Trends and shifts in mean annual inflow time series of hydropower plants in Brazil

Tendências e mudanças de nível em séries temporais de afluência média anual de usinas hidrelétricas no Brasil

ABSTRACT

Previous Brazilian streamflow trend detection studies didn´t fully covered the notable recent period of multiple drought episodes over the country. Most studies focused on detecting monotonic trends and few on abrupt changes of level (shifts). We provide an updated study on trends and shifts of long-term mean annual flow (LTMAF) based on the simplistic linear trend and on an S-shaped shift with flexible duration. We used a set of selected 52 time series of naturalized monthly inflows for Brazilian hydropower plants (HPPs) updated until DEC 2023. The selection considered only HPPs with unregulated headwaters and discarded early years of the record with plenty filled data. We employed t-test statistics for usual calculations of statistical significance levels. Regression slopes magnitudes of were interpreted qualitatively in terms of “practical significances”. The application of the S-shaped shift model found only long-lasting shift cases yielding essentially the same conclusions as the simpler linear trend model. Moderate and large linear trend rates were detected in 47 series, 33 cases with negative trend rates and 14 cases with positive trend rates. The discarding of early years of the historical records resulted in more dispersed and reduced streamflow trend rates, increasing the number of negative trend cases.

Keywords:
Hydropower; Streamflow; Trends and shifts; Fluviometric records; Statistical tests; Detection studies

RESUMO

Estudos anteriores de detecção de tendências em vazões realizados no Brasil não cobriram totalmente o recente período de múltiplos episódios de seca espalhados pelo país. A maioria dos estudos focaram a detectação de tendências monotônicas e poucos de mudanças abruptas de nível. Aqui, fornecemos um estudo atualizado de tendência e mudança de nível na vazão média anual de longo prazo (LTMAF) com base em avaliações do modelo simples de tendência linear e de um modelo de curva S para mudança de nível com duração flexível. Usamos uma seleção de 52 séries históricas de vazões afluentes mensais naturalizadas para usinas hidrelétricas (UHEs) brasileiras atualizadas até dezembro de 2023. A seleção considerou apenas séries temporais de UHEs com cabeceiras sem regularização e descartou os primeiros anos do registro histórico com muitos dados preenchidos. A metodologia considerou estatísticas t para os cálculos usuais de níveis de significância estatística. As magnitudes dos coeficientes de regressão foram interpretadas qualitativamente em termos de “significâncias práticas”. A aplicação do modelo de curva S encontrou apenas casos de mudança de nível com longa duração fornecendo essencialmente as mesmas conclusões que o modelo de tendência linear. Taxas de tendência moderadas e grandes foram detectadas em 47 séries, sendo 33 casos de taxas negativas e 14 casos de taxas positivas. O descarte dos primeiros anos dos registos históricos resultou em taxas de tendências mais dispersas e reduzidas, aumentando o número de casos de tendências negativas.

Palavras-chave:
Energia hidrelétrica; Vazão; Tendências e mudanças de nível; Registros fluviométricos; Testes estatísticos; Estudos de detecção

INTRODUCTION

While in 2023 the global electricity matrix hydropower had a share of 14.3%, in Brazil hydropower corresponded to 60.5% in 2023 (Ritchie & Pablo, 2024). Brazil rates second in hydropower generation countries with annual production of ~ 420 billion kilowatt-hours, only surpassed by China (Statista, 2024). In January 2025, the Brazilian National Interconnected System (SIN) had 214 hydropower plants (HPP) and 439 small hydropower plants (SHP) in operation with a total installed capacity of 109,062.74 MW. The current system has hundreds of small and micro HPPs not evaluated in this work.

Given the dependence of Brazilian power supply on hydro source, a major concern in Brazilian energy planning is the possibility that global climate changes provoke reductions in long-term average annual streamflow (LTMAF) of Brazilian HPPs by increasing the period, intensity and frequency of drought periods and ultimately affecting negatively country hydropower production (Silva & Garcia, 2022). Reductions in LTMAFs of Brazilian river basins can also impact other water uses in the country such as navigation in main river courses (Lima et al., 2024; Toloi et al., 2016), tourist activities around river and lakes (Melo et al., 2022) and water supply for irrigation, industry and municipalities, especially in the most vulnerable river basins of the Brazilian Central-Northeast and Northwest Amazon (Vieira et al., 2023; Gesualdo et al., 2024).

In general, LTMAFs can remain stable for hundreds of years, although presenting cyclical variations of considerable magnitude on various time scales following natural climatic fluctuations affecting their drainage area, mostly of them linked by causal relationships with ocean-atmosphere interaction oscillations, specially over the tropical Pacific and over tropical southern Atlantic regions (Intergovernmental Panel on Climate Change, 2013; National Academies of Sciences, Engineering and Medicine, 2016; Zhang et al., 2023; Su et al., 2018). In addition to the action of those natural climatic fluctuations, LTMAFs can be affected by human interventions in the drainage areas as the construction and operation of artificial regulation reservoirs (Hunt et al., 2025; Zhang et al., 2015; López-Moreno et al., 2014); massive water withdrawals for agricultural, industrial or residential water supply; and changes in land use and cover (deforestation of forests, advancement of agricultural frontiers, urbanization). Also, global climate changes caused by anthropogenic greenhouse gas (GHG) emissions tend to affect the entire hydrological cycle of the planet with different effects on LTMAFs of river basins (Intergovernmental Panel on Climate Change, 2023; Sukanya & Saby, 2023; Jiménez Cisneros et al., 2014). In Brazil, recent studies carried out by the Agência Nacional de Águas e Saneamento Básico (2024) projected water availability until the year 2100 for the 12 Brazilian hydrographic regions considering 4 different scenarios for total global GHG emissions pathways. The results projected trends with reductions in LTMAF for 10 hydrographic regions. Only the hydrographic regions of the South Atlantic and Uruguay showed projected trends with increasing LTMAFs.

Despite the challenges involved in the analysis of complex interactions between internal, forced and unforced climatic signals and plenty interferences from anthropogenic activities (Robson & Kundzewicz, 2000; Lehner & Deser, 2023; Puente et al., 2025) several research efforts have been directed to the detection of changes in watershed streamflow regimes and to the attribution of their driving forces (Su et al., 2018; Nabaei & Saghafian, 2021; Sharma & Singh, 2024).

