
Reactive power from shunt capacitor bank at bus j
Reactive power produced by RES
Reactive power produced by RES in deterministic case at bus j
Epigraph transformation factor at bus j
Voltage magnitudes at buses i and j
Objective function
POWER flow is the most important component of many power system problems such as planning and optimization problems for transmission and distribution systems. As modern power systems are evolving into highly complex entities owing to huge integration of renewable energy sectors and fast expansion of network applications, the selection of a proper power flow (PF) model is of key concern.
The most commonly used PF model is called alternating current PF (ACPF), which is non-linear. The traditional optimization problems based on ACPF would be non-convex, hard to solve, and time-consuming [
In order to achieve higher efficiency to solve PF and relevant optimization problems quickly, this paper chooses a previously analyzed linearized branch flow (LBF) model with fast computating speed and acceptable accuracy [
As for real radial distribution systems, various network equipments like transformers, capacitors, and distributed generation, mainly renewable energy source (RES), are present, and each branch not only has series impedance but also has shunt admittance. Several branch flow models proposed in [
There are three main previous studies [
This paper proposes an alternative LBF model with non-zero line shunts (LBFS), which is the equivalent circuit model for LBF. LBFS continues to hold the strength of short computing time under the same computing conditions as zero line shunts. Furthermore, LBFS is expanded to include transformers.
The “charging effect” caused by non-negligible capacitive susceptance of shunt admittance would lift the nodal voltage magnitude at certain nodes, potentially cause the risk of voltage violations, and affect the safe operation and reliability in modern distribution systems. Therefore, our essential purpose of applying Volt/VAr control (VVC) is to maintain nodal voltage in the acceptable range, avoid the violations of load power constraints, and provide additional reactive power from capacitors and RESs to the bulk distribution system [
Most VVC strategies implement a centralized controller which decides the optimum set points of controllable devices based on current network topology and net nodal injections [
As LBFS results in a linear mathematical structure, the storage memory space and solving time are hugely saved. Since LBFS shows the superiority in accuracy of voltage magnitude and computing efficiency with the modeling of non-zero line shunts and transformers, we construct a linear two-stage stochastic optimization formulation for VVC problem which is significantly faster than traditional methods and makes operating points free of voltage violations.
The schedules of the discrete controlling devices are held at fixed settings throughout the entire optimization horizon (an hour time window), while the inverter reactive powers are adjusted in the second stage, upon the final revelation of uncertainty according to the decision rule connecting active and reactive powers of RES.
The major contributions of the paper are two-fold:
1) LBFS is proposed as the first linear PF model with equivalent circuit to consider non-zero line shunts and shunt excitation admittance of the transformer. The distribution lines and transformers naturally admit a equivalent circuit model, but they are non-convex and cannot be applied directly. The strength of LBFS lies in its linear and simple mathematical structure. After introducing a system fixed factor in the voltage drop equation, the proposed LBFS model achieves an enhanced accuracy in nodal voltage magnitude and branch current magnitude compared with the widely used linear DistFlow and LBF model under the benchmark solved by ACPF, while maintaining the fast computating efficiency which is typical of LBF.
2) A two-stage VVC problem is proposed to control the nodal voltage magnitude and avoid the voltage violations caused by “charging effect” from shunt elements and nodal power injection from distributed RES units. A linear decision rule is introduced to adjust the real-time reactive power according to the active power forecasted errors. The whole optimization scheme based on LBFS results in a linear structure and therefore is easily implemented. The proposed VVC problem has a better performance in eliminating voltage violations than the traditional deterministic case and exhibits a fast computing efficiency compared with traditional VVC problems based on ACPF.
The rest of the paper is organized as follows. Section II introduces the model formulation. Section III constructs a two-stage linear stochastic VVC problem based on LBFS and describes the linearity and convexity of the framework. Section IV provides the numerical results on three test systems. The paper is concluded in Section V.
LBF model is adapted from linearized network model of direct current load flow [
(1) |
(2) |
(3) |

Fig. 1 Part of a radial distribution system.
