Abstract
This paper proposes a stochastic programming (SP) method for coordinated operation of distributed energy resources (DERs) in the unbalanced active distribution network (ADN) with diverse correlated uncertainties. First, the three-phase branch flow is modeled to characterize the unbalanced nature of the ADN, schedule DER for three phases, and derive a realistic DER allocation. Then, both active and reactive power resources are co-optimized for voltage regulation and power loss reduction. Second, the battery degradation is considered to model the aging cost for each charging or discharging event, leading to a more realistic cost estimation. Further, copula-based uncertainty modeling is applied to capture the correlations between renewable generation and power loads, and the two-stage SP method is then used to get final solutions. Finally, numerical case studies are conducted on an IEEE 34- bus three-phase ADN, verifying that the proposed method can effectively reduce the system cost and co-optimize the active and reactive power.
CURRENTLY, the optimal operation of active distribution networks (ADNs) plays an essential role in the effective integration of renewable-based generators, optimal energy management, power loss reduction as well as cost saving [
In practice, ADNs are generally unbalanced. For instance, the unbalanced line configuration including two- or single-phase laterals downstream of feeder backbones is common in ADN. meanwhile, the unbalanced demand consisting of single or two phases loads is prevalent [
To characterize the unbalanced ADN, it is necessary to implement the three-phase power flow on the scheduling framework. In [
In addition, based on the three-phase unbalanced system structure, active and reactive power flows for the three-phase system are supposed to be co-optimized. However, traditionally, the active and reactive power for ADN operation is optimized separately [
Apart from the comprehensive active and reactive power modeling, the key components in the ADN should also be characterized precisely. The ESS is a critical component in the ADN for both peak shaving and cost saving [
Finally, the uncertainties brought by RES and power load should be tackled in ADN operation. Broadly speaking, robust optimization (RO) and stochastic programming (SP) are the two main methods to address diverse uncertainties. The RO benefits from high computing efficiency, but the robust decisions hedging against the worst cases may suffer from over-conservativeness [
Given the insights above, this paper studies a two-stage SP method for the coordinated operation of the DER in unbalanced ADN considering diverse correlated uncertainties. The main research contributions are summarized as follows.
1) A comprehensive operation model for unbalanced AND, involving the linearized three-phase branch flow model, is proposed to schedule the DG generation separately among three phases and calculate the system cost more practically and precisely.
2) The unbalanced ADN operation method can optimally dispatch the active and reactive power simultaneously by applying the VVC scheme and regulating the voltage to derive a more realistic operation. In addition, the ESS degradation model is also included to model the aging cost through each charging/discharging event to obtain more accurate solutions.
3) Diverse uncertainties from the RES generation and power loads are addressed via the copula theory to capture the correlation relationship between the output from PVs, WTs, and the demand of each phase. Then all the uncertainties are tackled via the two-stage SP method.
The rest of this paper is organized as follows. The proposed unbalanced ADN model is presented in Section II. The mathematical formulation and solution methodology are discussed in Section III and Section IV, respectively. Numerical case studies are demonstrated in Section V, and finally, the conclusion is drawn in Section VI.
The framework of unbalanced ADN is provided in

Fig. 1 Framework of unbalanced ADN.
In general, the ADN is unbalanced. The relatively high degree of unbalance is caused by the naturally transposed architecture of three-phase distribution lines, the existence of single-phase laterals, single- and double-phase loads, or unbalanced three-phase loads [
In this paper, the linearized three-phase branch flow model developed in [
In this model, we first apply Kirchhoff’s voltage law for each line connected to an ordered buses pair and the voltage relationship can be obtained as:
(1) |
is expressed in (2), and * dentoes conjugate.
(2) |
Substituting (2) into (1) and multiplying both sides by their complex conjugate, we can obtain (3), where is the apparent branch power of bus pair . Operators and in the model denote the elementwise division and multiplication, respectively.
(3) |
where the last term is the higher-order term. To conduct the linear approximation of power flow, the following two assumptions are made in [
1) Line power losses are small, i.e., , which can be neglected in the model.
2) Voltages are nearly balanced, so we have:
(4) |
Therefore, substituting (4) into (3) and omitting the higher-order term , (3) can be simplified as:
(5) |
The impedance matrix , and is denoted as:
(6) |
Combined with the power balance constraints for the proposed unbalanced ADN operation model, the linearized three-phase power flow can be formulated in (7)-(9).
(7) |
(8) |
(9) |
Equations (
The life cycle of ESS units can be used to evaluate the ESS degradation through each charging or discharging process [
(10) |
(11) |
The relationship between expected average cycle and DoD for the Li-ion battery is shown in