Several studies had used simplistic models for describing apparent changes in streamflow records as persistent monotonic trends or of changes in level (shifts), the latter occurring abruptly at a one-time interval usually called as a step-change or a jump (Serinaldi et al., 2018). Examples of less simplistic modeling are (Hubert, 2000) that considered time series with multiple step-changes and (Damázio et al., 2011) that considered time series with a single S-shaped change of level occurring gradually over multiple time intervals. In general, these simplistic change models are used with accompanying null hypothesis significance tests (Chiew & McMahon, 1993; Kalra et al., 2008; Burn & Elnur, 2002) although this approach has been criticized as the underlying assumptions of the tests, e.g. the assumptions of normally distributed and IID (independent and identically distributed) random variables in t-tests and other parametric tests are generally not present in streamflow time series (Serinaldi et al., 2018; Koutsoyiannis & Montanari, 2007). Use of non-parametric tests to mitigate the effects of data non-normality are very popular. e. g. the Mann-Kendall and the Spearmans rho for testing monotonic trends in time series central values (Siegel & Castellan, 1988; Yue et al., 2002a; Mann, 1945) and the Pettitt´s test for testing an abrupt step-change in time series central values with the exact time of the step-change unknown (Pettitt, 1979). Measures for mitigating autocorrelation or long-term persistence effects are eventually used (Damázio et al., 2011; Cohn & Lins, 2005; Craigmile et al., 2004; Yue et al., 2002b; Souza & Reis Junior, 2022; Su et al., 2018; Muller et al., 1998), although these measures are also subjected to limitations and pitfalls (Serinaldi et al., 2018; Koutsoyiannis & Montanari, 2007). An alternative approach, used in this paper, is to interpret the results of statistical tests as evidences of apparent patterns in the recent evolution of the inflow time series, together with an assessment of their “practical significances” (Stahl et al., 2010; Yue et al., 2002a).

Since the end of the 90s, several analyzes of changes of hydrological regime in Brazilian streamflow time series using statistical tests identified positive and negative changes of LTMAFs in different regions of the country e.g. (Alves et al., 2013, Couto & Fichtner, 2023; Detzel et al., 2022; Damázio et al., 2011; Silva et al., 2019). Most studies focused on detecting monotonic trends, while few have focused on detecting step-changes (Guetter & Prates, 2002; Muller et al., 1998). Most studies used the historical naturalized time series of monthly inflows to SIN HPPs since 1931 available on the website of the Brazilian National System Operator (Operador Nacional do Sistema Elétrico, 2011). These time series are composed of reconstructed values of monthly inflows into the HPPs that would occur if no anthropogenic actions affecting the river regime had occurred in the contributing drainage areas.

Frequent updating of detection trends and shifts studies with Brazilian streamflow time series is currently especially recommendable given the notable recent period starting from 2015. This period presented higher frequency of intense and prolonged drought periods in several Brazilian river basins, especially in North, Northeast and regions (Brasil, 2021; Joint Research Centre, 2021), causing several socio-environmental impacts as reduction in Pantanal´s wetlands (World Wildlife Fund, 2024), closures of the navigation in Tietê-Paraná water-way (Toloi et al., 2016), isolation of amazonian communities (Lima et al., 2024), and a decrease in the country´s hydropower production (Cuartas et al., 2022; Damázio et al., 2024a). Given the coming water availability stresses to be faced in Brazil if the (Agência Nacional de Águas e Saneamento Básico, 2024) projections materialize, Brazilian water-resources planning can achieve substantial benefits from recurrent updates of these studies.

In the following items it is presented an update on previous published studies on detecting trends and shifts in LTMAFs of Brazilian HPPs based on time series of naturalized monthly flows provided by the ONS. It is not the intention to investigate the causes of the detected apparent changes, which should be addressed in subsequent studies. The study uses only a subset of the ONS times series chosen using criteria dedicated to avoiding as much as possible the errors involved in the approximations used in the process of reconstituting natural flows. The adopted approach prioritized the assessment of practical significances of trends and shifts.

MATERIALS AND METHODS

Site selection

The most important anthropic actions considered in the ONS process of obtaining the time series of naturalized monthly inflows since 1931 is the operation of upstream reservoirs (Operador Nacional do Sistema Elétrico, 2011). Upstream reservoir operation changes evapotranspiration tax and introduces a downstream flow regulation effect as a function of operational decisions regarding reservoir filling/draining regime. Other important considered anthropic actions are the abstractions of water for consumptive uses (irrigation, animal husbandry and urban, rural, and industrial supply), usually distributed throughout the river basin and whose temporal evolution has been causing an increasing impact on inflows in a large part of the country.

The process of obtaining natural inflows to HPPs uses as basic input operational data from HPPs (inflows, outflows, reservoir water storage), observed flows at nearby fluviometric stations, estimates of net evaporation at reservoirs and estimates of consumptive water use for upstream drainage areas (Operador Nacional do Sistema Elétrico, 2011). Several factors contribute to inaccuracies in the naturalized inflow time series (Machado et al., 2023). For periods with data availability inaccuracy contributions come from flow measurement errors, errors in the simulation of the propagation of outflows and errors in estimating evaporation and consumptive use. For record extensions towards 1931, the estimations of flows from older periods face the scarcity of data. Some fluviometric gaps have been resolved using rainfall-runoff models (when rainfall is available) or using spatial data transfer techniques (e.g.: regional regressions between stream flow and drainage areas; regressions with stream flows at neighboring stations, etc...). In general, the density of fluviometric and precipitation measurements in Brazil before 1970 is quite low.

In general, any streamflow trend analysis should use accurate data (Su et al., 2018; Pilo et al., 2000). Seeking to reduce the inaccuracy of naturalized inflow data included in our analysis, two measures were adopted:

1. to reduce the impacts of upstream anthropogenic actions it was only considered streamflow time series of HPPs located in unregulated headwater areas;

2. to reduce the impacts of data scarcity in filling gaps in the streamflow records we used a specific initial year for each river basin chosen according to the availability of nearby fluviometric data.

The selection process resulted in inflow time series of 52 Brazilian HPPs listed in Table 1 together with their installed capacity and drainage area. The distribution of these HPPs inside the Brazilian Hydrographic Regions is illustrated in Figure 1. Figure 2 presents the scattergram in log scales of drainage area and installed capacity. As expected, some positive correlation between the two variables is present, but the large variability of other factors of the drainage area besides its size (available water head, run-off and environment and economic feasibility conditions for development) induces a strong random scatter. It can be noted in Figure 2 the two extreme points with high drainage area and installed capacity from HPPs Belo Monte and Jirau.

