A method for generating water, wind and solar multi-resource scenarios considering temporal and spatial correlation

Abstract

The present invention belongs to the technical field of water, wind and solar multi-energy complementarity, and specifically discloses a method for generating water, wind and solar multi-resource scenarios taking into account temporal and spatial correlation. Firstly, based on the measured reservoir runoff, wind speed, light intensity and temperature data, a multi-series first-order regression model is used, while taking into account the temporal autocorrelation and spatial cross-correlation of each sequence, to generate a high-dimensional coupling scenario of reservoir runoff, wind speed, light intensity and temperature. Then, based on the basic scenario set generated by random simulation, the total amount of water, wind and light resources in each scenario is normalized. Finally, according to the characteristics of the extreme scenario considered, weights are set for a certain resource or multiple resource variables, and the total power generation potential of the resources under each scenario is obtained by weighting. Empirical frequency analysis is used to generate extreme scenarios of single resources or composite resources with different recurrence periods. This method can effectively generate high-dimensional coupling scenarios and extreme scenarios that take into account the temporal autocorrelation and spatial cross-correlation of multivariate water, wind, light and temperature.

Description

Technical Field

[0001] The present invention belongs to the technical field of water-wind-solar multi-energy complementarity, and in particular relates to a method for generating water-wind-solar multi-resource scenes taking into account temporal and spatial correlation. Background Art

[0002] The water-wind-solar multi-energy complementary system faces the influence of more complex hydrological and meteorological factors and more uncertainties. It is difficult to consider the uncertainty and spatiotemporal correlation of multiple resources and generate representative and extreme high-dimensional coupling scenarios. Existing research mainly uses the following methods to generate input scenarios for power systems: 1. Single variable scenario generation, that is, for runoff, wind power or photovoltaic power, based on the time series of historical data, single variable input scenarios are generated by using autoregression and other methods; 2. Two-variable scenario generation, that is, considering the spatiotemporal correlation of the two input variables, and generating wind power-photovoltaic two-variable input scenarios based on copula and other methods; the existing technology has the following disadvantages:

[0003] Existing scenario generation methods mostly focus on scenario analysis of wind and solar power output, and usually only consider the temporal correlation of a single variable or the spatial correlation of multiple variables on a single time section. There is a lack of coupled scenario generation methods that take into account the high-dimensional uncertainty and spatiotemporal correlation of water, wind, and light resources. The generation of extreme scenarios is also insufficiently considered, making it difficult to provide data support for the operation and management of water-wind-solar multi-energy complementary systems. Therefore, it is necessary to design a water-wind-solar multi-resource scenario generation method that considers spatiotemporal correlation to solve the above problems. Summary of the invention

[0004] The technical problem to be solved by the present invention is to provide a method for generating water, wind and light multi-resource scenarios taking into account the temporal and spatial correlation. The method normalizes the total amount of water, wind and light resources in each scenario, and then weights them according to the installed capacity of various water, wind and light power stations in the complementary system. The total power generation potential of water, wind and light resources in each scenario is calculated and ranked, and empirical frequency analysis is used to generate extreme scenarios that occur once in many years. High-dimensional coupling scenarios and extreme scenarios that take into account the temporal autocorrelation and spatial cross-correlation of multivariate water, wind and light can be effectively generated.

[0005] In order to achieve the above technical effects, the technical solution adopted by the present invention is:

[0006] A method for generating a water, wind and light multi-resource scene considering temporal and spatial correlation comprises the following steps:

[0007] Generate high-dimensional coupling scene of water, scenery and light:

[0008] A multi-series first-order regression model is constructed based on the measured reservoir runoff, wind speed, light intensity and temperature data. A high-dimensional coupling scenario set of water, wind and light is generated through random simulation of the multi-series first-order regression model.

[0009] Generate extreme water scenery scenes:

[0010] Generate extreme scenarios of water, wind and solar based on the principle of unfavorable total power generation potential of water, wind and solar resources:

[0011] Based on the high-dimensional coupled scene set of water, wind and solar power generated by random simulation, the total amount of water, wind and solar power resources in each scene is calculated and normalized;

[0012] Weights are set for runoff, wind speed, light intensity and temperature variables according to the characteristics of the extreme scenario considered, and the total amount of water, wind and solar resources in each scenario is weighted to obtain the total power generation potential of the current scenario;

[0013] The total power generation potential of each scenario is ranked, and the ranking results are subjected to empirical frequency analysis to calculate the extreme scenarios of single resources or composite resources with different return periods.

