Keywords

1 Introduction

With the increase permeability of photovoltaic, the randomness and uncertainty of distributed photovoltaic (DPV) output and the mismatch with load power, these problems make the voltage fluctuation of distribution network increase and lead to the problem of voltage exceeding the limit more prominent [1,2,3]. Traditional PV power supply usually works at the unity power factor and maximum power tracking control mode [4], which only plays the active power function of PV power supply. The revised scheme IEEE1547 and China’s technical requirements for grid-connected distributed generation (DG) (GB/T 33593-2017) stipulate that DG should participate in the voltage regulation actively and adjust the output of active and reactive power according to the requirement of voltage and power supply. The new PV grid-connected inverters basically have the function of controlling power remotely [5]. DPV participating in voltage regulation has many advantages, such as fast response, bidirectional adjustable reactive power, no mechanical wear and tear, etc. It plays an important role in solving the voltage problem of high-density PV access to the power grid.

DG voltage control mainly adopts three strategies: decentralized autonomous control, distributed control and centralized control. The first strategy [6,7,8] controls DG and other reactive power devices autonomously based on local measurement information, which has the advantages of no information communication, flexible configuration and strong real-time performance. However, due to the lack of overall synergy, it has limited global voltage control capability. Jia et al. [7, 8] proposed several DG active-reactive power coordination modes based on whether the PCC voltage exceeded the limit or not. Tonkoski et al. [9, 10] proposed voltage regulation strategies to prevent voltage from exceeding the limit by reducing the active output of PV power, but only based on local information, the active power cannot be minimized from the perspective of the whole network. Distributed control strategy [11] uses local intelligent controllers as nodes to build communication links. Controllers are relatively autonomous and cooperative, forming a peer-to-peer networked control system. The centralized control strategy adopts master-slave structure and optimizes the voltage level of the whole network at the control center. It has the advantages of wide control range and good voltage regulation effect, and mature application. Zhang et al. [12] proposes a reactive power optimization strategy combining DG with capacitors and transformers. Lv et al. [13] considered the coupling between active and reactive power of PV, coordinate and optimize the controllable active and reactive power resources in distribution network to regulate voltage, and adopt a multi-time scale optimal control scheme which combines day-ahead optimization and on-line correction. Huang et al. [14] aims at the problem of voltage limit caused by DG access to distribution network, a long-time scale reactive power optimization model for distribution network is established to synthetically optimize the DG reactive power output and reactive power compensation equipment. Fu et al. [15] proposes with controllable DG, energy storage, sectional switches and reactive power compensation equipment as control strategy, and with the objective of minimizing the operation cost and risk of limit the distribution network, an advanced optimization model of active distribution network is established. Ren et al. [16] optimizes coordination of DG, flexible load, energy storage, compensation capacitors and power transformers. A multi-time scale active-reactive power coordination optimization model is established to minimize the operation cost of distribution network.

PV inverters have large residual capacity in most operating time, and the active power output may be close to full capacity only in less time with very good weather conditions. Therefore, in the case of insufficient residual capacity, it is an economical and reasonable strategy to satisfy the voltage qualification by reducing active power. How to minimize active power reduction from the perspective of the whole network is a problem to be solved. In this paper, a voltage control method for distribution network considering DPV active power reduction strategy is proposed, and the optimization strategy is implemented according to whether voltage exceeds the limit.

The rest of the paper is organized as follows. Section 2 gives the complex affine model of the load and DPV. Sections 3 and 4 present the design plan of the coordination and optimization strategy for reactive-active power reduction with DPV. Section 5 shows how to set the model. And then the simulation results are shown in Sect. 6. Finally, Sect. 7 summarizes the conclusion.

2 Design of Complex Affine Model of DPV and Load

The active output of DPV fluctuates with weather conditions, and the load power consumption changes with time. Therefore, in the development of DPV grid-connected optimization strategy, it is necessary to consider the impact of the uncertainty of the two on the distribution network voltage. Among them, the most widely used is the probability-constrained planning method based on probability distribution [17], but it cannot change the PV output according to different weather conditions the next day. In this paper, the complex affine algorithm is used to establish the PV power supply and load uncertainty output model for each period. The affine mathematical method cannot only express this uncertainty, but also reduce the range of the solution in the calculation of power flow, while consider the completeness and accuracy of the solution.