Table 1
Selected Hydropower Plants (HPP). Numbers indicate points in Figure 1.

Name	Installed Capacity	Drainage Area	Name	Installed Capacity	Drainage Area
	(MW)	(km2)
				(MW)	(km2)
1- Cachoeira Caldeirão	219	30,510	27-Salto	116	10,961
2- Sto Antônio do Jari	370	52,517	28-Retiro Baixo	82	10,415
3- Balbina	250	14,949	29-Nova Ponte	510	15,300
4- Curuá Una	43	16,200	30-Guilman Amorim	140	4,060
5- Belo Monte	11,000	482,000	31-São Domingos	48	10,100
6- Boa Esperança	237	84,686	32-Candonga	140	8,900
7- Samuel	217	14,700	33-Rosal	55	1,774
8- Jirau	3,750	972,710	34-Camargos	46	6,230
9- Dardanelos	261	15,113	35-Barra do Brauna	39	7,180
10- Sinop	402	37,938	36-Caconde	80	2,560
11- Rondon II	74	3,230	37-Picada	50	1,702
12- Pedra do Cavalo	160	53,620	38-Salto Grande	102	38,800
13- Juruena	50	4,418	39-Jaguari	28	1,300
14- Serra da Mesa	1,275	51,300	40-Jurumirim	101	17,900
15- Manso	210	3,350	41-Paraibuna	87	4,150
16- Guaporé	120	1,260	42-Mauá	352	15,423
17- Jauru	118	2,230	43-Cap. Cachoeira	260	958
18- Queimado	105	3,657	44-Santa Clara PR	120	3,910
19- Corumbá IV	127	7,230	45-Foz do Areia	1,676	30,100
20- Irapé	399	5,819	46-Quebra Queixo	120	2,660
21- Itiquira I	61	5,230	47-Salto Pilão	192	5,440
22- Batalha	53	13,149	48-São Roque	142	4,341
23- Ponte de Pedra	176	3,997	49-Passo Fundo	226	1,940
24- Santa Clara MG	60	14,500	50-Barra Grande	690	11,900
25- Caçu	65	12,050	51-São José	51	9,595
26- Espora	32	4,050	52-Castro Alves	130	7,743

Figure 1
Brazilian Hydrographic Regions and Location of Selected HPPs.

Figure 2
Installed Capacity and Drainage Area Scattergram for the selected HPPs.

Figure 3 presents the histograms in logarithmic scale of installed capacities and drainage areas. Both variables occur along wide range intervals and are distributed with remarkable positive skewness.

Figure 3
Installed Capacity and Drainage Area Histograms for the selected HPPs.

The installed capacities spread between 28 MW and 11,000 MW with skewness equal to 5.87. Sixty percent of the HPPs have an installed capacity of less than 159 MW (31 cases). The remaining 21 HPPs are distributed in 17 cases with installed capacity between 160 and 1,000 MW, 3 cases with installed capacity between 1,000 and 4,000 MW (Serra da Mesa, 1,275 MW, Foz do Areia 1,676 MW and Jirau, 3,750 MW)), and 1 case with installed capacity greater than 4,000 MW (Belo Monte, 11,000 MW). The drainage areas spread from 958 km² to 972,000 km² with skewness equal to 5.47. Sixty percent of the HPPs have drainage area less than 11,000 km² (31 cases). The remaining 21 HPPs are distributed in 15 cases with drainage area between 11,000 km² and 40,000 km², 4 cases with drainage area between 40,000 km²and 90,000 km² (Serra da Mesa, 51,300 km², Santo Antonio do Jari, 52,517 km², Pedra do Cavalo, 53,620 km², and Boa Esperança, 84,686 km²), and 2 cases with drainage area far greater than 90,000 km² (Belo Monte, 482,000 km², and Jirau, 972,000 km²).

The analysis was carried out considering the HPPs grouped in Hydrograph Region and River Basin. The Amazon Hydrograph Region was divided in two parts: the Amazon1 Region, encompassing four northernmost HPPs (Balbina, Sto Antonio de Jari, Curuá-Una and Cachoeira Caldeirão), and Amazon2 Region encompassing the seven southernmost HPPs. Belo Monte HPP was included in the Amazon2 Region as its huge drainage area extends towards the South. The Tocantins-Araguaia, Parnaíba and São Francisco River Basins were aggregated to form a Hydrograph Region named as Northeast.

Trend/shift statistics

To detect trends in LTMAF, the classic two-tailed t-test can be used to assess the significance of the regression slope α in the simple linear regression of annual streamflow values on year:

$100\frac{Q\left(t\right)}{\overline{Q}}=\beta +\alpha t+{\epsilon }_t$ (1)

Q(t) is the annual streamflow of year t obtained as the arithmetic mean of the twelve monthly inflows data of year t, whereas $\overline{Q}$ is the arithmetic mean of all Q(t) in the time series. $\beta$ and $\alpha$ represents respectively the regression intercept and the regression slope whereas ${\epsilon }_t$ is an annual random departure from the trend whose quadratic sum is minimized.

This basic statistical test has been used for detection trend in the mean of streamflow time series, sometimes named as Spearman Correlation Coefficient test (Muller et al., 1998) or simply as Linear Gradient test (Robson et al., 2000). The regression slope $\alpha$ is the focus of the trend analysis. Its estimate is a trend statistic that measures the trend rate as a percentage of $\overline{Q}$ per year. Trend rate estimates equal to or near to zero are indications that no trend is present (the null hypothesis). Negative or positive departures from zero are considered as manifestations of persistent decrease/increase of LTMAF with time. Obviously, the larger the modulus of α, the greater the degree of the trend rate presence.

