Free Surface Flow Model

In this study, we consider that the only modeled fluid is water. Although certain substances contained in water have no influence on the flow, our mathematical model is based on Navier-Stokes equations for an incompressible Newtonian fluid [8][9]:

where u represents the velocity vector, P the pressure, ρ the water density (1000 kg/m³), ν the kinematic viscosity, and f external forces (mainly gravity).

Given the relatively low fluid velocities (at most a few m/s), we also neglect the Coriolis force compared to gravity. The system of equations can also be written in scalar form.

The pressure at any point in the fluid is decomposed into a reference pressure P₀ and a hydrostatic component:

With: x₃ elevation of point, z free-surface elevation.

The integration of the equations obtained over the vertical (between the bottom of elevation Z and the free surface z) allows the elimination of the vertical velocity.

This Vertical integration transforms the 3D system into a 2D model in the horizontal plane (x₁, x₂), known as the shallow-water model. This is the model that is actually solved to obtain the sizing discharges.

For any variable A in the 3D field, its depth-averaged value over the water column of thickness h [m] is defined as:

We note, moreover, that in the absence of inputs, the vertical velocity at the bottom is equal to:

Same type of expression at the free surface and the water depth his equal to z-Z.

The vertical velocity’s term then becomes:

where Tᵢ is the following expression:

This term decomposes into four distinct contributions t1,t2, t3 and t4:

t1: the term related to velocity dispersion over the vertical:

It is usually modeled as the sum of two terms:

And tᵢ and Dᵢ are time-constant coefficients representing, respectively, the dispersion coefficient and the horizontal turbulent diffusion coefficient.

t2: the term related to stresses inside the fluid (viscosity, turbulence)

It is replaced by a term:

t3: the bottom friction term:

t4: the surface friction term:

2.2.1. Boundary Conditions of the 2D Model

Boundary conditions adopted for the Sounda simulation are:

UPSTREAM: Qriv(t) = measured monthly flow at Sounda gauging station [m³/s], Qt imposed as inflow boundary condition time series 1952-2013.DOWNSTREAM : Free outflow normal or critical depth condition h_down = Hn (normal depth).BANKS: Impermeability zero normal-penetration condition u.n =0 (zero normal velocity at bank).BED: Manning-Strickler friction in t3 n_Manning ≈ 0.035 s/m^1/3.

2.2.2. Characteristic Flow Rates

Extraction of the scalar sizing flow

From the solved 2D field ( u ¯ 1 ( x 1 , x 2 ) , u ¯ 2 ( x 1 , x 2 ) , h ( x 1 , x 2 ) ), the velocity is integrated over the reference cross-section to obtain the volumetric flow Qriv. This discharge Qriv [m³/s] is the only scalar transferred from the 2D model to the 1D energy chain. All sizing formulas use exclusively this scalar together with the geometric (Hb, Hn) and mechanical (η) parameters of the plant.

Once the scalar flow rate Qriv is extracted from the 2D model, all power and energy calculations are performed through 1D algebraic relationships. These formulas constitute the complete sizing chain of the plant.

Minimum flows are required to account for other forms of water use; the turbine flow rate Qt used for sizing is the measured river flow rate Qriv from which we subtract the reserved flow rate Qres [8][10]-[12]:

Definition, Quantification and Justification of the Reserved Flow Qres

The reserved flow Qres (also called minimum ecological flow) is the minimum flow rate that must be maintained in the Kouilou-Niari River downstream of the Sounda intake at all times, to preserve aquatic ecosystem functions and satisfy downstream water uses (navigation, fisheries).

In this study, quantification adopted is based on the hydrological method of the 10th percentile of the natural daily flow series (Q10 method recommended by WMO Guide to Hydrological Practices, 2008 [11]), and consistent with practice in Central African river basins (CICOS/PEAC guidelines), Qres is set to 10% of the average annual flow:

This value ensures that even during the critical low-water months (July–September, when the mean is 402 - 533 m³/s), the reserved flow represents less than 23% of the natural flow, leaving ample turbined flow Q t = Q r i v − 92.85   m 3 / s .

Regulatory and ecological rationale:

The Republic of Congo’s Water Code (Law 14-2003) requires maintenance of a reserved flow on all regulated watercourses. The 10% rule is the minimum threshold recommended by Poff et al. [13] to prevent severe ecological degradation. The PEAC/CAPP pre-feasibility technical report for Sounda [1] adopts a reserved flow of 90 - 100 m³/s, consistent with the value retained here.

Furthermore, the total energy per kg of fluid is expressed by Bernoulli’s equation:

Hydraulic power is determined from:

This allows us, by applying Bernoulli’s theorem for incompressible fluids, to determine the flow rate:

2.2.3. Net Head

The net head Hn is the difference between the gross head Hb and the head losses Δh:

where Δh_l represents linear head losses and Δh_s represents singular head losses (valves, bends, grids, etc.).

2.2.4. Energy Equations of the Hydroelectric Power Plant

Maximum Gross Power (MGP)

Maximum gross power is calculated from the maximum diversion flow rate Qm and the gross head Hb, without taking into account head losses or machine efficiency:

Note: Q_max here equals the total rated discharge.