Fig. 2 Relationship between expected average cycle and DoD.
Based on the definitions above, the ESS degradation cost model for each charging/discharging event can be identified in (13).
(12) |
(13) |
The mathematical formulation of the proposed method is described based on a typical unbalanced ADN. The active power scheduling of DE, WT, PV, and ESS can be optimized, for the reactive power dispatch, operation schedules of the CB, OLTC, and the electronic converters for WT and PV would be decided. The overall deterministic ADN operation model is given as follows.
1) Objective function: the proposed unbalanced ADN operation aims at minimizing the system operation cost as:
(14) |
(15) |
(16) |
(17) |
(18) |
(19) |
(20) |
(21) |
(22) |
(23) |
2) Constraints:
(24) |
(25) |
(26) |
(27) |
(28) |
(29) |
(30) |
(31) |
(32) |
(33) |
(34) |
(35) |
(36) |
(37) |
(38) |
(39) |
(40) |
Formulas (
The mathematical function of the proposed model in (14)-(40) is nonlinear due to the nonlinear constraints (12), (13), and (21)-(23). Solving the nonlinear term directly is time-consuming and ineffective. To release the computation burden and improve the solution accuracy, the linearization and relaxation methods are adopted in this paper.
Equations (
(41) |
(42) |
(43) |

Fig. 3 Piecewise linear relaxation of a convex quadratic function.
In (41), each linear segment can be expressed by the slope and intercept . The binary variable denotes the binary state of each linear block, as shown in (43).
To describe the stochastic interdependence between uncertain variables (WT, PV, and power load), the copula function is demonstrated here to capture the correlations between them. The stochastic scenarios are then involved in the unbalanced ADN operation model to address uncertainties.
The copula is a multivariate cumulative distribution with uniform marginals of each variable on the interval [0, 1] [
(45) |
Hence, differentiating (45), the joint probability distribution function of variables can be obtained, as shown in (46), in which the copula density function is formulated as (47). The conditional density functions can be expressed as (48).
(46) |
(47) |
(48) |
For the high-dimensional models, various pair copulas will be constructed for scenario generation to reveal the hidden association of uncertain variables. In this paper, the Gaussian and Gumbel copula families are utilized whose cumulative distribution functions are expressed as (49) and (50). is the univariate standard normal distribution; is the bivariate normal distribution with zero means; is the unit variance and correlation parameter [
(49) |
(50) |
To handle the various uncertainties, the proposed operation model is converted to a two-stage SP problem indicated in