The most used shift detection tests, like the Pettitt´s test (Pettitt, 1979) or the Wilcoxon-Mann-Whitney test (Siegel & Castellan, 1988) are designed to identify in a time series a possible change point (step-change) of the mean or median of the time series. The null-hypothesis is the usual that there is no change in the mean or median of the time series and the alternative hypothesis is that the mean or median has suddenly changed at some time point (and for some test, at a specific known time point.) On the other hand, there are many situations in which the change in the mean of the time series does not occur abruptly, rather gradually over a limited interval of time. In Brazil, (Damázio & Costa, 2014) noted upward trend periods in annual maximum daily streamflow time series at Paraná valleys, Brazil Southeast, concentrated between early fifties and late eighties of the last century consistent with the increase of urbanization in these valleys after the Second War which stabilized around the nineties. In order to consider LTMAF gradual shifts, (Damázio et al., 2011) proposed to perform the two tailed t-test of regression slope using a linear regression of annual streamflow values on transformation of t, called Loess-Beta transformation, which can model S-shaped shifts with flexible duration:

$100\text{Q}\left(\text{t}\right)/\overline{Q}=\text{K}+\zeta \left[2\text{F}\left(\frac{\text{t}-\text{a}}{\text{L}}+\mathrm{0,5}|\text{p}\right)-1\right]+\epsilon$ (2)

where $\text{K}$ and $\zeta$ represent respectively the regression intercept and the regression slope and the Loess Beta term F(t|p) is the accumulated function of the symmetric normalized standard Beta distribution whose density is given by:

$$\text{f}\left(\text{t|p}\right)=\frac{1}{\text{B}\left(\text{p},\text{p}\right)}{\text{t}}^{\text{p}-1}{\left(1-\text{t}\right)}^{\text{p}-1},0<t<\mathrm{1,}p>0$$ (3)

Essentially the regression in (2) encompasses initial and shifted levels, defined by K ± $\zeta$, and a S-shaped curve linking them defined by three additional parameters (a, L, p). K and $\zeta$ are in percentual of $\overline{Q}$ units. Positive $\text{ζ}$ indicates increasing levels and negative $\text{ζ}$ decreasing levels, and K is just a location parameter. The parameter “a” represents the center of the shift period (in time unit), “L” is the shift length (also in time unit), and “p” is a dimensionless shape parameter. The shift regression curve maintains the initial level K - $\text{ζ}$ for any year t < a-(L/2). The shift starts in year a-(L/2) and ends in year a+(L/2) at the shifted level K+$\text{ζ}$, which is maintained for any year t > a+(L/2). The 5-uple of estimated parameters are found by a search algorithm which minimizes the ${\epsilon }_t$ quadratic sum. Note that the application of the calibration algorithm does not exclude the possibility of obtaining a best-fit curve with a shift that starts before the time series interval and/or a shift that ends after the time series interval.

The regression (2) parameter L provides flexibility to model abrupt or gradual shifts in LTMAF. A small L value indicates abrupt shifts and an L value equal to one is equivalent to a step-change. A step-change in a known year can be modelled by fixing the parameter L as 1 and the parameter “a” in the step change year minus 0.5. A large L value is an indication of a long-lasting gradual shift. Whenever L tends to infinity, the gradual shift tends to a linear trend.

The focus of the shift analysis is the regression slope $\zeta$. Its estimative is a shift statistic which doubled modulus measures the size of the shift in terms of percentage of $\overline{Q}$. Shift size estimates equal to or near to zero are indications that no shift is present (the null hypothesis). Again, negative or positive departures of shift size estimate from zero are considered as manifestations of decrease/increase of LTMAF and the larger the estimated shift size the greater the degree of shift presence. The quotient $2\zeta /L$ measures the variation of LTMAF during the shift in percentage of $\overline{Q}$ per year.

Statistical and practical significances

The classical estimation of statistical significance levels for the above proposed trend rate and shift size departures from zero (the null hypothesis) using tables of the Student´s t-distribution are subjected to the Normality and IID assumptions for the annual streamflow time series values Q(t) as discussed before in the Introduction. Marginal distributions of annual streamflow values, as they express average streamflow for a relatively longer time interval, can better resemble the normal distribution than marginal distributions of monthly or shorter time interval streamflow values do. Some researchers claim satisfactory fits to empirical annual streamflow frequency curves has been found using normal distribution (Yevdjevich, 1963; Stedinger & Taylor, 1982; Markovic, 1965; Chiew & McMahon, 1993) although Gamma or Lognormal distributions usually would fit equally satisfactorily. Gamma or Lognormal distributions would fit better than normal for catchments with annual streamflow characterized by small number of high flow events (Dykman et al., 2023). The assumption of independence between annual streamflow values is challenged by the possibility of annual water carryovers indicates that some degree of persistence between values in the annual streamflow time series are expected, especially in catchments with large water storage capacities (Yevdjevich, 1963). Also, cycling behavior induced through teleconnections of the watershed climate with ocean-atmosphere interactions oscillations referred above can be responsible for persistence in annual streamflow time series. The statistical significances obtained in published hydrology trend analyses even if adjusting procedures for persistence are applied had been criticized (Serinaldi et al., 2018).

We chose not to adopt any measures to mitigate non-normality or short and long-term persistence in the annual streamflow time series data, and to interpret the statistical significances of the slopes of Equations 1 and 2 only as indexes of their plausibility. Rather, in the same sense of Stahl et al. (2010), the magnitude of statistics $\alpha$ and $\zeta$ were interpreted qualitatively in terms of “practical” significances using the following arbitrary convention:

• small trend rate: $\left|\alpha \right|$ ≤ 0.1;

• medium trend rate: 0.1 < $\left|\alpha \right|$ ≤ 0.5;

• large trend rate: $\left|\alpha \right|$ > 0.5 ;

• small shift size: $\left|\frac{2\zeta }{L}\right|$ ≤ 0.1;

• medium shift size: 0.1 < $\left|\frac{2\zeta }{L}\right|$ ≤ 0.5 and

• large shift size: $\left|\frac{2\zeta }{L}\right|$ > $0.5$

A similar arbitrary convention was used by Souza & Reis Junior (2022) to classify the magnitude of trend rates.

RESULTS

Linear trend statistics

Parameter $\text{α}$ varied in the interval (-1.80% per year, 1.19% per year) around the average -0.27% per year. Tables 2 to 5 present for all the analyzed annual inflow time series the selected period of the time series used by the analysis and the obtained α together with its qualitative practical significance class and its statistical significance in terms of p-level.

Table 2
Linear Trend Rate Statistics. Amazon Region.