[0014] Preferably, generating a high-dimensional coupling scene of water, wind and light includes:

[0015] Based on the measured reservoir runoff, wind speed, light intensity and temperature data, the original sequence of each variable is standardized:

[0016] ;

[0017] In the formula, i=1,2,3,4, representing runoff, wind speed, light intensity and temperature data respectively; and Respectively represent the original and standardized sequences of runoff, wind speed, light intensity and temperature, t represents the time period, Representation sequence The standard deviation of Representation sequence The mean of

[0018] For each standardized sequence, the normalization of the sequence is achieved based on the Wigner-Hough transform:

[0019] ;

[0020] In the formula, represents the normalized sequence, Is a sequence The skewness coefficient of

[0021] Each sequence is standardized and normalized to construct a multi-series first-order regression model of runoff, wind speed, temperature and light intensity:

[0022] ;

[0023] In the formula, are the normalized data of runoff, wind speed, light intensity and temperature to be simulated in the tth period; are the normalized data of runoff, wind speed, light intensity and temperature to be simulated in the t-1 period; matrices A and B are the parameters of the multi-sequence first-order regression model, , , , They represent the random variables of runoff, wind speed, light intensity and temperature to be simulated in the tth period that obey the standard normal distribution;

[0024] The multi-series first-order regression model is used to randomly simulate the sequences of runoff, wind speed, light intensity and temperature with a sample size of m. The simulated values ​​are then inversely transformed into standardized and normalized values ​​to obtain a high-dimensional coupled scenario set of runoff, wind speed, light intensity and temperature.

[0025] Preferably, the matrix A The matrix reflects the time autocorrelation and cross-correlation characteristics of the runoff, wind speed, light intensity and temperature series with a lag period of 1. B The time-delay-free cross-correlation characteristics between the runoff, wind speed, light intensity and temperature series are reflected by using matrices A and B to reflect the spatiotemporal correlation between the series. The calculation methods of matrices A and B are as follows:

[0026] ;

[0027] ;

[0028] in:

[0029] ;

[0030] ;

[0031] In the formula, is the mutual correlation coefficient between the sequence x and the sequence y with a time lag of K, the number of x and y is m, and K is 0 or 1; The characteristic matrix constructed for the cross-correlation coefficient with time lag 1, is the characteristic matrix constructed for the cross-correlation coefficient with time lag 0. The superscript -1 indicates the inverse matrix.

[0032] Preferably, based on the high-dimensional coupling scene set of water, wind and light generated by random simulation, the total amount of water, wind and light resources in each scene is calculated and normalized, including:

[0033] ;

[0034] In the formula, represents the total amount of resources of the ith variable in the simulation period under the jth scenario, It represents the total amount of resources of the i-th variable after normalization of the j-th high-dimensional coupling scenario, i=1,2,3,4, representing runoff, wind speed, light intensity and temperature respectively; min is the minimum function, and max is the maximum function.

[0035] Preferably, weights are set for runoff, wind speed, light intensity and temperature variables according to the characteristics of the extreme scenario under consideration, and the total amount of water, wind and solar resources in each scenario is weighted to obtain the total power generation potential of the current scenario, including:

[0036] ;

[0037] In the formula, are the weights of runoff, wind speed, light intensity and temperature, respectively. is the total power generation potential of resources in the jth scenario; The total amount of runoff resources corresponding to the jth high-dimensional coupling scenario after normalization; The total amount of resources corresponding to the normalized wind speed of the jth high-dimensional coupling scenario; The total amount of resources corresponding to the normalized light intensity of the j-th high-dimensional coupling scene; The total amount of resources corresponding to the normalized temperature of the jth high-dimensional coupling scenario; the total power generation potential of the scenario is positively correlated with the risk of power abandonment in the hydro-wind-solar complementary system, and negatively correlated with the risk of insufficient output of the hydro-wind-solar complementary system.

