1. Introduction
Integrating high penetrations of variable renewable sources into the electric power grid presents a range of unprecedented grid operations and planning challenges requiring new study approaches and tools so that the impacts of a proposed wind power plant (WPP) can be properly assessed prior to interconnection [
1,
2,
3,
4,
5]. In particular, increasing attention has been drawn to power quality concerns: harmonics and the output harmonic impedance of wind turbines (WTs) and the aggregating WPP are stochastic due to the inherent intermittency of the wind and wide control bandwidth of the power electronics interfaces [
6,
7]. These wideband stochastic harmonic emissions from modern converter-based WTs may adversely interact with the grid via vast underground or submarine cable systems, leading to many series and parallel resonance points that significantly change with the harmonic filters, feeder configuration and reactive power compensation devices associated with the number of WTs in operation [
1,
7,
8,
9].
Probabilistic aspects of harmonics in electric power systems especially due to dynamic nonlinear loads have long been observed, and numerous research efforts suggest that statistical representations and analyses of time-varying harmonics are desired and present methods for determining the statistical distributions of harmonics and limits [
10,
11,
12,
13,
14]. However, little effort for representing harmonic characteristics of the renewable generation has been exerted in system planning studies. Recent analyses of harmonic emissions from WTs by type interestingly claim that we may statistically characterize harmonic components with certain probabilistic distributions independent of the wind power generation (i.e., operating condition) in contrast with the fundamental current, which should be proportional to the wind power [
7,
8,
15]. Resonance concerns in line with these harmonics arise when one of the harmonics coincides with the resonance frequency. It is well understood that the resonance points vary with changes in the system operating conditions; for example, the number of operating WTs due to wind maintenance or failure and the operation of filters and switched shunts [
1,
16,
17]. An IEEE working group collaboratively investigated these harmonics and resonance issues within a WPP and summarized the compliance, mitigation measures and analysis methods, such as the harmonic impedance scan [
18]. It is interesting to note the efforts to identify the spectral characteristics of the WPP by aggregating those of WTs [
19,
20].
Although harmonics and resonance issues have been investigated and well reported as above, the grid planners are increasingly challenged due to the lack of a comprehensive and practical planning study process and supporting tools, to the extent of authors’ knowledge and experience. They still rely on deterministic analysis often based on the worst case scenarios without effectively incorporating observations and knowledge gained through extensive efforts recently [
19,
20]. The study scope and results could thus be limited and possibly lead to unnecessary capital investment. Pursuing comprehensiveness, flexibility and practicality for implementing a supporting tool of the approach, this research proposes a statistical modeling and planning study methodology for investigating the stochastic harmonics and resonance concerns of WPPs. We observe that existing methods, such as random harmonic phasor summation [
21] or harmonic summation introduced in the standard IEC 61400-21, are unable to fully capture the harmonic behavior of modern WPPs, and the results from both methods could be misleading [
22,
23,
24]. By incorporating the statistical distribution of each harmonic component of interest from WTs and representing the overall time-varying harmonic behavior of WPPs, system planners can synthesize credible operating scenarios and investigate and collectively quantify harmonics and resonance concerns. The deterministic approach or its variation by incorporating certain correlations in the generators, for example operating closely, can also be taken in this proposed framework for comparison. System operators can exploit both stochastic and deterministic approaches, as well, to investigate unforeseen problematic operating conditions and, thus, systematically develop mitigation or correction measures. The proposed modeling and analysis framework is presented in
Figure 1 and will be elaborated in the paper.
This paper is structured as follows: Methods for statistic modeling are reviewed in
Section 2.
Section 3 presents a proposed method designed for generating harmonic emissions of the WTs. Next, a modeling of a WPP is presented in
Section 4. Harmonics for a single turbine model are assessed in
Section 5 followed by a practical harmonic evaluation for a realistic test system designed based on an offshore WPP (100 MW) under development in Korea in
Section 6.
2. Statistical Modeling
As illustrated in
Figure 1, the approach develops statistical models of fundamental and harmonic currents from WTs and WPPs based on prior theoretical and empirical knowledge based on actual harmonic measurements of the WTs and WPP at the point of connection (PoC), which should be the case in operations planning. In grid planning, simulation results from high-fidelity models and data from the manufacturers can be used for this purpose because field measurements are not generally available. Measurements from already operating sites may also be considered for use as long as similar WTs and operating conditions are to be adopted: the offshore WPPs under development and in planning in Korea should be the case.
To capture the statistical characteristics of each harmonic component, we then fit probability density functions (PDFs) to the experimental data and incorporate them in the model as the noted frequency domain analysis in
Figure 1. The maximum likelihood estimation (MLE) and chi-square goodness-of-fit test are finally applied to determine the best fit for each harmonic component among feasible PDFs.
2.1. Maximum Likelihood Estimation
The characteristics of harmonic components are observed to be stochastic in terms of their magnitudes and locations. Proprietary converter topology and controls and various filter types should be contributing factors on top of variable wind. Well-known parametric PDFs could be fitted to each harmonic component. Custom mathematical and even nonparametric distribution functions may be investigated as needed. This research applies the MLE method to determine the parameters of the PDFs [
25,
26].