Hydrographic Region	River Basin	Selected Period	HPP	α	Class	p-level
				(% p.a.)
Amazon1	Uatumã	1976-2023	Balbina	0.97	large	0.00*
(4 HPP)	Jari	1976-2023	Sto.Ant. de Jari	0.84	large	0.01*
	Curuá-Una	1976-2023	Curuá-Una	1.19	large	0.00*
	Xingu	1976-2023	Cach. Caldeirão	0.45	medium	0.05
Amazon2	Madeira	1976-2023	Rondon ll	0.30	medium	0.03*
			Samuel	0.23	medium	0.17
			Guaporé	0.04	small	0.47
			Dardanelos	-0.04	small	0.82
(8 HPP)
			Jirau	-0.31	medium	0.04*
	Tapajós	1976-2023	Sinop	-0.21	medium	0.14
			Juruena	-0.46	medium	0.00*
	Xingu	1976-2023	Belo Monte	-0.31	medium	0.14

• ^*
  Statistically significant at 5%.

Table 5
Linear Trend Rate Statistics. Southeast/South Regions.

Hydrographic Region	River Basin	Selected Period	HPP	α	Class	p-level
				(% p.a.)
Southeast Atlantic	Doce	1964-2023	Candonga	-0.47	medium	0.11
			Guilman-Amorim	-0.52	large	0.02*
			Salto Grande	-0.49	medium	0.06
	Itabapoana	1964-2023	Rosal	-0.06	small	0.81
(8 HPP)
	Paraíba do Sul	1964-2023	Jaguari	-0.61	large	0.00*
			Paraíbuna	-0.43	medium	0.01*
			Picada	-0.39	medium	0.06
			Barra do Brauna	-0.38	medium	0.15
Uruguay	Pelotas	1960-2023	Barra Grande	0.24	medium	0.29
(5 HPP)	Canoas	1960-2023	São Roque	0.16	medium	0.56
	Passo Fundo	1960-2023	Passo Fundo	0.18	medium	0.48
	Chapecó	1960-2023	Quebra-Queixo	0.23	medium	0.37
	Ijuí	1960-2023	São José	0.51	large	0.06
South Atlantic	Itajaí-Açu	1941-2023	Salto Pilão	0.62	large	0.00*
(3 HPP)	Capivari	1941-2023	Capivari-Cachoeira	0.34	medium	0.00*
	Jacuí	1941-2023	Castro Alves	0.69	large	0.00*

• ^*
  Statistically significant at 5%.

In Table 2 the Amazon1 sub-region presents 3 cases of large positive linear trend rates together with one case of medium positive linear trend rate. In the Amazon2 sub-region, the results for the Madeira River Basin are distributed in two cases of medium positive linear trend rates and two cases of small linear trend rates (one negative and one positive). The Tapajós and Xingu River Basins include 3 cases of medium negative linear trend rate. The analysis of Table 3 (Northeast, East Atlantic and Paraguay Regions) indicates the predominance of large negative linear trend rates (nine cases) together with two cases of medium negative linear trend rates. Table 4 (Paraná Region) is dominated by the 11 cases of negative linear trend rates (6 large, 4 medium and 1 small) together with two out-of-place positive linear trend rates (1 medium and 1 small). These two exceptions are in the Paranapanema River Basin, well known by its transitional climatological regime between Brazilian Southeast and South regions regimes. Table 5 encompasses Southeast Atlantic Region 8 cases of negative linear trend rates (2 large, 5 medium, 1 small), Uruguay Region 5 cases of positive linear trend rates (1 large and four medium), and the South Atlantic Region 3 cases of positive linear trend rates (two large and 1 medium).

Table 3
Linear Trend Rate Statistics. Northeast/East Atlantic/Paraguay Regions.

Hydrographic Region	River Basin	Selected Period	HPP	α	Class	p-level
				(% p.a.)
Northeast	Tocantins	1993-2023	Serra da Mesa	-0.85	large	0.09
	Parnaíba	1970-2023	Boa Esperança	-0.75	large	0.00*
(4 HPP)
	S. Francisco	1956-2023	Queimados	-0.80	large	0.00*
			Retiro Baixo	-0.53	large	0.03*
East Atlantic	Paraguaçu	1985-2023	Pedra do Cavalo	-1.80	large	0.04*
	Jequiti-nhonha	1985-2023	Irapê	-1.35	large	0.03*
(3 HPP)
	Mucuri	1985-2023	Santa Clara MG	-1.14	large	0.00*
Paraguay	Manso	1985-2023	Manso	-0.73	large	0.00*
(4 HPP)	Itiquira	1985-2023	Itiquira I	-0.34	medium	0.06
	Correntes	1985-2023	Ponte de Pedra	-0.40	medium	0.04*
	Jauru	1985-2023	Jauru	-0.66	large	0.00*

• ^*
  Statistically significant at 5%.

Table 4
Linear Trend Rate Statistics. Paraná Hydrographic Region.

Hydrographic Region	River Basin	SelectedPeriod	HPP	α	Class	p-level
				(% p.a.)
Paraná	Verde	1985-2023	S. Domingos	-0.04	small	0.80
	Paranaíba	1976-2023	Batalha	-1.14	large	0.00*
		1976-2023	Caçu	-0.85	large	0.00*
		1976-2023	Corumba IV	-1.11	large	0.00*
		1976-2023	Nova Ponte	-1.03	large	0.00*
		19762023	Salto	-0.68	large	0.00*
(13 HPP)
		1976-2023	Espora	-0.23	medium	0.04*
	Grande	1966-2023	Camargos	-1.09	large	0.00*
		1961-2023	Caconde	-0.47	medium	0.04*
	Paranapa-nema	1961-2023	Mauá	0.23	medium	0.36
		1961-2023	Jurumirim	0.08	small	0.72
	Iguaçu	1980-2023	Santa Clara	-0.32	medium	0.46
		1980-2023	Foz do Areia	-0.14	medium	0.76

• ^*
  Statistically significant at 5%.

Summing up, the analyzed set contains 47 cases out of 52 with detected large or medium linear trend rates, 33 cases with negative linear trend rates indicating reductions in LTMAF and 14 with positive linear trend rates indicating increases in LMTAF. In general, cases of large linear trend rates tend to be statistically significant at 5% and cases of small linear trend rates tend not to be. In our results, all five cases of small linear trend rates are not statistically significant at 5%, but two of the 23 cases of large linear trend rates are also not statistically significant at 5% (Serra da Mesa and São José). Most of the 24 cases of medium linear trend rates aren´t statistically significant at 5%, with only 8 of them statistically significant at 5%.