Fig. 4 Two-stage structure for unbalanced ADN operation.
The day-ahead ADN operation covers a longer timescale, with the given information of input prediction and operation parameters. In this paper, the day-ahead decisions include ESS charging/discharging power, on/off status of DEs, tap position of OLTC, and position levels of CB.
The intra-day ADN dispatch is conducted in a much shorter operation timescale, normally no more than one hour. The intra-day decisions will be made after the realization of various uncertainties and the operational decisions from the day-ahead stage [
The classification of the two stages depends on the different roles each unit plays and its response speed. ESS schedule is decided in the first stage since it mainly contributes to peakshaving and flexibility improvement over long-time periods. Besides, charging or discharging frequently would incur a higher degradation cost (19) and shorten its lifespan. The CB status and OLTC status are also scheduled in the first stage, as they cannot respond fast and the frequent movements could reduce their lifetime dramatically [
To deal with diverse uncertainties, massive scenarios can be generated considering correlations among all the uncertainty sources in Section III-A. However, too many scenarios would lead to an excessively high computational burden. To improve the solution efficiency, the scenario reduction method and simultaneously backward reduction (SBR) technique can be utilized to select a rather small but representative scenario set [
(51) |
In (51), D(m) is the objective of the day-ahead stage related to (17), (19), (22), and (23), and m denotes all the corresponding decisions discussed in Section III-B; s is the index of the representative scenarios; is the constraint set, including (28)-(32) and (36)-(39), related to the decision m; is the number of total scenarios; is the probability of scenario; L() is the object of intra-day ADN dispatch after revealing the uncertainties, consisting of (16), (18), (20), and (21), in which is the intra-day decision variable; and is the constraint set, involving (7)-(9), (24)-(27), and (33)-(35), corresponding to the decision . It is noteworthy that the proposed two-stage SP method has some limitations, including the fixed stochastic variations and limited scenario numbers, which can be regarded as the future research direction of the SP method.
The proposed method is validated via an IEEE 34-bus distribution system, whose topology is shown in

Fig. 5 Topology of IEEE 34-bus distribution system.
The bus voltage limit is set to be [0.95 p.u., 1.05 p.u.]; the substation voltage is 1 p.u.; the tap range of substation is set to be 5% with 20 tap positions, so and . There are five CBs installed at all three phases of the buses 812, 850, 824, 862, and 834, with the same capacity of 300 kvar for each unit [
The energy tariffs [
Parameter | Price ($/kW) |
---|---|
DE maintenance cost | 0.0288 |
RES maintenance cost | 0.0093 |
RES curtailment cost | 0.0050 |
Power transaction | 0.0768 (during 00:00-06:00, 23:00-24:00) |
0.1276 (during 06:00-08:00, 11:00-17:00) | |
0.1696 (during 08:00-11:00, 17:00-22:00) |

Fig. 6 Predictions of RES generation and power demands. (a) Active power demands. (b) Reactive power demands. (c) PV output. (d) WT output.
All the case studies are conducted on an Inte
Based on the historical data of RES generation and power loads, 1000 scenarios are generated by applying copula theory and reduced to 10 representative scenarios by the SBR technique [

Fig. 7 Comparison of 300 samples and 500 samples via copula theory.
The OLTC tap positions and the voltage profile of the reference bus are presented in

Fig. 8 OLTC tap positions and voltage profile of reference bus.
To demonstrate the impact of an unbalanced ADN on day-ahead operation decisions, the power output results of ESS are shown in

Fig. 9 Power output results of ESS.
The solution time of the proposed day-ahead operation is 1656.41 s, which indicates that the proposed method is compatible with real-world applications and efficient enough for the day-ahead operation.
The intra-day ADN dispatch is conducted on hourly bases with the realization of RES and load uncertainties. The simulation results are demonstrated in

Fig. 10 Intra-day active power dispatch results for three phases. (a) Phase a. (b) Phase b. (c) Phase c.
The reactive power balance is presented in

Fig. 11 Intra-day reactive power dispatch results for three phases. (a) Phase a. (b) Phase b. (c) Phase c.
Moreover, the hourly root branch three-phase active power unbalance are depicted in

Fig. 12 Hourly root branch three-phase active power unbalance.

