Abstract
A photovoltaic (PV)-rich low-voltage (LV) distribution network poses a limit on the export power of PVs due to the voltage magnitude constraints. By defining a customer export limit, switching off the PV inverters can be avoided, and thus reducing power curtailment. Based on this, this paper proposes a mixed-integer nonlinear programming (MINLP) model to define such optimal customer export. The MINLP model aims to minimize the total PV power curtailment while considering the technical operation of the distribution network. First, a nonlinear mathematical formulation is presented. Then, a new set of linearizations approximating the Euclidean norm is introduced to turn the MINLP model into an MILP formulation that can be solved with reasonable computational effort. An extension to consider multiple stochastic scenarios is also presented. The proposed model has been tested in a real LV distribution network using smart meter measurements and irradiance profiles from a case study in the Netherlands. To assess the quality of the solution provided by the proposed MILP model, Monte Carlo simulations are executed in OpenDSS, while an error assessment between the original MINLP and the approximated MILP model has been conducted.
Set of phases,
L Set of lines
N Set of nodes
S Set of stochastic scenarios
T Set of time intervals
, Indices of phases, ,
, Indices of lines, ,
Indices of nodes,
s Index of scenario,
t Index of time interval,
, Parameters used for linearization of voltage magnitudes
The maximum line current limit
Power factor of photovoltaic (PV) inverters
Expected active power generation of PV systems
Rated power capacity of PV inverter
, Expected active and reactive power consumptions of customers
Resistance and reactance of lines
Length of time intervals
, The maximum and minimum voltage magnitudes
Real and imaginary parts of voltage magnitude at estimated operational point
Customer export limit
Real and imaginary parts of current injection of PV inverters
Real and imaginary parts of current injection of customers
Real and imaginary parts of current injections of lines
Net power injection of customers
Active power injection of PV inverters (AC side)
Active power export limit of customers
Real and imaginary parts of voltage magnitude
, Auxiliary variables used for linearization of voltage magnitude
Variable associated with export limit constraint
THE world-wide installed photovoltaic (PV) capacity is continuously increasing, with a total of 102.4 GW added only in 2018 [
Several solutions are available to cope with the technical issues as a result of the high penetration of PV systems in LV distribution networks. These solutions can be roughly classified in coordinated and locally-implemented approaches. Coordinated approaches require either a centralized [
With the above consideration, this paper presents a mixed-integer nonlinear programming (MINLP) model to define the optimal customer export limit of a PV-rich LV distribution network. This model aims to minimize the PV generation curtailment while considering the technical operation of the LV distribution network through a three-phase power flow formulation. Here, the operation of voltage regulators, capacitor banks, or switches has not been considered. Nevertheless, the proposed model can be easily extended following the modeling approach of such devices and using the models available in [
To linearize the proposed MINLP model, a new set of approximations based on the Euclidean norm is introduced to turn the model into an MILP model, which can be solved with a reasonable computational effort. An extension to consider stochastic scenarios is also presented, which enables the DSO to control the robustness of the solution provided by the proposed model. The proposed model is tested in a real LV distribution network using smart meter and irradiance profiles for a case study in the Netherlands. To assess the operation of the distribution network under the optimal customer export limit provided by the proposed model, Monte Carlo simulations in OpenDSS are executed. Although results are presented only for one case of study, the proposed model can be used first with a set of representative networks [
The main contributions of this paper are as follows. An MINLP model is presented to define the net customer power export limit in an LV distribution network with high PV penetration, considering the operational constraints of network. An accurate MILP formulation is presented, which approximates the original MINLP problem into a mathematical formulation that can be solved using commercial optimization solvers.
To ensure that individual net power injections defined in (1), i.e., PV generation minus consumption , do not violate technical constraints, a power limit can be defined. Imposing this power limit on the net power injection, as shown in (2), must trigger power curtailment on the PV systems. Notice that in this case, customers are modelled as non-flexible loads, thus consumption curtailment is not considered as an option. For simplicity, this power limit can be defined as a percentage of the rated power capacity of the PV inverter installed in phase , i.e., [
(1) |
(2) |
(3) |
Notice that such an export limit in (3) has the same value for all the residential customers within the distribution network, while its value can change dynamically with time. In order to better understand the net power generation export limit,