Shift statistics

The two-phase least squares method described in Damázio et al. (2011) was adopted for the estimation of model (2). In the first phase, parameters (a, L, p) were estimated by exhaustive search using instead of the original annual streamflow time series values ​​a description of trends observed in the time series obtained through smoothing by local regressions. The shift center (parameter “a”) varied from 1957 to 2014 (average 1997.83) and the shift length (parameter “L”) varied between 63 and 65 years (average 63.54) indicating that abrupt LTMAF shifts are not prevailing in the recorded time series. Shift start (a-(L/2)) varied between 1925 and 1983 (average 1966.06), and shift end (a+(L/2)) varied between 1990 and 2046 (average 2029.60). In several cases, the start and/or the end of the shift resulted to be located outside the time range of the record. Because this work focuses on describing prevailing change patterns in the record we maintained these best fitted shift curves for further analysis.

The shape parameter “p” varied between 1.50 and 2.0 (average of 1.98). In phase 2, K and $\zeta$ were estimated by classic linear regression of annual inflow on the Loess Beta transformation of t. Parameter K varied between 71.60% and 113.24% (average 94.06%) and parameter $\zeta$ varied between -47.17% and 34.78% (average -7.55%). The shift coefficients $2\zeta /\text{L}$ varied around the average -0.24% per year, close to the average -0.27% per year of the linear trend rate. Its interval of variation (-1.45%, 1.07%) was slightly narrower than the linear trend rate variation interval (-1.4%, 1.19%).

Tables 6 to 9 present for all the analyzed annual streamflow time series, the obtained shift period $\left(\text{a}-\text{L}/\mathrm{2,}\text{a}+\text{L}/2\right)$ and the obtained quotient $2\zeta /L$ together with its qualitative practical significance class and its statistical significance in terms of p-level.

Table 6
Shift Size Statistics. Amazon Region.

Hydrographic Region	River Basin	HPP	Shift Period	$2\zeta /L$	Class	p-level
				(% p.a.)
Amazon1	Uatumã	Balbina	1958-2023	0.795	large	0.00*
	Jari	Sto.Ant. de Jari	1982-2046	0.858	large	0.01*
(4 HPP)
	Curuá-Una	Curuá-Una	1961-2026	1.011	large	0.00*
	Xingu	Cach. Caldeirão	1977-2042	0.398	medium	0.04*
Amazon2	Madeira	Rondon ll	1983-2046	0.398	medium	0.00*
(8 HPP)
		Samuel	1983-2046	0.264	medium	0.09
		Guaporé	1983-2046	0.040	small	0.43
		Dardanelos	1954-2017	-0.020	small	0.91
		Jirau	1954-2017	-0.295	medium	0.04*
	Tapajós	Sinop	1983-2046	-0.217	medium	0.10
		Juruena	1963-2026	-0.361	medium	0.00*
	Xingu	Belo Monte	1955-2018	-0.298	medium	0.12

• ^*
  Statistically significant at 5%.

Table 7 Shift Size Statistics. Northeast/East Atlantic/Paraguay Regions.

Hydrographic Region	River Basin	HPP	Shift Period	$2\zeta /L$	Class	p-level
				(% p.a.)
Northeast	Tocantins	Serra da Mesa	1983-2046	-0.600	large	0.09
	Parnaíba	Boa Esperança	1963-2028	-0.583	large	0.00*
(4 HPP)
	S. Francisco	Queimados	1982-2046	-1.359	large	0.00*
		Retiro Baixo	1983-2046	-1.264	large	0.00*
East Atlantic	Paraguaçu	Pedra do Cavalo	1983-2046	-1.437	large	0.03*
(3 HPP)
	Jequiti-nhonha	Irapê	1983-2046	-1.053	large	0.03*
	Mucuri	Santa Clara MG	1977-2042	-1.451	large	0.00*
Paraguay	Manso	Manso	1983-2046	-0.627	large	0.00*
(4 HPP)	Itiquira	Itiquira I	1983-2046	-0.342	medium	0.02*
	Correntes	Ponte de Pedra	1983-2046	-0.417	medium	0.01*
	Jauru	Jauru	1983-1962	-0.515	large	0.00*

*

Statistically significant at 5%.

Table 8 Shift Size Statistics. Paraná Hydrographic Region.

Hydrographic Region	River Basin	HPP	Shift Period	$2\zeta /L$	Class	p-level
				(% p.a.)
Paraná	Verde	S. Domingos	1983-2046	-0.109	medium	0.40
	Paranaíba	Batalha	1976-2041	-1.014	large	0.00*
		Caçu	1983-2046	-0.868	large	0.00*
(13 HPP)		Corumba IV	1976-2041	-0.984	large	0.00*
		Nova Ponte	1966-2031	-0.865	large	0.00*
		Salto	1983-2046	-0.714	large	0.00*
		Espora	1983-2046	-0.308	medium	0.00*
	Grande	Camargos	1980-2045	-1.035	large	0.00*
		Caconde	1983-2046	-0.683	medium	0.01*
	Paranapa-nema	Mauá	1939-2002	0.445	medium	0.17
		Jurumirim	1939-2002	0.443	medium	0.13
	Iguaçu	Santa Clara	1958-2021	-0.283	medium	0.45
		Foz do Areia	1958-2021	-0.093	small	0.81

*

Statistically significant at 5%.

Table 9
Shift Size Statistics. Southeast/South Regions.

Hydrographic Region	River Basin	HPP	Shift Period	$2\zeta /L$	Class	p-level
				(% p.a.)
Southeast Atlantic	Doce	Candonga	1983-2046	-0.532	large	0.04*
		Guilman-Amorim	1983-2046	-0.713	large	0.01*
		Salto Grande	1983-2046	-0.698	large	0.02*
(8 HPP)
	Itabapoana	Rosal	1942-2005	-0.102	medium	0.71
	Paraíba do Sul	Jaguari	1967-2032	-0.543	large	0.00*
		Paraíbuna	1971-2034	-0.376	medium	0.01*
		Picada	1954-2009	-0.340	medium	0.06
		Barra do Brauna	1942-2005	-0.418	medium	0.18
Uruguay	Pelotas	Barra Grande	1938-2001	0.421	medium	0.15
	Canoas	São Roque	1938-2001	0.354	medium	0.35
(5 HPP)	Passo Fundo	Passo Fundo	1938-2001	0.353	medium	0.28
	Chapecó	Quebra-Queixo	1938-2001	0.581	large	0.08
	Ijuí	São José	1938-2001	0.893	large	0.01*
South Atlantic	Itajaí-Açu	Salto Pilão	1928-2001	1.044	large	0.00*
(3 HPP)	Capivari	Capivari-Cachoeira	1958-2023	0.333	medium	0.00*
	Jacuí	Castro Alves	1925-1990	1.070	large	0.00*

• ^*
  Statistically significant at 5%.