[0038] Preferably, the total power generation potential of each scenario is ranked, and the ranking results are subjected to empirical frequency analysis to calculate the extreme scenarios of single resources or composite resources with different return periods, including:

[0039] ;

[0040] ;

[0041] In the formula, is the total resource power generation potential corresponding to the scenario with a frequency of P%, and T is the extreme scenario at this time The total power generation potential with different frequencies P% is set, and the extreme resource scenarios of the water-wind-solar complementary system under different return periods are calculated.

[0042] The beneficial effects of the present invention are as follows:

[0043] 1. This method for generating water-wind-solar multi-resource scenarios can take into account the temporal autocorrelation and spatial cross-correlation of key input factors of the water-wind-solar complementary system, such as reservoir runoff, wind speed, light intensity and temperature, thereby generating credible high-dimensional coupling scenarios and extreme scenarios of water-wind-solar multi-resources, which can provide data support for the operation planning of the water-wind-solar multi-energy complementary system.

[0044] 2. The present invention is based on a set of high-dimensional coupling scenarios generated by random simulation, normalizes the total amount of water, wind and light resources in each scenario, and then weights them according to the installed capacity of each type of water, wind and light power stations in the complementary system, calculates the total power generation potential of water, wind and light resources in each scenario and ranks them, and uses empirical frequency analysis to generate extreme scenarios that occur once in many years; this method can effectively generate high-dimensional coupling scenarios and extreme scenarios that take into account the temporal autocorrelation and spatial cross-correlation of multivariate water, wind and light; it solves the problem that the existing scenario generation method of the water-wind-light complementary system only considers the temporal correlation of a single variable and the spatial correlation of multiple variables on a single time section, lacks a high-dimensional coupling scenario generation method that takes into account the temporal autocorrelation and spatial cross-correlation of multivariate water, wind and light, and does not give enough consideration to the generation of extreme scenarios. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 It is a flowchart of the method of the present invention;

[0046] Figure 2 A high-dimensional coupled scene set of runoff, wind speed, light intensity and temperature in an embodiment of the present invention;

[0047] Figure 3 This is an extreme flood-strong light scene with a once-in-a-millennium and once-in-a-century event in the embodiment of the present invention. DETAILED DESCRIPTION

[0048] Embodiment 1:

[0049] like Figure 1 As shown, a method for generating a water, wind and light multi-resource scene considering temporal and spatial correlation includes the following steps:

[0050] Generate high-dimensional coupling scene of water, scenery and light:

[0051] A multi-series first-order regression model is constructed based on the measured reservoir runoff, wind speed, light intensity and temperature data. A high-dimensional coupling scene set of water, wind and light is generated through random simulation of the multi-series first-order regression model.

[0052] Generate extreme water scenery scenes:

[0053] Generate extreme scenarios of water, wind and solar based on the principle of unfavorable total power generation potential of water, wind and solar resources:

[0054] Based on the high-dimensional coupled scene set of water, wind and solar power generated by random simulation, the total amount of water, wind and solar power resources in each scene is calculated and normalized;

[0055] Weights are set for runoff, wind speed, light intensity and temperature variables according to the characteristics of the extreme scenario considered, and the total amount of water, wind and solar resources in each scenario is weighted to obtain the total power generation potential of the current scenario;

[0056] The total power generation potential of each scenario is ranked, and the ranking results are subjected to empirical frequency analysis to calculate the extreme scenarios of single resources or composite resources with different return periods.

[0057] Preferably, generating a high-dimensional coupling scene of water, wind and light includes:

[0058] Based on the measured reservoir runoff, wind speed, light intensity and temperature data, the original sequence of each variable is standardized:

[0059] ;

[0060] In the formula, i=1,2,3,4, representing runoff, wind speed, light intensity and temperature data respectively; and Respectively represent the original and standardized sequences of runoff, wind speed, light intensity and temperature, t represents the time period, Representation sequence The standard deviation of Representation sequence The mean of

[0061] For each standardized sequence, the normalization of the sequence is achieved based on the Wigner-Hough transform:

[0062] ;

[0063] In the formula, represents the normalized sequence, Is a sequence The skewness coefficient of , t represents the time period;

[0064] Each sequence is standardized and normalized to construct a multi-series first-order regression model of runoff, wind speed, temperature and light intensity:

[0065] ;

[0066] In the formula, are the normalized data of runoff, wind speed, light intensity and temperature to be simulated in the tth period; are the normalized data of runoff, wind speed, light intensity and temperature to be simulated in the t-1 period; matrices A and B are the parameters of the multi-sequence first-order regression model, , , , They represent the random variables of runoff, wind speed, light intensity and temperature to be simulated in the tth period that obey the standard normal distribution;

[0067] The multi-series first-order regression model is used to randomly simulate the sequences of runoff, wind speed, light intensity and temperature with a sample size of m. The simulated values ​​are then inversely transformed into standardized and normalized values ​​to obtain a high-dimensional coupled scenario set of runoff, wind speed, light intensity and temperature.