Let
, ...,
be independent, identically-distributed samples of a random variable
with a PDF,
, where
is a
vector of parameters that characterize
. The joint PDF of the
n samples can be obtained as shown in Equation (
1):
A likelihood function is specified as shown in Equation (
2), and the parameters are continuously updated until the likelihood is maximized:
The parameters are estimated by differentiating Equation (
2). This process is often carried out with the logarithm of the likelihood Equation (
2):
The PDF of a harmonic component is often observed to be the normal distribution as specified below [
7,
8,
9,
15]:
where
is the average and
is the standard deviation. Both parameters can be estimated by Equations (
1) and (
2), as illustrated in
Figure 2.
2.2. Chi Square Test
The statistical behavior of a particular harmonic component could be represented by a variety of PDFs. It is then necessary to evaluate how well the dataset matches a specific distribution and whether a dataset is consistent with different probability distributions. This research thus conducts a chi-square goodness-of-fit test to identify the PDF that most appropriately reflects the harmonic characteristics [
27].
The chi-square test determines whether the estimation of the probability is suitable or not by using the error between the real value and the estimated value. If
and
are a real value and an expected value, respectively, the chi-squared value is defined as:
According to the degree of freedom, which is defined as
, where
k is the number of categories existing in the whole set, the
p-value is defined with the approximate chi-squared distribution.
The
p-value indicates the probability that the chi-squared distribution under a certain degree of freedom is larger than the chi-squared value obtained from Equation (
5). If the
p-value is less than 0.05, for example, the assumption of the probability is proven to be false.
However, it is challenging to identify PDFs that fit the measured harmonics from WTs or WPPs and pass this test. Therefore, the chi-square test can be adapted to identify the distribution with the least error: the chi-square value indicates the overall error of the distribution. Thus, the distribution with the least chi-square value can be considered as the best fit among several distributions.
5. Harmonics Assessment for a 100-MW Offshore Wind Power Plant
5.1. System Configuration
Figure 8 provides a test system configuration for the case study. Based on the WPP under development in Korea, the system is configured with 20 units of 5-MW full converter WTs from a single manufacturer with transformers and harmonic filters (
Figure 3), a substation transformer, submarine cables connecting the WPP to the grid or connecting the WTs to each other and the equivalent grid impedance connected to the infinite bus.
The internal grid of the WPP adopts 33-kV line-to-line voltage and three types of submarine cables with different capacities as differentiated by the thickness of the lines connecting WTs in
Figure 8. The equivalent
π model is used for the submarine cable with parameters provided by the manufacturer. The frequency scan results of the two different cases are provided in
Figure 9.
Harmonic resonance occurs when a harmonic frequency coincides with a power system natural frequency. There are two forms of resonance, i.e., parallel and series resonances, and they can substantially distort voltage and amplify current [
1,
30]. As shown in
Figure 9, without considering the PCS filter, the dominant resonance points are 600 Hz and 1610 Hz. However, they are moved to 270 Hz and 770 Hz, respectively, due to the effect of the PCS filters, which confirms that the PCS filters should be influential and carefully reflected in the harmonic resonance study. If the frequency of the external harmonic component coincides with the resonance frequency, the harmonic voltage can be magnified due to the parallel resonance. Harmonic data are collected at the PoC in
Figure 8. Each harmonic magnitude is assumed to follow the normal distribution with its uniquely-identified parameters. In order to investigate the interactions of time-varying harmonics, resulting in harmonic cancellation or amplification, the changes of the standard deviation and interval size of the uniform distribution of the phase angle are taken into consideration. The details of the parameters are shown in the
Appendix A. The random harmonic generation and FFT analysis are repeated every 0.2 s. The simulation study of this WPP scenario is carried out for 10 min (600 s), and 3000 samples of each harmonic component are analyzed.
5.2. Simulation Results of a Base Case
A base case model of the WPP is built based on the following scenarios: The standard deviation of each harmonic is 10% of its average value, and the phase angle is uniformly distributed with its interval size of
. Very short time harmonic values (aggregation of 15 samples) and short time harmonic values (aggregation of 3000 samples) are obtained and assessed by the requirements in IEEE Std519-2014 [
31]. Since there is no statement in the standard about inter-harmonics over 120 Hz, the recommended harmonic limit for the integer multiple of the fundamental frequency is applied to the inter-harmonic assessment. The 99th percentile very short time individual harmonic values from the base case are shown in
Figure 10.
Two largest harmonic components at the 4.5th and 12.5th order result from the two resonance points of the WPP with PCS filter.
Figure 11 illustrates the histogram and the cumulative probability of the very short time voltage total harmonic distortion (THD
) values. The THD
varies in the range from 0.4676% to 0.6829%, and the 99th percentile THD
value is 0.6482%, which satisfies the requirement of the IEEE standard.