2.1 Complex Affine Model of DPV

The DPV active output range is mainly related to the fluctuation of illumination intensity which has a great relationship with the change of clouds in the sky [18]. According to the weather information, the conditions of each time period can be obtained, and the affine expression of the cloud layer coefficient and the atmospheric layer illumination intensity is obtained by using the affine and interval conversion formula, which can be expressed as follows [18]:

$$ \hat{J}_{t} = J_{0t} + h\varepsilon_{{{\text{j}}t}} $$
(1)
$$ \hat{G}_{{{\text{a}}t}} = G_{0t} + k\varepsilon_{\text{gt}} $$
(2)

where \( \hat{J}_{t} \) and \( \hat{G}_{at} \) are the complex affine value of cloud layer coefficient and atmospheric layer illumination intensity in t period, \( J_{0t} \) and \( G_{0t} \) are the central value of change of cloud layer coefficient and atmospheric layer illumination intensity in t period, h and k are the ratio of change of cloud layer coefficient, \( \varepsilon_{jt} \) and \( \varepsilon_{\text{gt}} \) are the noise element of cloud layer coefficient and atmospheric layer illumination intensity in t period, the range of \( \varepsilon_{jt} ,\varepsilon_{\text{gt}} \), are [−1,1].

2.2 Complex Affine Model of Load

The output of power load is characterized by uncertainty and time series variation. Therefore, the complex affine method is used to build the output of active and reactive power models for each period which can be expressed as follows:

$$ \hat{P}_{{{\text{L}}it}} = P_{{{\text{L}}it}} + \delta_{{{\text{L}}t}} P_{{{\text{L}}it}} \varepsilon_{{{\text{L}}it}} $$
(3)
$$ \hat{Q}_{{{\text{L}}it}} = Q_{{{\text{L}}it}} + \delta_{{{\text{L}}t}} Q_{{{\text{L}}it}} \gamma_{{{\text{L}}it}} $$
(4)

where \( \hat{P}_{{{\text{L}}it}} \) and \( \hat{Q}_{{{\text{L}}it}} \) are complex affine value of active power and reactive power of node i load in t period, \( P_{{{\text{L}}it}} \) and \( Q_{{{\text{L}}it}} \) are real values of active power and reactive power of node i load in t period, \( \delta_{{{\text{L}}t}} \) is uncertainty rate, reflecting the relative magnitude of uncertainty, which is related to time t. \( \varepsilon_{{{\text{L}}it}} \) and \( \gamma_{{{\text{L}}it}} \) are noise element of active power fluctuation and reactive power fluctuation of node i load in t period.

3 The Coordination and Optimization Strategy for Reactive-Active Power Reduction with DPV

DPV usually employs the MPPT controller to ensure the maximum active power of PV. However, when there is a risk of over-limit voltage in distribution network, the output of active power should be reduced and the reactive power should be increased.

In this paper, a voltage control strategy for distribution network is proposed, which coordinates DPV reactive and active power reduction, and a step-by-step voltage and reactive power optimization strategy and model are established to fully ensure that the whole network voltage is qualified and voltage deviation is minimum. The optimization of reactive and active power reduction coordination strategy with DPV is shown in Fig. 1.

Fig. 1
figure 1

Flow chart of reactive-active reduction coordination optimization strategy with DPV participation

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4 The Coordination and Optimization Model for Reactive-Active Power Reduction with DPV

4.1 Objective Function

The objective function is to minimize the total voltage deviation in each period.

$$ F_{1} = \hbox{min} \left( {\sum\limits_{i = 1}^{n} {\left| {U_{it} - U_{0} } \right|} } \right) $$
(5)

where \( U_{it} \) is median value of node i voltage interval in t period, \( U_{0} \) is expected value of node voltage; m is number of divided periods; n is number of nodes.

The objective function is to maximize affine value of total active power of PV.

$$ F_{2} = \hbox{max} \left( {\sum\limits_{f = 1}^{{N_{\text{PV}} }} {\hat{P}_{{{\text{PV}}ft}} } } \right) $$
(6)

where \( N_{\text{PV}} \) is number of PV power supply and \( \hat{P}_{{{\text{PV}}ft}} \) is complex affine value of PV active power output in t period.