Fig. 1 Representation of power flow balance of residential customer .
Thus, if the net power injection is lower than the export limit, no PV generation curtailment is applied, then . In the case that is greater than the export limit, the PV system adjusts its generation to fulfill the export limit requirement, as shown in (4). Here, the efficiency of the PV inverter is considered when estimating from .
(4) |
As all the PV inverters connected to the distribution network will contribute to the aggregated reverse power flow and the increased voltage profile of the feeder, defining a unique export limit is considered as a social fair approach. Moreover, if such a limit is allowed to dynamically adapt to the operational conditions, e.g., low demand periods, high generation periods, of the distribution network, a lower amount of PV generation curtailment can be ensured, when compared with the case of a static value. Notice that although one export limit is defined system-wide, the total amount of curtailed active power depends on the nominal PV rate of each user. Finally, the definition of the same export limit for all customers might facilitate its implementation by the DSOs, which are required to be updated only if the number of PV installations increases or if the number of overvoltage issues starts to increase.
In this section, an MINLP problem formulation is presented to properly define the customer export limit for an LV distribution network. Then, a set of new approximations and linearization procedures is introduced in order to transform the MINLP model into MILP formulation that can be solved using commercial solvers. Finally, an extension considering multiple scenarios is also presented.
The definition of the custom export limit can be done using the MINLP model given by (1), (2), and (5)-(20). The objective function in (5) aims at minimizing the PV generation curtailment for the time horizon , which is equivalent to maximize the amount of active power provided by the PV systems.
(5) |
s.t.
(6) |
(7) |
(8) |
(9) |
(10) |
(11) |
(12) |
(13) |
(14) |
(15) |
(16) |
(17) |
(18) |
(19) |
(20) |
To model the output power of the PV systems at the AC side of the PV inverter, i.e., , described by the expression in (4), a binary variable is used. Thus, (4) can be replaced by the set of expressions in (6)-(9), where is a parameter with a large positive value. Notice that if , , i.e., no PV curtailment is applied; whereas if , , enforcing the export limit and performing PV generation curtailment. In both cases, is limited by the current PV generation (at the same time, which is a function of the current irradiance) at the DC side of the PV inverted by (8).
The unbalanced distribution network is modeled using the AC three-phase power flow formulation shown in (10)-(13). Constraints (10) and (11) model the real and imaginary line current balances, respectively. Constraints (12) and (13) model the real and imaginary voltage drop in lines, respectively. The active and reactive power consumptions of customers are modeled using (14) and (15), respectively, while the active and reactive PV generations of customers are modeling using (16) and (17), respectively. Notice that in (17), it is assumed that the PV inverter operates with unity power factor. Constraints (18) and (19) enforce the voltage magnitude limits and the thermal limits of lines, respectively. Finally, (20) defines the boundaries for the customer export limit .
Mathematical formulations such as that presented in Section III-A are difficult to solve, due to the nonlinear expressions used to model the active and reactive power consumption in (14) and (15), the PV generation of customers in (16) and (17), as well as the constraints used to enforce voltage magnitude and current limits in (18) and (19), respectively. Hence, precise linearization and approximations are used to transform the original MINLP formulation into an accurate MILP problem. Notice that although (6) and (7) are linear expressions, they can be more easily understood if they are expressed as:
(21) |
(22) |
To linearize (14) and (15), they are re-written as the functions and in (23) and (24), respectively.
(23) |
(24) |
Then, these can be approximated as linear expressions using a first-order Taylor series expansion around an estimated operational point , as explained in [
(25) |
(26) |
To linearize the voltage magnitude constraint (18), an approach based on the Euclidean norm is used. Recall that as [
(27) |
where and ; and and are the parameters. Hence, the square of voltage magnitude can be approximated as:
(28) |
where and are the parameters that depend on the maximum voltage deviation angles of the distribution system and are obtained following the fitting procedure previously proposed in [
(29) |

Fig. 2 Representation of voltage of phase in terms of its real and imaginary parts.
To apply the same approximation to phases , a linear rotation transformation is applied to have the same angle reference as phase .
In order to avoid the calculation of the nonlinear term in (29), two new continuous variables and are defined as in (30) and (31), respectively. These expressions are derived based on the fact that can be either or . Notice that only one of these expressions approximates the voltage magnitude correctly, corresponding to .
(30) |
(31) |
To enforce the voltage magnitude constraint in (18), variables and are forced to be within the required voltage limits and , as expressed in (32) and (33), respectively. Thus, (18) is approximated by adding (32) and (33) to the MILP formulation.
(32) |
(33) |
Regarding the cases of , it has been shown in [
Similar to the linearization of the voltage magnitude, to linearize (19), the Euclidean norm of a vector can be approximated using the expression in (34) [
(34) |
where ; ; and . Thus, (34) can be extended to estimate the square of current magnitude as [
(35) |
where ; and . Notice that to avoid the estimation of and , as being nonlinear functions due to the max and min operators and the absolute values, all the expressions derived among all the possible combinations, whereas if and are equal to or , can be added to the MILP model and limited by . These new sets of constraints shown in (36)-(43) enforce the maximum current magnitude constraint in (19).
(36) |
(37) |
(38) |
(39) |
(40) |
(41) |
(42) |
(43) |
Thus, the optimal customer export limit can be calculated solving the next MILP formulation, where we minimize (5) subject to: (1), (2), (6)-(9), (10)-(13), (20), the linearized versions of (23) and (24)-(26), (30)-(33), and (36)-(43). An assessment of the error introduced by the proposed linearizations/approximations are presented in Section IV-C.
To cope with multiple scenarios of PV generation and load consumption of customers, the MILP formulation presented in Section III-B can be extended as a scenario-based stochastic model. Thus, different irradiance profiles (used to estimate the PV generation profile of each customer based on their PV system rate ) and active power consumption are considered. For each scenario , the proposed MILP model is solved and an optimal customer export limit is defined. The final customer export limit can be defined by the DSO based on a robustness criterion. Therefore, if the DSO desires to comply the technical constraints (voltage and current magnitude limits) for all the 100% scenarios in the set , can be defined as:
(44) |
The proposed MILP model is used to study one case in the Netherlands. The LV distribution network used corresponds to a real distribution network, provided by a Dutch DSO. This LV distribution network, as shown in