In (1), the magnitude of complex load power is set as the branch current magnitude, so the current injections satisfy Kirchoff’s current law (KCL). In (2), is the product of the magnitude of line impedance and branch current, satisfying Ohm’s law. Besides, the accuracy and application scope analyses of LBF model for radial distribution systems are proposed by [
Next, consider the branch flow model of a radial distribution system with equivalent circuit model shown in

Fig. 2 Part of a radial distribution system with equivalent circuit model.
According to the LBFS model shown in
(4) |
Then, the voltage magnitude at bus i can be obtained as:
(5) |
Therefore, the formulation of LBFS model is given as:
(6) |
(7) |
(8) |
Obviously, (6) obeys the KCL, while (7) represents the Ohm’s law. The LBFS model preserves the mathematical structure as LBF model by setting the coefficient of equal to a parameter denoted by , known as the system fixed factor. It is important to note that the fixed parameter of the system depends on the line impedance and shunt admittance magnitudes, and hence remains fixed for different operating conditions. LBFS still holds its simplicity without involving iterative and complex parts.
This paper models the shunt capacitors with . Also, we assume the value remains constant for a particular operating state, which is reasonable and makes the model formulation simple. The reactive power compensation from capacitors can supply leading var and maintain the voltage in the stable region.
The traditional distribution systems usually have transformers located in the substation nodes to change feeder voltage levels [
The original equivalent circuit of an ideal transformer located between buses i and j is visualized in
(9) |
(10) |

Fig. 3 Original equivalent circuit of ideal transformer in part of a radial distribution system.
Also, the voltage change at both sides is given as:
(11) |
The following equations are obtained by combining (10) and (11):
(12) |
A equivalent circuit is assumed to exist between buses i and j, which is defined by the following pair of equations:
(13) |
By combining (12) and (13), we can obtain:
(14) |
where shows the reciprocity of passive circuit of transformers; and . Thus, the transformer is modeled within the LBFS by considering the effect of the nominal tap ratios on the equivalent circuit model, as shown in

Fig. 4 equivalent circuit of ideal transformer in part of a radial distribution system.
According to the LBFS model with transformers shown in
(15) |
Then, we can obtain the voltage magnitude at bus i:
(16) |
Like the LBFS model, we set the coefficient of equal to , known as the system fixed factor. Therefore, the LBF model with line shunts and transformers is given as:
(17) |
(18) |
(19) |
Obviously, (17) obeys the KCL, while (18) represents the Ohm’s law. LBFS with transformers still holds its simplicity without involving non-linear parts.
As a traditional optimization problem, VVC is a minimization problem, whose constraints are mostly inequality type. Our goal is to find the global solution quickly without violating the constraints and limits:
(20) |
(21) |
represents the minimization of the difference between 1 p.u. and nodal voltage magnitude, in other words, the minimization of voltage deviation around the nominal voltage profile. We use the epigraph transformation with of the objective function, as shown in , to avoid the absolute value sign for obtaining the linearity.
We set up a two-stage linear stochastic VVC problem based on the linear VVC framework. In the first stage, the slow controls, shunt capacitor banks, and the set points of transformer voltage ratio are “here and now” decisions, which should be made before the uncertainty of RES active power output is revealed, and kept fixed throughout the entire optimization interval (an hourly time window). As for the second stage, the fast controls viz. the reactive power from RES inverters are allowed to be “wait and see” decisions. The second stage adjusts the reactive power from the RES in real time according to a pre-defined rule. This pre-defined rule is chosen to be a linear equation relating the RES reactive power to its active power, whose coefficients are also available from the first stage. Previously, another linear rule adjusting the RES reactive power has been used in [
(22) |
(23) |
The following assumptions are enforced while carrying out the optimization.
1) As for the discrete variables, i.e., reactive power from shunt capacitor banks and transformer voltage ratio , they would make the optimization problem a mixed-integer linear programming framework. To avoid this, we regard them as the continuous variable in the linear programming and subsequently round the final values to the nearest discrete values.
