(1) Bidirectional Reactive Power Regulation Capabilities of Photovoltaic Inverters

Currently, many inverters operate under a constant power factor (CPF) mode. Taking a power factor of 0.95 as an example, the output reactive power varies between ±31.33%. When using a variable power factor control mode, during nighttime when the photovoltaic output is zero, the theoretical value of reactive power output can reach ±100%. At other times, the reactive power output of the inverter is influenced by the active power output [Samadi et al. (2012)].

(2) The technical feasibility of reactive power loss reduction by photovoltaic inverters.

The losses in distribution lines are shown in Formula 1: Δ P = P 2 + Q T 2 U 2 R = P 2 + Q L − Q S 2 U 2 R (1)

Where Q T , Q L , Q S are the variables represent the reactive power transmitted through the line, the reactive power of the load, and the reactive power local compensated by the inverter. By balancing the reactive power demand of the load on-site, the inverter can reduce the active power losses caused by reactive power transmission along the line. Additionally, reactive power compensation can also increase the voltage of each node, thereby further reducing losses.

When photovoltaic inverters participate in reactive power loss reduction, both the benefits and costs increase, as shown in Table 1. The benefits of distributed photovoltaics participating in reactive power loss reduction are divided into two parts, namely, direct benefits and indirect benefits. Direct benefits refer to the energy-saving benefits brought by the loss reduction during the statistical period. Indirect benefits refer to the electricity price discounts obtained by the users when the power factor is improved.

When photovoltaic inverters participate in reactive power loss reduction, it will lead to an increase in their operating costs, including: the capital investment costs of equipment that should be allocated to the use of reactive power regulation functions, the renovation costs incurred from developing regulation functions; the loss costs associated with the use of reactive power regulation functions; and the lifespan deterioration costs caused by utilizing reactive power regulation functions.

When the comprehensive benefits are positive, it is economically feasible for photovoltaic inverters to participate in reactive power loss reduction, and the maximization of comprehensive benefits should be pursued.

(1) Investment Costs

When the reactive power output function is enabled, it occupies the active output capacity of the photovoltaic equipment. The capital investment cost of distributed photovoltaic equipment should be allocated between active and reactive power transmission functions.

First, the capital investment cost of the inverter is allocated to the investment cost for each period according to its designed service life, as shown in Formula 2: R = λ * D − M − v (2)

Where D represents the capital investment cost of the inverter; R is the investment cost allocated for each period; M is the cumulative allocated cost for the current period; v is the estimated residual value; and λ is the depreciation rate.

Next, the opportunity cost for each period is calculated as shown in Formula 3: F = D * 1 + C a v m − D (3)

Where F represents the opportunity cost for each period; C _av is the average investment return in society; and m is the current interest period.

Finally, the investment cost allocated for the reactive power loss reduction function for each period is calculated as Formula 4 and recorded as C ₁. C 1 = β F + R (4) Where β is the allocation coefficient.

(2) Transformation Costs

The existing distributed photovoltaic equipment only needs to upgrade the inverter control software without replacing hardware, and can have bidirectional reactive power output function.

First, the total transformation cost is calculated as shown in Formula 5: C 2 ∑ = o * d * b T + E (5) Where T represents the number of distributed photovoltaics in a certain distribution network; o is the number of personnel involved in the upgrade; d is the number of working days; b is the labor cost per person per day; and E is the software upgrade fee for a single distributed photovoltaic.

Next, the transformation cost for each period is calculated as shown in Formula 6: C 2 = 2 N * C 2 ∑ − M 2 ∑ − v 2 ∑ (6) Where N represents the total number of allocation periods; M 2 ∑ and v 2 ∑ represent the cumulative allocated cost and the cumulative residual value, respectively. Using the accelerated depreciation principle and considering the decline in equipment value or production efficiency due to technological and market factors, it is necessary to allocate more costs in the earlier periods. Therefore, the depreciation rate is taken as 2/N.