Fig. 3 Real Dutch residential LV distribution network.

Fig. 4 Mean and standard deviations of set of 350 irradiance scenarios.
The proposed MILP model is implemented in Python language, using the optimization language Pyomo, and solved with CPLEX. To obtain the results presented below, first, a relaxed version of the original MINLP model is solved, obtained after relaxing all the integer variables. The solution of this relaxed model provides the estimated operational point needed to solve the proposed MILP model. Then, the proposed MILP model is solved for each scenario independently, and the optimal customer export limit is defined, as shown in

Fig. 5 Optimal customer export limit obtained after solving proposed MILP model for all Monte Carlo scenarios.

Fig. 6 Convergence of mean and standard deviations of for time step at 12:30 for Monte Carlo simulation.
To analyze the obtained results in terms of the optimal customer export limit ,

Fig. 7 Experimental CDF of for time step at 12:30.
The observed peak value (close to ) in

Fig. 8 Optimal customer export limit obtained from MILP model using 350 stochastic scenarios.
In order to estimate the impact of the introduction of the optimal customer export limit on the operation of the distribution system,
Robustness criterion (%) | PV generation (kW) | Mean power loss (kW) | Mean exported active power (kW) |
---|---|---|---|
100 | 9114.81 | 281.67 | 7939.65 |
95 | 9154.60 | 285.03 | 7976.09 |
90 | 9163.28 | 285.77 | 7984.02 |
To show the operation of a customer under the defined export limit,

Fig. 9 Active power consumption, PV generation (with and without a definition of an export limit), power export limit, and total active power exported by user 179 in a scenario of high irradiance and low consumption simulated in OpenDSS.

Fig. 10 Voltage magnitude of user 179 with and without export limit in a scenario of high irradiance and low consumption simulated in OpenDSS.
To show the effectiveness of the definition of the customer export limit under different robustness criteria, and ensure that the voltage magnitude values are within the expected limits in the defined maximum number of scenarios,

Fig. 11 CDF of voltage magnitude of user 179 obtained from Monte Carlo simulation process in OpenDSS considering different robustness criteria for customer export limit.
To assess the error introduced by the proposed linearizations/approximations, a comparison of the results with the solutions obtained by different models is presented in
Phase | Voltage magnitude (p.u.) | Error (%) | |||
---|---|---|---|---|---|
MILP model | SOCP model | NLP model | OpenDSS | ||
A | 1.0805 | 1.0805 | 1.0813 | 1.0836 | 0.074 |
B | 1.0809 | 1.0809 | 1.0817 | 1.0848 | 0.077 |
C | 1.0816 | 1.0817 | 1.0823 | 1.0853 | 0.062 |
According to the results presented in
Finally, in terms of computational time, to solve the proposed MILP model, a total of 4.28 s (CPU time) is required, while a total of 3.40 s is required for the NLP model (after fixing the binary variables). These results show that the proposed model can be solved fast enough for practical applications.
An MINLP model to define the customer export limit in PV-rich LV distribution networks is presented. A new set of accurate linearizations is introduced in order to turn the proposed model into an MILP formulation that can be solved using commercial solvers. An extension to consider stochastic scenarios is also presented. The proposed model is tested in a real distribution network using a database of real residential smart meter measurements. To assess the quality of the obtained solution, Monte Carlo simulations are executed in OpenDSS. According to the obtained results, the proposed model is able to successfully estimate the optimal customer export limit that guarantees the minimum PV curtailment and comply with the technical constraints. Nevertheless, as defining a customer export limit might discourage the installation of new and large-size PV systems or even be seen as an inefficient approach, this should not be seen as a long-term solution. Instead, the proposed model can be used by the DSOs to characterize and identify the networks with the lowest export limit. Then, long-term actions aiming to increase the PV hosting capacity of the distribution networks might be implemented. In this sense, a more comprehensive approach, comparing the long-term economic impact of imposing an export limit or implementing a local or coordinated voltage control strategy, is required. Finally, the comparison results between the MINLP and the proposed MILP model are also presented to assess the accuracy of the proposed linearizations. Negligible errors are obtained when comparing the proposed MILP model with the original MINLP formulation.
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