DISCUSSIONS

Comparisons between linear trend and loess-beta shift model

The estimated Loess-Beta shift models resulted in long-lasting shifts (L>60 years) practically coincident with the estimated Linear Trend models in most of the data period. Figure 4 compares the fitted curves of the two models for some cases.

Figure 4
Model Comparison.

The two sets of obtained angular coefficients (α and $2\zeta /\text{L})$are very close and all angular coefficient pairs showed sign agreement. There are seven cases of changing the practical significance classification: three cases of small trend rates classified as medium shift sizes (São Domingos, Jurumiri and Rosal), four cases of median trend rates classified as large shift sizes (Caconde, Candonga, Salto Grande and Quebra Queixo) and one case, shown in Figure 4d, of medium trend rate classified as small shift size (Foz do Areia).

General results are very close: the results of the shift model contain only 2 extra cases of detected large or medium-sized statistics. Balances of statistically significant cases also are close: All three cases analyzed of small shift sizes are not statistically significant at 5% and only two cases out of the 26 large shift size cases (Serra da Mesa and Quebra Queixo) are not statistically significant at 5%. Eleven of the 23 cases of medium shift size are statistically significant at 5%.

General results obtained for each Hydrographic Region are almost the same for the two models. For example, large positive cases of both angular coefficients together with one case of medium positive of both angular coefficients appears at the Amazon1 sub-region. In the Northeast, East Atlantic and Paraguay Regions, one can find in both models the predominance of large negative angular coefficients together with two cases of medium shift size angular coefficients.

Figure 4 shows that the separations between the trend curves of the two models occurs only in the early period or in the late period of the time series. These separations address interesting questions like whether the LTMAF of Castro Alves (session d of Figure 4) was stable in the past in a lower level and after an upward trend period, is now stabilized; or like whether the observed upward trend is only part of a long-term internal climate cyclical variation and the near future will show a downward trend. But, considering the description of prevailing patterns of the streamflow time series evolution in the analyzed record, the two trend models can be considered equivalent for all cases. In the following we will consider only the results of the Linear Trend Model.

Comparisons between complete time series versus selected period

The estimation of the linear trend model for the set of chosen HPP was repeated using complete ONS naturalized inflow time series.

Figure 5 and its companion Table 10 compare for four cases fitted linear trend curves and rates obtained by using the complete ONS naturalized inflow time series and using only the selected period. In some cases, noticeable differences were noted, including change of the rate sign, as the case of Foz do Areia illustrated in session d of Figure 5.

Figure 5
Linear Trend Curves - Complete Period versus Selected Period.

Table 10
Linear Trend Rates - Complete versus Selected Period.

HPP	Complete period		Selected period
	$\overline{Q}$	α	$\overline{Q}$	α
	(m3/s)	(% per year)	(m3/s)	(% per year)
Belo Monte	7,693	-0.120	7,838	-0.311
		(medium)		(medium)
Jurumirim	223.1	0.368	240.2	0.084
				(small)
		(medium)
Castro Alves	156.3	0.625	159.6	0.688
				(large)
		(large)
Foz do Areia	662.7	0.378	736.1	-0.142
		(medium)		(medium)

The scattergram of Figure 6 and the boxplots of Figure 7 illustrates the general impact on the linear trend rate of the different periods. Figure 6 plots for each of the 52 cases its linear trend rate of the complete period analysis in the abscissa and of the selected period analysis in the ordinate. It can be noted that most of the points are below the line of equal trend rates. 27 cases out of the 28 cases with negative trend rates in the complete period analysis (left of the zero vertical line) maintained the negative direction of its trend rate in the selected period analysis and most of them had its negative trend rate deepened. In the other side, 17 cases out of the 25 cases with positive trend rates in the complete period analysis changed the direction of its trend rate in the selected period analysis. The selected period analysis resulted in a greater fraction of negative trend cases (34 cases versus 27 cases) and in a lower mean trend rate (-0.25% p.a. versus -0.02% p.a.). Figure 7 shows that the dispersion of trend rates was larger in the selected period analysis than in the complete period analysis, which can be in part attributed to the shorter period length. Further statistical analysis is needed to ascertain about it.

Figure 6
Scattergram of Linear Trend Rates.

Figure 7
Box Plots of Linear Trend Rates.

Hydrological trends in Brazilian hydrographic regions

Table 11 presents for the different Brazilian regions and for the whole country percentages of detected large/medium trend in terms of number of HPP, installed capacity and drainage area.

Table 11
Percentages of Large/Medium Trend.

Hydrograph Regions	Number of HPPs	Installed Capacity	Drainage Area
Amazon	83%	94%	40%
	(10/12)	(15,755/16,755)	(654/1,645)
Northeast/East Atlantic/Paraguay	100%	100%	100%
	(11/11)	(2,883/ 2,883)	(236/236)
Paraná	85%	77%	68%
	(11/13)	(2,058 / 2,679)	(102/149)
Southeast Atlantic/Uruguay/South Atlantic	94%	51%	75%
	(15/16)	(1,250 / 2,451)	(84/112)
Total:	88%	88%	50%
	(47/52)	(21,765/24,768)	(1,078 / 2,145)

Figure 8 presents for each hydrograph region the average linear trend rate $\alpha$. As can be seen, the analysis indicates increasing LTMAFs for 3 Hydrographic Regions (Uruguay, South Atlantic and Amazon 1), and reductions in LTMAFs for other 6 (Amazon 2, Paraguay, Southeast Atlantico, Parnaíba, Paraná, Northeast, East Atlantico).

Figure 8
Hydrograph Region Average Linear Trend Rates.

Table 12 compares the general sign of the trends in inflow time series for each hydrographic region obtained here with results of other trend detection studies in Brazilian streamflow time series. It’s worth pointing out that the set of streamflow time series, its data periods and selected statistical procedures are not the same along the studies.