2) The load power equation considering the active power from RES units has the following expression:
(24) |
Also, we set fictitious buses in the branches where transformers are located to form the ideal transformer between the sending-end bus and fictitious bus. Thus, the two-stage linear stochastic VVC problem is shown as follows.
(25) |
s.t.
(26) |
(27) |
(28) |
(29) |
(30) |
(31) |
(32) |
(33) |
(34) |
(35) |
The objective function is the minimization of voltage magnitude violation. The feasible region is demarcated by the constraints (26)-(35) and is defined by the vector of optimization variables .
The two-stage linear stochastic VCC problem is proposed to decrease the probability of voltage violation with the uncertain nodal net injections from RES under the condition of “charging effect” from shunt elements. The large-size VVC problem, caused by the consideration of all generated scenarios, can be solved because of the proposed model. The first stage (deterministic case when ) ignores the forecasted errors of RES as and gets the slow control settings of , , and . As for the second stage, Gaussian scenarios along with are generated randomly and used to solve the optimization problem, while the optimal solutions from the first stage, , , and , are kept fixed. The real-time from RES inverters is optimally adjusted based on and the forecasted errors of RES. Notice that only contains a single value corresponding to each RES unit for all scenarios, which means that the fixed value of this slope can handle all scenarios, even for the worst one. Thus, the optimal control of reactive power from RES inverters is obtained through the two-stage linear stochastic VCC problem.
We test the proposed LBFS model and two-stage linear stochastic VVC problem in three radial distribution systems. All tests are carried out in MATLAB on a laptop with an Intel Core i5-10500 3.10 GHz CPU and 8 GB of RAM, where the two-stage stochastic optimization is solved using the CPLEX 12.10 optimization studio. The description of test systems is given below.
1) The IEEE 33-bus system: the voltage level is 12.66 kV and the base power is 10 MVA. Other data and network topology for this system are available from Matpower 7.0. The test system is modified with the consideration of non-zero shunt admittance: the shunt conductance is considered the same for the whole system as p.u.; the shunt susceptance is 1/3 of the series impedance magnitude of the same branch [
2) The 95-bus UK generic distribution system (UKGDS95) [
3) IEEE 123-bus standard test system (case123): five distributed RES units are installed with the forecasted output of 0.5 MW at nodes 114, 47, 3, 35, and 83. The system originally contains 4 shunt capacitors at nodes 83, 88, 90, and 92. Shunt capacitors can supply a maximum of 250 kvar. Two transformers are integrated in the original system including a substation transformer. Moreover, overhead lines are replaced by underground cables, thus the value of line shunts is increased to 5 times the original one [
In the three test systems, the distributed RES units are set in their maximum power point tracking (MPPT) schedule to achieve the highest active power output. The forecasted value of RES active power in each scenario is generated from the probability distribution function between its minimum and maximum values. As for the forecasted value of real-time system load and forecasted error of RES active power, they obey the Gaussian distribution. All the test systems have a transformer located at the substation node, with the tap changer placed on the high voltage winding of transformers. Each tap changer has 13 set points (one at center rated tap and six to increase and decrease the turn ratio). As for shunt capacitor banks, these are made up of a combination of ten capacitor steps connected in parallel. All the test systems are expanded to 100 scenarios with different load power (generated from Gaussian distribution for each scenario) to verify the performance of linear models in large-scale distribution systems at light-, normal-, and heavy-load levels.