4.2 Constraint Condition

  • Equality Constraints

PV output, load power and reactive power of compensation device should satisfy the following power flow equation:

$$ \left\{ {\begin{array}{*{20}l} {\hat{P}_{it} = \hat{U}_{it} \sum\limits_{j \in i} {\hat{U}_{jt} } \left( {G_{ij} \cos \theta_{ij} + B_{ij} \sin \theta_{ij} } \right)} \hfill \\ {\hat{Q}_{it} = \hat{U}_{it} \sum\limits_{j \in i} {\hat{U}_{jt} } \left( {G_{ij} \cos \theta_{ij} - B_{ij} \sin \theta_{ij} } \right)} \hfill \\ \end{array} } \right. $$
(7)
$$ \left\{ {\begin{array}{*{20}l} {\hat{P}_{it} = \hat{P}_{{{\text{PV}}t}} - \hat{P}_{{{\text{L}}it}} } \hfill \\ {\hat{Q}_{it} = \hat{Q}_{{{\text{PV}}t}} + Q_{{{\text{c}}it}} - \hat{Q}_{{{\text{L}}it}} } \hfill \\ \end{array} } \right. $$
(8)

where \( \hat{P}_{it} \) and \( \hat{Q}_{it} \) are complex affine value of node i active power and node i reactive power in t period, \( \hat{P}_{{{\text{PV}}t}} \) and \( \hat{Q}_{{{\text{PV}}t}} \) are complex affine value of PV active power and PV reactive power in t period, \( Q_{{{\text{C}}it}} \) is output of node i reactive power compensation capacitor banks in t period, \( G_{ij} \) and \( B_{ij} \) are conductance and susceptance between node i and j, \( \theta_{ij} \) is angular phase difference of voltage between node i and j.

  • Node Voltage Constraints

$$ \left[ {U_{it} } \right] \subseteq \left[ {U_{\hbox{min} } ,U_{\hbox{max} } } \right] $$
(9)

where \( \left[ {U_{it} } \right] \) is voltage amplitude interval of the node i in the period, \( U_{\hbox{max} } \) and \( U_{\hbox{min} } \) is upper and lower limits of voltage to satisfy operation requirements, respectively.

  • Inverter Operation Constraints

The reactive power generated by DPV is determined by the operation capacity of PV inverters and the active power output by PV, which can be expressed as follows:

$$ 0 \le Q_{ft} \le \sqrt {S^{2} - P_{ft}^{2} } $$
(10)

where \( Q_{ft} \) is reactive power of PV in t period, S is installed capacity of PV inverters, \( P_{ft} \) is active power of PV inverters after reduction.

5 Model Solving Method

Ybus gauss iterative power flow calculation method has low requirement for initial value and does not need fast decoupling and its convergence will not be restricted by R/X [19]. LinWPSO algorithm overcomes the premature and oscillation phenomenon of the basic particle swarm optimization algorithm near the optimal solution, and has good convergence [13]. Therefore, this paper uses Ybus gauss iterative power flow calculation method and LinWPSO algorithm to solve the optimization model. The specific solution process is shown in Fig. 2.

Fig. 2
figure 2

Flow chart of active reduction optimization model solution

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6 Cases Analysis

The IEEE33-bus system is selected as a simulation example to verify the effectiveness of the optimization strategy proposed in this paper. The system topology is shown in Fig. 3. An on-load tap changer is connected between node 0 and 1, with a variable ratio from 0.95 to 1.05, a total of 9 gears and a regulation step of 1.25%. DPVs are connected at nodes 7 and 12 respectively, with an installed capacity of 500 kW for each DPV, and a reactive compensation capacitor bank is connected to nodes 17 and 32, with a single capacity of 150 kvar, totaling 8 units. The parameters of algorithm are set as follows: population size of LinWPSO is 50, learning factor c1 and c2 are 2.0, dimension D is 5, inertial weight \( \omega \) is 0.8, \( \omega \) is between [0.4, 0.9] with an algebraic linear decline, and maximum number of iterations T is 60.

Fig. 3
figure 3

IEEE33 node system structure diagram

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In this paper, one day’s PV output and load are analyzed. The output power complex affine model of PV power supply and load is known from 2.1 to 2.2 sections. Cloud layer coefficient affine value under different weather conditions are obtained from actual meteorological information, as shown in Table 1.