Fig. 13 DE power outputs.
The intra-day ADN dispatch results verify the effectiveness of the proposed method of coordination of all the DER units and reactive power devices to attain both optimal active and reactive power managements. The final cost of intra-day ADN dispatch over 24 hours is $8978.32 and the solving time for the intra-day operation is 6.24 s for each hour. This solution performance also validates that the proposed method is compatible with the intra-day operation.
To illustrate the validness and effectiveness of the proposed method, other unbalanced distribution system operation benchmarks are compared.
Method A (MA): this is the deterministic operation method and the forecast of the RES generation and load are regarded as accurate [
Method B (MB): the centralized VVC scheme is not involved. This means the coordination between active and reactive power dispatch does not exist and all the reactive power is satisfied by only renewable inverters and the main grid [
Method C (MC): the ESS degradation is not considered where the ESS capacity assumes to be constant during each charging/discharging event [
The optimization results of these benchmarks are demonstrated in Table II. Besides, one of the comparing criteria called the hourly unbalance rate , which is the average of three-phase unbalance , is defined in (52).
(52) |
Method | First stage | Second stage | ||||
---|---|---|---|---|---|---|
Operation cost ($) | Solution time (s) | Operation cost ($) | Power loss cost ($) | Hourly unbalance rate (p.u.) | Solution time (s) | |
MA | 8395 | 23 | 8778 | 244 | 0.023 | 6.37 |
MB | 8835 | 812 | 9128 | 458 | 0.045 | 6.72 |
MC | 8534 | 424 | 8856 | 243 | 0.039 | 6.08 |
Proposed | 8633 | 1656 | 8978 | 244 | 0.042 | 6.24 |
From the comparison of simulation results in Table II, it can be inferred that:
1) For MA, the system operation cost and power loss cost are lower compared with all methods with the least solution time. This is rational as only one scenario is considered with a smaller problem dimension. However, for this method, it is not practical because in reality, the accuracy of uncertainty predictions cannot be guaranteed completely.
2) For MB, without the modeling of the VVC scheme, the reactive power balance is only supported by the main utility. The transmission of a great amount of power to a single bus will scarify higher power loss costs. It can be observed that the power loss cost in MB is twice as high as the proposed method. Hence, to reduce the power losses and build up a cost-effective model, the coordination of active and reactive power is necessary to be considered in the ADN operation.
3) For MC, the constraints in terms of ESS degradation are ignored, and the operation cost is lower than the proposed method with a better computational performance since the linearization of ESS degradation involves many binary variables. However, the neglection of ESS degradation leads to imprecise and idealistic results.
4) Compared with MA to MC, the proposed method has lower power loss and reasonable operation costs. of the proposed method is similar to those of MB and MC. Because the three-phase imbalance is mainly about active power dispatch and if the reactive power optimization is ignored in MB, it would not influence the active power scheduling and the three-phase unbalanced power. From Table II, the obtained results for the second stage are slightly higher than the first-stage operation cost results. Those differences are brought by the impact of the individual scenario in the second-stage simulation. The results from the first stage are the expected value of all the various covered scenarios, while the second stage is significantly affected by the revealed uncertainty.
To further indicate the effectiveness of the proposed method, the sensitivity analysis is conducted based on different numbers of scenarios. As the proposed method is solved with 10 scenarios reduced from 1000 scenarios, for the comparison, 5, 15, and 20 scenarios are utilized. The comparison of different numbers of scenarios is listed in Table III, where S5 to S20 means we use 5 to 20 scenarios for simulation.
Item | Power loss cost ($) | Hourly unbalance (p.u.) | Solution time (s) |
---|---|---|---|
S5 | 327.76 | 0.0443 | 598 |
S10 | 244.86 | 0.0421 | 1656 |
S15 | 246.94 | 0.0417 | 19519 |
S20 | 248.82 | 0.0435 | 65499 |
The results show that there is a significant difference in system operation cost and power loss cost between S5 and S10. This is because for fewer scenario cases, each scenario has a critical impact on the simulation results and a single extreme case will greatly affect the system cost. With the increasing number of scenarios, the results are similar but a longer solution time is taken. From the sensitivity analysis results, it can be observed that 10 scenarios can be enough to balance the solution accuracy and solution time.
To illustrate the impact of changing the power unbalance limits on the system operation, we have selected four extra cases (cases 1-4) whose three-phase power unbalances are 20% and 10% lower than the proposed method; and 20% and 10% higher than the proposed method, for which Taking case 1 as an example, and are set to be 20%, which are lower than the proposed method, and .. The results of this sensitivity analysis are presented in