Table 12
Sign of Trends/Shifts in Historical Streamflow Time Series obtained in Different Studies.

Hydrograph Region	This Study	Couto & Fichtner (2023)	Silva et al. (2019)	Alves et al. (2013)	Damázio et al. (2011)
	Statistics	Statistics	Statistics	Statistics	Statistics
	LT, LBS	MK	MK	MKS, MM, LT, LOWT	LBS
	Period 1941-2023 1993-2023	Period 1931-2021	Period 1941-1999 1941-2007	Period 1931-2010	Period 1931-2005
Amazon1	Increase	Increase	-	-	-
Amazon2	Reduction	Reduction	Reduction	-	Reduction
Toc./Araguaia (Northeast)	Reduction	Reduction	Reduction	Reduction	-
Parnaíba (Northeast)	Reduction	Reduction	-	-	-
São Francisco (Northeast)	Reduction	Reduction	Reduction	-	-
East Atlantic	Reduction	Reduction	-	-	-
Paraguay	Reduction	Increase	-	-	-
Paraná	Reduction	Increase	Increase	Increase	-
Southeast Atlantic	Reduction	Reduction	-	-	-
Uruguay	Increase	Increase	-	-	-
South Atlantic	Increase	Increase	Increase	-	-

• Note: LT = Linear Trend; LBS = Loess Beta Shift; MK = Mann-Kendall; MKS = Mann-Kendall-Sen; MKTS = Mann-Kendall-Theis-Sen; MM = Median Mobile; LOWT = Lowess Trend.

Sign disagreements found in the literature appeared in the Paraguay and Paraná regions. Previous studies had detected positive trends for these regions, usually prevailing in periods which were discarded in this study (see session d of Figure 5).

Finally, it is worth to highlight that the LTMAF tendencies found in our study coincides with the projected tendencies found in Agência Nacional de Águas e Saneamento Básico (2024).

FINAL CONSIDERATIONS

Recent studies on global climate change impacts on water availability for Brazilian water basins (Agência Nacional de Águas e Saneamento Básico, 2024) projected reduction of streamflow for most of the country hydrographic regions with the only exception of the Uruguay and South Atlantic regions highlighting the importance of providing Brazilian water planning agencies with updated information about observed trends and shifts in country streamflow time series. This paper aimed an update of previous studies on detecting trends and shifts in LTMAFs of Brazilian HPPs based on time series of naturalized monthly flows provided by the ONS updated until DEC 2023. Previous studies were focused on monotonic trends or abrupt shifts levels and used mostly non-parametric tests whereas. This study was based on the simplistic linear trend model and on the Loess Beta shift with flexible duration model proposed in Damázio et al. (2011) and prioritized the assessment of trends and shifts practical significances rather than statistical significance.

The application of the flexible Loess Beta shift model did not find abrupt shift cases and produced essentially the same conclusions as the simpler linear trend model. The exercise results indicate the presence of large or medium trends for most of them. Summing up, the analyzed set contains 47 cases out of 52 with detected large or medium linear trend rates, 33 cases with negative linear trend rates indicating reductions in LTMAF and 14 with positive linear trend rates indicating increases in LMTAF.

The reduction of record lengths due to discarding of filled records resulted in more dispersed and generally reduced streamflow trend rates. The reduction of trend rates can be attributed to a real cycling trend in Brazilian streamflow with a positive trend rate phase followed by a negative trend rate. Further studies on this issue should consider the uncertainties in the reconstruction of naturalized streamflow for the early periods.

The analysis indicates increasing LTMAFs for 3 Hydrographic Regions (Uruguay, South Atlantic and Amazon 1), and reductions in LTMAFs for other 6 (Amazon 2, Paraguay, Southeast Atlantico, Parnaíba, Paraná, Northeast, East Atlantico). Sign disagreements with the other published studies appeared in the Paraguay and Paraná regions which can be attributed to discarding of early period criterium adopted in our study (see Figure 5d). The effect of including the year 2022 was not remarkable and definitively not the cause of disagreement of results with the results of Couto & Fichtner (2023). Remarkable, also, the inclusion of the 2015-2022 period was not the cause of the few disagreements in the direction of the prevailing trends for the Paraguay and Paraná Hydrographic Regions also found for the other tree studies analyzed in Table 6. Again, the discarding of early period was determinant for these disagreements.

The observed variation of trend rates in LTMAF over the Brazilian Hydrographic Regions raises a question about the impact of the hydropower production in each of these regions and about the net impact on total hydropower production in SIN, the Brazilian national interconnected power system. As a matter fact, SIN encompasses 175,000 km of transmission lines which effectively distribute electric energy production throughout the country. The study in Damázio et al. (2024b) carried out trend analysis in historical time series of aggregated energy content of naturalized inflows for the Brazilian hydropower system using data from 1931-2018. The trend detection study with this time series found evidence of cyclical regime with a positive trend period from 1931 to 1990 followed by a negative trend. Further studies should be done in this direction considering the inaccuracy of early data in the streamflow record.

The locations where trends in naturalized annual inflow series were detected cover 13 of the 27 states of the federation and the Federal District, respectively: Amazonas, Pará, Rondônia, Mato Grosso, Mato Grosso do Sul, Goiás, Bahia, Maranhão, Minas Gerais, São Paulo, Paraná, Santa Catarina and Rio Grande do Sul. Become clear that further investigations about the condition of the Brazilian HPP system will be appreciated specially in the Hydrograph Regions of Paraná, Southeast Atlantic and São Francisco where reductions of LTMAFs were detected. Apart the concern about the reduction of hydropower production, these regions are highly populated and include the most important Brazilian states in terms of economic performance São Paulo, Rio de Janeiro and Minas Gerais, producing respectively ~32%, ~11% and ~10% of the national gross domestic product.

ACKNOWLEDGEMENTS

The three anonymous reviewers are thanked for critically reading the manuscript and suggesting substantial improvements.

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Edited by

• Editor-in-Chief:
  Adilson Pinheiro

• Associated Editor:
  Carlos Henrique Ribeiro Lima

Publication Dates

• Publication in this collection
  26 May 2025
• Date of issue
  2025

History

• Received
  13 Oct 2024
• Reviewed
  31 Jan 2025
• Accepted
  15 Mar 2025