[0068] Preferably, the matrix A The matrix reflects the time autocorrelation and cross-correlation characteristics of the runoff, wind speed, light intensity and temperature series with a lag period of 1. B The time-delay-free cross-correlation characteristics between the runoff, wind speed, light intensity and temperature series are reflected by using matrices A and B to reflect the spatiotemporal correlation between the series. The calculation methods of matrices A and B are as follows:

[0069] ;

[0070] ;

[0071] in:

[0072] ;

[0073] ;

[0074] In the formula, is the mutual correlation coefficient between the sequence x and the sequence y with a time lag of K, the number of x and y is m, and K is 0 or 1; The characteristic matrix constructed for the cross-correlation coefficient with time lag 1, is the characteristic matrix constructed for the cross-correlation coefficient with time lag 0. The superscript -1 indicates the inverse matrix.

[0075] Preferably, based on the high-dimensional coupling scene set of water, wind and light generated by random simulation, the total amount of water, wind and light resources in each scene is calculated and normalized, including:

[0076] ;

[0077] In the formula, represents the total amount of resources of the ith variable in the simulation period under the jth scenario, It represents the total amount of resources of the i-th variable after normalization of the j-th high-dimensional coupling scenario, i=1,2,3,4, representing runoff, wind speed, light intensity and temperature respectively; min is the minimum function, and max is the maximum function.

[0078] Preferably, weights are set for runoff, wind speed, light intensity and temperature variables according to the characteristics of the extreme scenario under consideration, and the total amount of water, wind and solar resources in each scenario is weighted to obtain the total power generation potential of the current scenario, including:

[0079] ;

[0080] In the formula, are the weights of runoff, wind speed, light intensity and temperature, respectively. is the total power generation potential of resources in the jth scenario; The total amount of runoff resources corresponding to the jth high-dimensional coupling scenario after normalization; The total amount of resources corresponding to the normalized wind speed of the jth high-dimensional coupling scenario; The total amount of resources corresponding to the normalized light intensity of the j-th high-dimensional coupling scene; The total amount of resources corresponding to the normalized temperature of the jth high-dimensional coupling scenario; the total power generation potential of the scenario is positively correlated with the risk of power abandonment in the hydro-wind-solar complementary system, and negatively correlated with the risk of insufficient output of the hydro-wind-solar complementary system.

[0081] Preferably, the total power generation potential of each scenario is ranked, and the ranking results are subjected to empirical frequency analysis to calculate the extreme scenarios of single resources or composite resources with different return periods, including:

[0082] ;

[0083] ;

[0084] In the formula, is the total resource power generation potential corresponding to the scenario with a frequency of P%, and T is the extreme scenario at this time The total power generation potential with different frequencies P% is set, and the extreme resource scenarios of the water-wind-solar complementary system under different return periods are calculated.

[0085] Embodiment 2: Figure 2 The high-dimensional coupled scene set of runoff, wind speed, light intensity and temperature generated in this embodiment is shown. The high-dimensional coupled scene set is also regarded as a basic scene set.

[0086] like Figure 3 As shown, considering that extreme scenarios usually cover multiple resource situations such as water, wind, and light, this embodiment will consider two situations: one is when water, wind, and light resources are sufficient, due to the limitations of transmission capacity and hydropower station regulation capacity, the complementary system will generate a large amount of abandoned power; the other is when water, wind, and light resources are relatively scarce, the power system is difficult to ensure the reliability of power supply, thus posing a challenge to energy security. Based on these two considerations, this embodiment mainly generates extreme scenarios based on the principle that the total power generation potential of water, wind, and light resources is unfavorable.