Fig. 14 Results of sensitivity analysis for three-phase power unbalance limits.
From
This paper proposes an optimal coordinated operation method for the unbalanced ADN. The active and reactive power can be jointly optimized by utilizing the VVC scheme to regulate the voltage and reduce the power losses. The ESS degradation model is also considered to characterize the aging impact of each charging/discharging event. Copula theory is applied for handling uncertainty by recognition of correlations of all the uncertainty sources. Finally, numerical case studies are conducted to illustrate the following aspects.
1) The linearized three-phase branch flow model can schedule DER output differently among three phases and derive a more precise and realistic operation cost. The ESS aging cost also verifies that the degradation process has a neglectable impact on system cost.
2) The proposed method enables the joint optimization of the active and reactive power effectively in the unbalanced ADN. Besides, the power losses can be effectively reduced.
3) The copula-based two-stage SP method could correctly reveal the correlation between dependent uncertain variables, which effectively addresses the diverse uncertainties.
Nomenclature
Symbol | —— | Definition |
---|---|---|
A. | —— | Sets and Indices |
b, i, p | —— | Indexes for branches, buses, and phases |
—— | Numbers of phases and candidate buses | |
—— | Index and number of dispatch period | |
B. | —— | Parameters |
, | —— | Rated charging life, life cycle, and rated life cycle of battery energy storage system (ESS) |
—— | Maintenance costs of diesel generator (DE), wind turbine (WT), photovoltaic (PV), and ESS | |
—— | Emission conversion from diesel generator (DE) | |
—— | Wind turbine (WT) and photovoltaic (PV) power curtailment costs | |
—— | Unit power loss cost | |
—— | The minimum and maximum allowed charging power | |
—— | The minimum and maximum allowed discharging power | |
—— | Decay rate, charging efficiency, and discharging efficiency of ESS | |
—— | Start-up and shut-down costs of each DE unit | |
—— | Degradation cost of battery ESS | |
—— | Rated depth of discharge (DoD) | |
—— | The minimum and maximum energy stored in each ESS unit | |
—— | Rated battery ESS capacity | |
—— | Unit investment cost of ESS | |
—— | Electricity purchasing and selling prices | |
—— | Number of battery ESS life cycle | |
—— | The minimum and maximum power output rates of DE | |
, | —— | Active and reactive power demands |
—— | Available energy resources from WT and PV | |
—— | Hourly maximum three-phase active power unbalance limit | |
—— | The minimum and maximum allowed reactive power of capacitor bank (CB) | |
—— | The minimum and maximum allowed reactive power of PV | |
—— | The minimum and maximum allowed reactive power of WT | |
—— | Ramping-up and ramping down rates of DE | |
—— | The maximum transformer power limit | |
—— | Unit adjustment levels of CB and on-load top changer (OLTC) | |
—— | Total active three-phase power unbalance limit at the root branch | |
—— | The minimum and maximum voltage square limits | |
—— | Voltage of substation | |
, | —— | Parameters of life cycle curve fitting |
C. | —— | Variables |
—— | Binary variable for DE on and off statuses | |
—— | RES curtailment and power loss costs | |
—— | Gas emission and battery aging costs | |
—— | Power transaction and maintenance costs | |
—— | Start-up and shut-down costs | |
—— | DoD of ESS units | |
—— | Energy stored in battery ESS units | |
—— | Three-phase line current vector, | |
—— | Functional relationship of ESS life cycle in terms of DoD | |
—— | DE, WT, and PV power outputs | |
—— | Charging and discharging power of ESS | |
, | —— | Active and reactive power flowing on the lateral branch of branch b+1 |
,, | —— | Root branch power flow for phase a, b, and c |
—— | Purchasing and selling power between distributed system and power grid | |
—— | Hourly root branch three-phase active power unbalance | |
—— | Reactive power output from CB | |
—— | WT and PV reactive power outputs | |
—— | Position levels of OLTC and CB | |
—— | The minimum and maximum tap change ratios | |
—— | Voltage vectors, and | |
—— | Bus voltage squares for phases a, b, and c | |
—— | Hourly three-phase power unbalance rate | |
, | —— | Three-phase voltage vectors at buses i and j, and |
—— | Line impedance from buses i to j | |
—— | Line impedance matrix, denoted by the complex form of line resistance and reactance , |
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