5.3. Change of the Standard Deviation
A change in the standard deviation and its effect on the harmonics at the PoC are investigated. The ratio of the standard deviation of each harmonic PDF to its average value is changed from 0% to 20% with its phase angle interval size of
.
Figure 12 shows the maximum value (circle), minimum value (star), 99% very short time harmonic value (triangle) and short time harmonic value (square) of the largest individual harmonic distortion (IHD, 12.5th order) and the voltage THD at the PoC.
The 99% IHD value is in the range from 0.5185% to 0.5582%, and IHD values in all cases are about 0.42%. The subtraction of the minimum from the maximum of THD ranges from 0.197% to 0.223%, and THD values are about 0.56%. All harmonic values in all of the cases comply with the requirements of the IEEE standard. Overall, IHD and THD in all cases vary in a similar range, even for the increased standard deviations, which results from the assumption that each harmonic generation is independent, and no correlation among harmonic components exits between the WTs. Stochastic and independent harmonic generation based on the normal distribution in 20 WTs causes the averaging effect on the harmonics at the PoC. Further studies incorporating the correlation among the WTs or harmonics in the same modeling framework may show different perspectives.
5.4. Change of the Interval Size of the Uniform Distribution of the Phase Angle
The phase angle of harmonics is still assumed to be uniformly distributed, and the interval size is now allowed to vary from zero to
with 10% of the standard deviation over the average. For example, 90 degrees of the interval size means that the harmonic phase angle is generated in the range from zero to
.
Figure 13 shows the results of varying the interval size while other parameters are kept the same as in
Figure 12.
The interval size variation is more influential on the harmonics at the PoC than the standard deviation change. IHD drops from 2.203% in the zero interval size to 0.4154% in the interval size, and THD decreases from 2.407% to 0.5589%. The difference between the maximum and minimum value steadily increases in both IHD and THD. Furthermore, with a zero interval size, which represents the worst case, the maximum IHD value is 2.246%, and THD is 2.407% close to the IEEE standard limit. Moreover, IHD values in the 0 and 90 interval sizes are over the IEEE limit. That is because there is no harmonic cancellation, and the harmonics are simply added at the PoC in the zero or 90 degree interval size. The wider the interval size, the higher the effect of harmonic cancellation. Therefore, the highest interval size shows the lowest harmonic magnitude.
6. Discussion
The proposed framework is designed to adopt the statistical information from the field measurements, if any. However, this framework can still be utilized in planning studies with, for example, simulation results with high-fidelity models. It is also worth noting the assumptions made in the study below: First, the correlation between the WT output power and harmonic magnitudes or between harmonic components is ignored because there is no consensus about the correlation related to the harmonics [
7,
8,
9,
15]. If any correlation between the harmonics is identified through measurements in operations, this needs to be incorporated in harmonic generation. Second, the uniform distribution is applied to the phase angle modeling. Although there is arguably no common rule for the phase angle modeling, the proposed framework can allow for different statistic models or a narrow interval size in a uniform distribution for phase angles. As shown in
Figure 13, the narrow variation range of the phase angle results in the large harmonics at the PoC. Since the phase angle of the harmonic component is an important characteristic, illustrating natural harmonic cancellation at the PoC, the field measurements of the phase angles are desired and can improve the accuracy and validity of study result once they become available in operations. It is worth mentioning that the proposed framework can also synthesize the worst-case scenarios for investigation as needed by conditioning PDFs for magnitudes and phase angles of harmonic components, under which the harmonics from the WPP exceeded the limits in the case study.
7. Conclusions
This paper presented a study framework for capturing and evaluating stochastic harmonics and resonance concerns of wind integration. By exploiting the experimental data from the WT manufacturers or prior statistical information about harmonics from credible sources, we designed a harmonic current generator that represents the statistical characteristics of the harmonic currents from the WT and WPP. By incorporating the feeder configuration and the impact of harmonic filters, the variability of the wind and contingencies, etc., we can synthesize future operating scenarios and explore more credible cases, thus ensuring the grid compatibility of the WPP, especially when it is to be expanded. System operators can also comprehensively investigate the problematic operating conditions, in particular those that are unforeseen in the planning, and thus, systematically develop mitigation or correction measures.
Based on the framework above, we investigated harmonics and resonance issues of integrating a 100-MW offshore WPP consisting of 20 units of 5-MW full converter WTs benchmarking a WPP under development in Korea and concluded that the WPP would comply with the IEEE standards at the PoC. Based on the lessons learned from this study, the stochastic characteristics of the harmonics from the WTs could be exploited in taking any preventive measures against harmonics and resonance at this planning stage.
High penetrations of variable renewable sources into the electric power grid present a range of unprecedented grid operations and planning challenges requiring new study approaches and tools, so that the impacts of the proposed WPP can be properly assessed prior to interconnection. The proposed framework of interfacing statistical harmonics generation in the frequency domain and representation in the time domain simulation environment should be conceptually extended to other planning studies, especially those having harmonics and resonance issues. The research should help take advantage of prior knowledge to handle variability and offer opportunities for striking a balance between the technical and economic aspects of grid planning and operations.