[0087] (1) Based on the basic scenario set generated by random simulation, the total amount of water, wind and light resources in each scenario is calculated and normalized accordingly:

[0088] ;

[0089] In the formula, represents the total amount of resources of the ith variable in the simulation period under the jth scenario, represents the total amount of resources of the i-th variable after normalization of the j-th high-dimensional coupling scenario, i=1,2,3,4, representing runoff, wind speed, light intensity and temperature, respectively, that is, Respectively j The total amount of resources of runoff, wind speed, light intensity and temperature during the simulation period under each scenario, They are the normalized total resources of runoff, wind speed, light intensity and temperature corresponding to the jth high-dimensional coupling scenario; min is the minimum function, and max is the maximum function.

[0090] (2) According to the characteristics of the extreme scenario under consideration, weights are set for runoff, wind speed, light intensity and temperature variables, and the total amount of each resource is weighted to obtain the total power generation potential of the scenario:

[0091] ;

[0092] In the formula, are the weights of runoff, wind speed, light intensity and temperature, respectively. is the total resource power generation potential of the jth scenario, The total amount of runoff resources corresponding to the jth high-dimensional coupling scenario after normalization; The total amount of resources corresponding to the normalized wind speed of the jth high-dimensional coupling scenario; The total amount of resources corresponding to the normalized light intensity of the j-th high-dimensional coupling scene; The total amount of resources corresponding to the normalized temperature of the jth high-dimensional coupling scenario. The greater the total power generation potential of the scenario, the greater the risk of power abandonment faced by the hydro-wind-solar hybrid system. The smaller the total power generation potential, the greater the risk of energy supply security faced by the hydro-wind-solar hybrid system.

[0093] Specifically, if only extreme scenarios of a single water, wind or light resource are considered, the weight of the corresponding resource variable is set to 1. When only extreme scenarios of drought or flood are considered, the weight of runoff is set to 1. When considering extreme scenarios of strong wind or strong light, light wind or dim light, the weight of wind speed or light intensity is set to 1, and the weights of other variables are all 0. If a composite extreme scenario of multiple resources is considered, the weights of the corresponding variables can be set according to the emphasis on different resources. For example, when considering the extreme scenario of high temperature and drought, the weights of runoff and temperature can be set separately according to the emphasis. and , when considering extreme scenarios of flood and strong light, the weights of runoff and light can be set separately and , when considering the extreme scene of breeze and dim light, set the weights of wind speed and light separately and , when considering the compound drought of water, wind and light resources, the weights of runoff, wind speed and light can be set and ; or the weights of each variable are set according to the installed capacity ratio of hydropower stations, wind power stations and photovoltaic power stations in the water-wind-solar complementary system to represent the total power generation potential of the resources affecting the complementary system.

[0094] (3) Finally, the total power generation potential of each scenario is ranked and an empirical frequency analysis is performed to obtain extreme scenarios of single resources or composite resources with different return periods:

[0095] ;

[0096] ;

[0097] In the formula, is the total resource power generation potential corresponding to the scenario with a frequency of P%, and T is the extreme scenario at this time By setting the total power generation potential at different frequencies P%, we can get the extreme resource scenarios of the water-wind-solar hybrid system under different return periods.