A linearized branch flow model with line shunts (LBFMS) from [
Test model | IEEE 33-bus system | UKGDS95 | IEEE 123-bus standard test system | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Ve | Ie | Ple | Time (s) | Ve | Ie | Ple | Time (s) | Ve | Ie | Ple | Time (s) | ||||
LinDist | 4.580 | 9.180 | 24.700 | 26.400 | 1.27 | 1.870 | 5.140 | 19.300 | 25.800 | 4.75 | 0.740 | 1.160 | 10.700 | 18.600 | 5.40 |
LBF | 6.010 | 10.100 | 27.900 | 29.600 | 1.34 | 1.670 | 5.000 | 21.600 | 24.800 | 4.88 | 0.720 | 1.080 | 12.100 | 14.800 | 5.27 |
LBFMS | 1.680 | 5.030 | 13.400 | 15.800 | 1.64 | 1.120 | 2.030 | 8.130 | 8.640 | 5.21 | 0.680 | 1.130 | 6.470 | 8.130 | 6.22 |
LBFS | 1.690 | 4.170 | 14.100 | 15.400 | 1.53 | 0.430 | 1.120 | 6.780 | 8.330 | 4.93 | 0.510 | 1.080 | 8.380 | 10.100 | 5.99 |
ACPF | 0.979 | 0.983 | 0.115 | 0.196 | 24.70 | 1.019 | 1.024 | 0.059 | 0.150 | 58.80 | 0.996 | 0.994 | 0.038 | 0.014 | 67.20 |
According to
With regard to the average errors of branch current and power loss, although LBFS still has a relatively higher error, around 10%-15%, it is noted that it does improve a lot from LinDist and LBF, which is nearly 60% enhancement. We believe that this error can be accepted if the load condition is normal or light, and the accuracy of branch current is not the major concern of the fast calculation and estimation of large distribution systems. Perhaps a direct way to improve this shortcoming of accuracy in branch current is modifying the basic LBF model with the consideration of phasor angles instead of just adapting the magnitude parts, which will be discussed in our future work. Moreover, LBFMS shows a higher accuracy than LBFS in branch flow (power loss), but lower accuracy in voltage magnitude. Thus, LBFS shows its own strength of calculation accuracy compared with another linear model with line shunts.
According to
The effect of network reverse PF is a non-negligible issue for LBF models. Some warm-start LBF models only consider the single-direction branch flow from former nodes to latter nodes. The reverse PF would potentially cause huge errors of nodal voltage and branch flow (power or current) in these models. In this case, the effect of network reverse PF should be tested to verify the performance of LBFS.
The test is mainly performed in the IEEE 123-bus standard test system, which has several generation power injections at different nodes, and thereby has reverse PFs. For the purpose of the proper comparison, we modify the IEEE 123-bus standard test system by turning off all the RES units. Thus, the reverse PFs are prevented in the modified test system. The simulation results are shown in

Fig. 5 Error comparison of nodal voltage magnitude of LBFS with and without reverse PF in IEEE 123-bus standard test system.
According to
Therefore, the reverse PF has little influence on the calculation accuracy of LBFS.
As previously described in Section I, the “charging effect” caused by the capacitive susceptance of shunt admittance would lift the nodal voltage magnitude, potentially cause voltage violations at certain nodes and affect the safe operation and reliability in power distribution systems. This claim is tested in the IEEE 33-bus system with and without line shunts. As shown in

Fig. 6 Nodal voltage magnitude of IEEE 33-bus system.
The two-stage scenario-based VVC problem based on LBFS is a linear program, thus the problem with a large number of scenarios, , can be solved directly within a single optimization framework. By setting such a big number of scenarios, we ensure that all possible real-time RES output uncertainties are handled though the two-stage optimization problem.
System | Case | (p.u.) | (p.u.) | (p.u.) | |||
---|---|---|---|---|---|---|---|
IEEE 33-bus system | Deterministic | 0.252 | 0 | 0.174 | 0 | 0.182 | 0 |
Adjustable | 0.079 | 0.098 | 0.092 | ||||
UKGDS95 | Deterministic | 0.128 | 0 | 0.185 | 0 | 0.309 | 0 |
Adjustable | 0.098 | 0.086 | -0.107 | ||||
IEEE 123-bus standard test system | Deterministic | 0.246 | 0 | 0.041 | 0 | 0.127 | 0 |
Adjustable | -0.103 | 0.085 | 0.132 |
System | (p.u.) | (p.u.) | ||
---|---|---|---|---|
IEEE 33-bus system | 0.1918 | 0.20 | 1.0009 | 1.001 |
0.0389 | 0.04 | |||
0.0188 | 0.02 | |||
UKGDS95 | 0.1782 | 0.20 | 1.0014 | 1.001 |
0.0597 | 0.06 | |||
0.0925 | 0.10 | |||
IEEE 123-bus standard test system | 0.2000 | 0.20 | 1.0019 | 1.002 |
0.0467 | 0.04 | |||
0.0724 | 0.08 |

Fig. 7 Nodal voltage magnitude of IEEE 33-bus system before and after implementing VVC.