(3) Operating Loss Costs

When participating in reactive power regulation, a larger ripple current increment is generated on the DC side, leading to an increase in the heating losses of electrolytic capacitors and equipment.

The power loss caused by reactive power transmission is calculated as Formula 7: Δ P Q = P 2 + Q 2 U N − P U N 2 × R C (7) Where P and Q represent the active operating value and the reactive operating value of the inverter, respectively. U _N is the rated voltage of the grid, and R _C represents the equivalent resistance.

The reactive power operating loss costs for each period is calculated as Formula 8: C 3 = Δ P Q * T * X P (8) Where T represents the statistical cycle, and X _P is the feed-in tariff for photovoltaic.

(4) Lifespan Loss Costs

The limiting factor affecting the lifespan of distributed photovoltaics is the lifespan of the DC-side capacitors, which is significantly influenced by temperature.

The lifespan of the electrolytic capacitors before participating in reactive power loss reduction is calculated as shown in Formula 9: K = K 0 * 2 T max − T a 10 (9) Where T _max and K ₀ represent the maximum rated temperature and the corresponding estimated lifespan, respectively. T _a is the actual operating temperature of the capacitor.

The temperature of the inverter is calculated as shown in Formula 10: T a = T s + R * P s (10) Where T _s is the room temperature, R is the thermal resistance between the inverter and the atmosphere, and P _s is the loss power.

The temperature increment of the inverter caused by reactive power output is calculated as shown in Formula 11: Δ T a = T s + R * Δ P Q (11)

The useable lifespan after participating in reactive power loss reduction is calculated as shown in Formula 12: K ′ = K 0 * 2 T max − T a − Δ T a 10 (12)

The lifespan loss cost per period is calculated as shown in Formula 13: C 4 = 2 × t K ′ D C − M C − v C − 2 × t K D C − M C − v C (13) Where D _C is the purchase price of the electrolytic capacitors, M _C is the already allocated cost, and v _c is the residual value, t is selected as the statistical interval.

In summary, the cost function for the distributed photovoltaic participating in reactive power loss reduction is established as Formula 14 and denoted as C: C = C 1 + C 2 + C 3 + C 4 (14)

(1) Benefit of loss reduction

First, the source-load operating scenarios are defined, and the loss reduction effect of inverter reactive power compensation is modeled under each scenario. The active power of distributed photovoltaics are categorized into large-scale, medium-scale, and small-scale scenarios. The loads are divided into large-scale and small-scale scenarios, leading to a total of six combined operating scenarios.

In these six scenarios, the line loss rate of the distribution network before the inverter participates in the regulation is calculated as σ _n0 (where n represents different scenario numbers). Then, the line loss rate of the distribution network after the inverter participates without compensation is calculated as σ _n .

The line loss rate of the distribution network varies depending on the reactive power compensation capacity. The function mapping relationship between reactive power of photovoltaic inverters and line loss rate is established using polynomial fitting, as shown in Formula 15. The goodness-of-fit is defined as the ratio of the regression sum of squares to the total sum of squares. When the goodness-of-fit is greater than or equal to 0.99, the fitting is complete. σ n = f n Q (15)

Next, the loss reduction quantity in the n-th scenario is calculated as Formula 16: Δ L n = T L P n 1 − σ n 0 * σ n 0 − P n 1 − σ n * σ n (16) Where P _n is the grid-connected power in the n-th scenario, T _L is the duration of the n-th scenario.

Finally, the total loss reduction benefit during the statistical period is calculated as Formula 17: B 1 = ∑ n = 1 6 Δ L n * X S A L E (17) Where X _SALE represents the electricity price during the statistical period.