Table 1 Complex affine value of cloud layer coefficient
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According to Eq. (11), complex affine values of PV active power in each period can be obtained.

$$ \hat{P}_{{{\text{PV}}t}} = P_{STC} \frac{{\hat{G}_{{{\text{T}}t}} }}{{G_{STC} }}\left[ {1 - 0.005\left( {T_{\text{a}} + C\hat{G}_{{{\text{T}}t}} - 25} \right)} \right] $$
(11)

where \( P_{STC} \) is the maximum test power of PV system under standard test conditions taken as 408 kW; \( \hat{G}_{{{\text{T}}t}} \) is the complex affine value of illumination intensity in time period; \( G_{\text{STC}} \) is the illumination intensity under standard test conditions taken as 1000 W/m2; \( T_{\text{a}} \) is the ambient temperature, in units of ℃, C is 0.03.

Uncertainty of load data should be considered in power flow calculation. This paper assumes that uncertainty rates of load are basically the same in the same period, that means \( \delta_{t} \) is the same. Then, the affine values of different node load in each period are calculated according to the Eqs. (3) and (4).

6.1 Analysis of Reactive Power Optimization Results

On the basis of known DPV and load complex affine output, reactive power optimization is carried out by coordinated control of PV inverters, On-load tap changer and reactive power compensation capacitors. The optimization results show that the reactive power optimization taking into account the PV reactive power regulation ability can improve voltage level. However, the problem of voltage over-limit condition occurs in some nodes during the 11–14 period, as shown in Fig. 4a–c is voltage amplitude range of each node in 11–12, 12–13 and 13–14 periods, respectively.

Fig. 4
figure 4

Voltage results of the 11–14 period after the reactive power optimization

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The reason of voltage over-limit is that the output of PV active power is larger, even more than 70% of the installed capacity, which reduces the effective utilization capacity of reactive power regulation. Even if all the remaining capacities of the inverters are used for reactive power regulation, the node voltage cannot be adjusted to the normal range.

6.2 Analysis of DPV Active Power Reduction Optimization Results

The aim of active power reduction optimization is to ensure the maximum absorption of PV power supply on the premise of qualified voltage. The reactive power optimization results of 11–14 periods which do not satisfy the requirements are selected. The reactive power reduction optimization of PV power supply is carried out based on DPV-participated reactive and active power reduction coordination optimization model. The results are shown in Table 2.

Table 2 Photovoltaic power active output value
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From Table 2, we can see that the output value of active power reduction optimization is reduced from 20 to 40 kW compared with the initial median value of PV power supply. The active power reduction value of PV 1 at node 7 is more than that of PV 2 at node 12, which shows that the influence of PV 1 on voltage is greater than that of PV 2. Therefore, PV 1 can be operated preferentially when a successful reduction process is implemented.

6.3 Analysis of Reactive Power Optimization Results After Increasing Reactive Power Adjustable Capacity

The new range of PV reactive power output is obtained by solving the data substitution Eq. (10) in Table 3, and the reactive power optimization is carried out again. The reactive power value of PV supply in 11–14 period is obtained, as shown in Table 3. The optimized voltage results are shown in Fig. 5.

Table 3 Photovoltaic power reactive output value
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Fig. 5
figure 5

Voltage result of period 11–14 after active-reactive power coordination optimization

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Table 3 shows that all the additional residual capacity of PV inverters is used for reactive power optimization and voltage regulation, and the minimum reactive capacity is used to achieve voltage lower than the upper limit.

From Fig. 5, it can be seen that node voltages in Fig. 4 which do not satisfy the requirements fall into the [1.01, 1.05] range after PV active power reduction and reactive power optimization. Therefore, the coordinated optimal control strategy of reactive and active power reduction proposed in this paper can effectively solve the problem of over-voltage caused by inadequate reactive power capacity, and ensure the maximum output of PV.

7 Conclusion

A voltage coordination optimization method considering DPV active power reduction is proposed. DPV reactive power optimization model and active power reduction optimization model is established according to whether the voltage exceeds the limit or not, and the optimization strategy of DPV reactive and active power reduction coordination is established. The residual capacity of the inverters is used to optimize the voltage and reactive power. When the voltage requirement cannot be satisfied, minimizing DPV active output from a network-wide perspective, which ensures the voltage security and new energy absorption of the high permeability distribution network under complex and uncertain operation scenarios. The uncertain power output model of each period is established by complex affine method, and the adjustable range is given according to the weather conditions of the next day, so that the control strategy can better adapt to the uncertainties of the optimization scenario.