Claims

1. A method for generating water, wind and light multi-resource scenes considering spatiotemporal correlation, characterized in that: The following steps are involved: Generate high-dimensional coupling scene of water, scenery and light: A multi-series first-order regression model is constructed based on the measured reservoir runoff, wind speed, light intensity and temperature data. A high-dimensional coupling scene set of water, wind and light is generated through random simulation of the multi-series first-order regression model. Generating high-dimensional coupling scenarios of water, wind and light includes: Based on the measured reservoir runoff, wind speed, light intensity and temperature data, the original sequence of each variable is standardized: ； In the formula, i=1,2,3,4, representing runoff, wind speed, light intensity and temperature data respectively; and Respectively represent the original and standardized sequences of runoff, wind speed, light intensity and temperature, t represents the time period, Representation sequence The standard deviation of Representation sequence The mean of For each standardized sequence, the normalization of the sequence is achieved based on the Wigner-Hough transform: ； In the formula, represents the normalized sequence, Is a sequence The skewness coefficient of Each sequence is standardized and normalized to construct a multi-series first-order regression model of runoff, wind speed, temperature and light intensity: ； In the formula, are the normalized data of runoff, wind speed, light intensity and temperature to be simulated in the tth period; are the normalized data of runoff, wind speed, light intensity and temperature to be simulated in the t-1 period; matrices A and B are the parameters of the multi-sequence first-order regression model, , , , They represent the random variables of runoff, wind speed, light intensity and temperature to be simulated in the tth period that obey the standard normal distribution; The multi-series first-order regression model is used to randomly simulate the sequences of runoff, wind speed, light intensity and temperature with a sample size of m. The simulated values ​​are then inversely transformed into normalized and normalized values ​​to obtain a high-dimensional coupled scenario set of runoff, wind speed, light intensity and temperature. Generate extreme water scenery scenes: Generate extreme scenarios of water, wind and solar based on the principle of unfavorable total power generation potential of water, wind and solar resources: Based on the high-dimensional coupled scene set of water, wind and solar power generated by random simulation, the total amount of water, wind and solar power resources in each scene is calculated and normalized; Weights are set for runoff, wind speed, light intensity and temperature variables according to the characteristics of the extreme scenarios considered, and the total amount of water, wind and solar resources in each scenario is weighted to obtain the total power generation potential of the current scenario; The total power generation potential of each scenario is ranked, and the ranking results are subjected to empirical frequency analysis to calculate the extreme scenarios of single resources or composite resources with different return periods.

2. According to the method for generating water, wind and light multi-resource scenes considering spatiotemporal correlation according to claim 1, it is characterized in that: matrix A The matrix reflects the time autocorrelation and cross-correlation characteristics of the runoff, wind speed, light intensity and temperature series with a lag period of 1. B The time-delay-free cross-correlation characteristics between the runoff, wind speed, light intensity and temperature series are reflected by using matrices A and B to reflect the spatiotemporal correlation between the series. The calculation methods of matrices A and B are as follows: ； ； in: ； ； In the formula, is the mutual correlation coefficient between the sequence x and the sequence y with a time lag of K, the number of x and y is m, and K is 0 or 1; The characteristic matrix constructed for the cross-correlation coefficient with time lag 1, is the characteristic matrix constructed for the cross-correlation coefficient with time lag 0. The superscript -1 indicates the inverse matrix.

3. According to the method for generating water, wind and light multi-resource scenes considering spatiotemporal correlation according to claim 1, it is characterized in that: Based on the high-dimensional coupling scene set of water, wind and solar power generated by random simulation, the total amount of water, wind and solar power resources in each scene is calculated and normalized, including: ； In the formula, represents the total amount of resources of the ith variable in the simulation period under the jth scenario, It represents the total amount of resources of the i-th variable after normalization of the j-th high-dimensional coupling scenario, i=1,2,3,4, representing runoff, wind speed, light intensity and temperature respectively; min is the minimum function, and max is the maximum function.

4. The method for generating water, wind and light multi-resource scenes considering spatiotemporal correlation according to claim 3 is characterized in that: According to the characteristics of the extreme scenario considered, weights are set for runoff, wind speed, light intensity and temperature variables, and the total amount of water, wind and solar resources in each scenario is weighted to obtain the total power generation potential of the current scenario, including: ； In the formula, are the weights of runoff, wind speed, light intensity and temperature, respectively. is the total power generation potential of resources in the jth scenario; The total amount of runoff resources corresponding to the jth high-dimensional coupling scenario after normalization; The total amount of resources corresponding to the normalized wind speed of the jth high-dimensional coupling scenario; The total amount of resources corresponding to the normalized light intensity of the j-th high-dimensional coupling scene; The total amount of resources corresponding to the normalized temperature of the jth high-dimensional coupling scenario; the total power generation potential of the scenario is positively correlated with the risk of power abandonment in the hydro-wind-solar complementary system, and negatively correlated with the risk of insufficient output of the hydro-wind-solar complementary system.

5. The method for generating water, wind and light multi-resource scenes considering spatiotemporal correlation according to claim 4 is characterized in that: The total power generation potential of each scenario is ranked, and the ranking results are subjected to empirical frequency analysis to calculate the extreme scenarios of single resources or composite resources with different return periods, including: ； ； In the formula, is the total resource power generation potential corresponding to the scenario with a frequency of P%, and T is the extreme scenario at this time The total power generation potential with different frequencies P% is set, and the extreme resource scenarios of the water-wind-solar complementary system under different return periods are calculated.

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