(2) Benefit of power factor improvement

Based on the principle of electricity price adjustment according to the power factor, the formula for calculating the electricity price adjustment coefficient is as Formula 18: α = − 0.0075 + 96 − 100 ⁡ cos ⁡ θ × 0.0015   cos ⁡ θ ∈ 0.9 , 1 91 − 100 ⁡ cos ⁡ θ × 0.005   cos ⁡ θ ∈ 0.8 , 0.9 (18)

In the n-th scenario, the functional relationship between the reactive power capacity and the power factor is established as shown in Formula 19: cos ⁡ θ n = g n Q (19)

According to Formula 19, the power factors before and after the reactive power regulation of the photovoltaic inverter for the n-th scenario are calculated and denoted as cosθ _n0 and cosθ _n , respectively. Then, using Formula 18, the electricity price adjustment coefficients before and after the reactive power regulation are calculated and denoted as ɑ _n0 and ɑ _n .

The benefit of power factor improvement allocated to distributed photovoltaic enterprises for each period is calculated as Formula 20: B 2 = ε * L n * X * α n 0 − α n (20) Where L _n represents the total electricity consumption of industrial users during the statistical period, X is the benchmark electricity price for industrial users, and ɛ is the allocation coefficient.

In summary, the benefit function for distributed photovoltaics participating in reactive power regulation is established as Formula 21: B = B 1 + B 2 (21)

Based on the cost model and the benefit model, the comprehensive benefit of distributed photovoltaics participating in reactive power loss reduction is calculated and denoted as Formula 22: H = B − C (22) Where H, B, and C are all functions of Q. The optimization objective of this paper is to maximize the benefit. The value of Q is constrained by its maximum reactive power capacity and the inverter capacity, as shown in Formula 23: Q ≤ Q max Q ≤ S 2 − P 2 (23)

The genetic algorithm (Ji et al., 2022) is adopted to solve this model, and the flowchart is shown in Supplementary Figure S1.

In the simulation case, the specific parameters are shown in Table 2. According to Formulas 2–13, various costs are calculated. C 1 = β F + B = 395.9792 ¥   C 2 = 2 N * C 2 ∑ − M 2 ∑ − v 2 ∑ = 92.8889 ¥ C 3 = = 0.2028 × P 2 + Q 2 220 2 − P 220 2 C 4 = 24 × 2 Q 15000 − 1

The reactive power output of the converters is adjusted in the different scenarios, and the corresponding line loss rates and power factors are obtained through simulation.

Based on Table 3, the function of the line loss rate with respect to the reactive power output in Scenario 1 is fitted as follow: σ 1 = f 1 Q = 4 × 10 − 13 Q 3 + 6 × 10 − 9 Q 2 − 3 × 10 − 5 Q + 0.2502

Based on Table 3, the function of the power factor with respect to the reactive power output in Scenario 1 is fitted as follow: cos ⁡ θ 1 = g 1 Q = 3.3534 × 10 − 12 Q 3 − 1.1353 × 10 − 8 Q 2 − 9.1088 × 10 − 5 Q + 0.89028

Based on Table 4, the function of the line loss rate with respect to the reactive power output in Scenario 2 is fitted as follow: σ 2 = f 2 Q = − 2.123 × 10 − 10 × Q 3 + 2.07 × 10 − 7 × Q 2 − 0.0000526 × Q + 0.01371

Based on Table 4, the function of the power factor with respect to the reactive power output in Scenario 2 is fitted as follow: cos ⁡ θ 2 = g 2 Q = − 1.02795 × 10 − 6 × Q 2 + 0.001028 × Q + 0.7334

Based on Table 5, the function of the line loss rate with respect to the reactive power output in Scenario 3 is fitted as follow: σ 3 = f 3 Q = 10 − 13 Q 3 + 4 × 10 − 9 Q 2 − 2 × 10 − 5 Q + 0.0424

Based on Table 5, the function of the power factor with respect to the reactive power output in Scenario 3 is fitted as follow: cos ⁡ θ 3 = g 3 Q = − 1.15732 × 10 − 8 × Q 2 + 0.000127 × Q + 0.739667