System | Computing time (s) | |
---|---|---|
Deterministic case | Adjustable case | |
IEEE 33-bus system | 91.843 | 1044.32 |
UKGDS95 | 186.752 | 1754.98 |
IEEE 123-bus standard test system | 222.445 | 1934.61 |
Three indices are introduced to further demonstrate the effectiveness of the proposed two-stage linear stochastic VVC problem and are defined as follows.
1) Scenario failure rate (SFR) is the ratio of scenarios with voltage violations among all scenarios:
(34) |
2) The maximum voltage violation value is the difference between the maximum/minimum voltage violation point and the upper/lower limit.
3) Voltage violation probability (VVP) is the probability of all nodal voltage violations of all scenarios:
4) Conservative index (CI) is defined as the increase in the expected power loss for the stochastic two-stage VVC schedule from that of the base schedule:
The values of the security constraint indices are given in
System | Case | NsVV | NsIV | SFR (%) | (p.u.) | VVP (%) | Power loss (p.u.) | CI(%) |
---|---|---|---|---|---|---|---|---|
IEEE 33-bus system | Deterministic | 783 | 134 | 7.83 | 0.0130 | 3.47 | 1.32 | 4.54 |
Adjustable | 81 | 33 | 0.81 | 0.0012 | 0.14 | 1.38 | ||
UKGDS95 | Deterministic | 1015 | 182 | 10.15 | 0.0180 | 4.12 | 1.59 | 5.35 |
Adjustable | 69 | 35 | 0.69 | 0.0014 | 0.28 | 1.68 | ||
IEEE 123-bus standard test system | Deterministic | 1123 | 201 | 11.23 | 0.0240 | 4.41 | 1.66 | 6.02 |
Adjustable | 140 | 79 | 1.40 | 0.0011 | 0.39 | 1.76 |
This paper proposes an LBFS in radial distribution systems. LBFS, as the first linear equivalent circuit model considering shunt elements of lines and transformers, exhibits linearity and high computating efficiency compared with traditional ACPF and other models with line shunts like LBFMS, and achieves enhanced accuracy compared with linear DistFlow and LBF models. Extensive simulation tests are conducted to prove the improvement of LBFS in calculating nodal voltage magnitude (the precise requirement in distribution systems). Besides, the network reverse PF is tested to have nearly no influence on LBFS. Moreover, a linear two-stage linear stochastic VVC problem based on LBFS considering the uncertainty in active power output of RES is formulated in this paper. The primary purposes of implementing VVC method are limiting the voltage magnitude within stable operating region and avoiding the violation in voltage magnitude caused by interactions of distributed generators (RES units) and the “charging effect” caused by non-negligible capacitive susceptance, which are achieved through this modelling framework. The proposed two-stage linear VVC problem can adjust the reactive power output according to active power fluctuations through the linear decision rule of fast control devices viz. RES inverters, in addition to deciding the set points of the classical voltage control devices. Numerical studies prove that the proposed two-stage linear stochastic VVC problem exhibits a better performance in eliminating voltage violations than the traditional deterministic case, while it has some acceptable increase in conservativeness (system power loss). The two-stage linear stochastic VVC problem based on LBFS preserves linearity and convexity, thus the computing efficiency is significantly improved when compared with traditional ACPF-based stochastic VVC methods.
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