Based on Table 6, the function of the line loss rate with respect to the reactive power output in Scenario 4 is fitted as follow: σ 4 = f 4 Q = 0.042367 + 5 × 10 − 13 Q 3 + 10 − 8 Q 2 − 8 × 10 − 6 Q

Based on Table 6, the function of the power factor with respect to the reactive power output in Scenario 4 is fitted as follow: cos ⁡ θ 4 = g 4 Q = − 8.65055 × 10 − 8 × Q 2 + 0.0001065 × Q + 0.972385

Based on Table 7, the function of the line loss rate with respect to the reactive power output in Scenario 5 is fitted as follow: σ 5 = f 5 Q = 0.1006 + 1 × 10 − 12 Q 3 − 1 × 10 − 9 Q 2 − 1 × 10 − 5 Q

Based on Table 7, the function of the power factor with respect to the reactive power output in Scenario 5 is fitted as follow: cos ⁡ θ 5 = g 5 Q = − 8.293368 × 10 − 9 × Q 2 + 6.83512 × 10 − 5 × Q + 0.871616

Based on Table 8, the function of the line loss rate with respect to the reactive power output in Scenario 6 is fitted as follow: σ 6 = f 6 Q = 0.0067 + 10 − 8 × Q 2 − 8 × 10 − 6 Q + 3 × 10 − 12 Q 3

Based on Table 8, the function of the power factor with respect to the reactive power output in Scenario 6 is fitted as follow: cos ⁡ θ 6 = g 6 Q = − 2.56766 × 10 − 7 × Q 2 + 0.000318 × Q + 0.916213

The functions f 1 Q ∼ f 6 Q are substitute into Formulas 15–17 to calculate the loss reduction benefit B ₁ for the statistical period.

The functions g 1 Q ∼ g 6 Q are substitute into Formulas 18–20 to calculate the power factor improvement benefit B ₂ .

With the goal of maximizing comprehensive benefits, global optimization is performed through the rotation of six scenarios over a 24-h period. The step size is set to 2 h. The optimization results are shown in Supplementary Figure S2.

An average reactive power adjustment in Scenario 6 is 290.67 kvar, which is from 0:00 to 6:00. From 6:00 to 10:00 is Scenario 4, with a reactive power adjustment of 2,332 kvar; from 10:00 to 12:00 is Scenario 2, with a reactive power adjustment of 3,000 kvar; from 12:00 to 14:00 is Scenario 1, with a reactive power adjustment of 2022 kvar; from 14:00 to 20:00 is Scenario 3, with a reactive power adjustment of 1769.67 kvar; from 20:00 to 24:00 is Scenario 6, with a reactive power adjustment of 557 kvar.

Before the participation of photovoltaic inverters in reactive power loss reduction, the line loss rates for the six periods were 25.02%, 1.37%, 4.24%, 4.24%, 10.05% and 0.67%, respectively. The power factors were 0.8904, 0.7434, 0.7437, 0.9708, 0.8726 and 0.9096, respectively. After the participation of photovoltaic inverters in reactive power loss reduction, the line loss rates for the six periods were 21.79%, 1.37%, 2.13%, 4.09%, 7.09% and 0.53%, respectively. The power factors were 0.9982, 0.9985, 0.9800, 0.9990, 0.9882 and 0.9990.

Compared with not participating in reactive power loss reduction, the cost of the inverter increased by 536.8681¥, while the line losses decreased in all six periods and the power factors improved. The comprehensive benefit amounted to 3,370.3577 ¥. When the photovoltaic inverter participates in loss reduction at the maximum reactive power value of 3,000 kvar throughout the day, its comprehensive benefit is calculated to be 2,655.8340¥, which is less than the optimized result described in this paper.

As shown in Table 3: The line loss rate in Scenario 1 before reactive power regulation reaches as high as 25.02%, and the reduction in losses after reactive power regulation is the most significant. Therefore, it is recommended to focus on reactive power loss reduction control for this scenario during dispatch operations.