Average Installed Electrical Power (AIEP)

The average power of a power plant planned for a given site can be calculated with the following formula:

Maximum available power (MAP)

Maximum available power represents an estimate of the actually exploitable power, taking into account head losses and machine efficiency η:

Normal Gross Power (NGP)

Normal gross power is calculated from the average usable flow rate Qm over an annual cycle (considering reserved flow rate) and gross head:

With: Qt_ann = mean(Qt) over 1952-2013 = 835.68 m³/s (= 928.53 − 92.85 m³/s after subtracting Qres).

Normal Available Power (NAP)

Normal available power is calculated from the average usable flow rate, net head, and machine efficiency:

Annual Theoretical Energy Production (ATEP)

Annual theoretical energy production is calculated from normal available power for an estimated annual operating time of 7500 h (one year has 8760 h), to account for scheduled and unscheduled outages:

The study is based on a hydrological database from the Kouilou-Niari measurement station at Sounda, covering the period 1952-2013, representing 62 years of continuous observations. This long time series allows for robust statistical analysis of characteristic flows and their interannual variability. The data includes average monthly flows measured in m³/s distributed by period, enabling identification of seasonal trends and quantification of variations in available water resources [5][6][11][13].

The hydrometric series was obtained and covers monthly average flows rate [m³/s] at the Sounda gauging station (Kouilou–Niari River) from January 1952 to December 2013, yielding N = 744 monthly values. Inspection of the raw dataset (cf. Appendix 1) confirms that all 62 years are complete with no missing months (0 gaps out of 744).

Quality-Control Procedure

The following four-step procedure [5][6] was applied:

Range plausibility check any value below 100 m³/s or above 4000 m³/s was flagged for manual verification. The observed minimum is 201.0 m³/s (September 2004) and maximum is 3303.5 m³/s (February 1961), both physically plausible for the 57,900 km² catchment. Outlier detection Grubbs’ test (α = 0.05) was applied to the annual mean series: no statistically significant outliers were detected. Rating-curve change assessment two rating-curve updates were documented for the Sounda station (in 1965 and 1991 following gauge resets); values in the affected transition months were cross-checked against the neighbouring Kakamoéka station records and found consistent to within 5%. Trend and homogeneity test—the Pettitt non-parametric test detected a statistically significant change-point in 1972 (p = 0.03), coinciding with the 1970s Sahelian drought that affected Central Africa; both the pre-1972 sub-period (mean = 1012 m³/s) and post-1972 sub-period (mean = 893 m³/s) were retained as they represent natural hydro-climatic variability relevant to long-term plant sizing.

The average flow rates calculated over 62 years of hydrological data from the Kouilou-Niari River are shown in the following Figures 1-4:

Figure 1. Average monthly flows in m3/s from 1952 to 1965.

Figure 2. Average monthly flows in m³/s from 1966 to 1980.

Figure 3. Average monthly flows in m³/s from 1981 to 1995.

Figure 4. Average monthly flows in m³/s from 1996 to 2013.

The hydrological regime of the Kouilou-Niari River is directly influenced by rainfall in the Republic of Congo-Brazzaville, which presents two distinct seasons [3][5][11]-[13]:

A rainy season (flood period) from October to May (8 months) characterized by abundant precipitation;A dry season (low-water period) from June to September (4 months) marked by significant precipitation decrease.

Seasonal Mean Computation

The flood season (from October to May, 8 months) and low-water season (from June to September, 4 months) were defined based on the hydrograph shape analysis and regional rainfall seasonality (equatorial bimodal regime). For each season, the average flow rate was computed as the simple arithmetic average of the corresponding monthly values over all 62 years (cf. Formula in Appendix 2):

Q_flood = average (Oct, Nov, Dec, Jan, Feb, Mar, Apr, May) over 1952-2013 = 1129.50 m³/s (σ = 298 m³/s);Q_low = average (Jun, Jul, Aug, Sep) over 1952-2013 = 526.60 m³/s (σ = 112 m³/s). The average annual Q_ann = 928.53 m³/s (σ = 197 m³/s)

The seasonal analysis results of the characteristic flow rates of the Kouilou-Niari River, derived from the exploitation of 62 years of measurements (1952-2013) at the Sounda gauging station, are presented in Table 1.

Table 1. Characteristic flow rates of the Kouilou-Niari river at Sounda (1952-2013).

The analysis reveals strong seasonal variability with a flood/low-water ratio of 2.14, indicating that the flow rate during the flood period is more than double that during the low-water period. This characteristic has a direct impact on the plant’s energy production management.

3.2.1. Simulation of Hydrodynamic Parameters

The simulation curves of the energy capacity of the sounda hydroelectric power plant on the Kouilou-Niari river, produced using matlab software, are presented in the following Figures 5-11 [2][9][11]:

Figure 5.Maximum gross power curve from 2002 to 2013.

Figure 6. Maximum available power curve from 2002 to 2013.

Figure 7.Normal gross power curve from 2002 to 2013.

Figure 8. Normal available power curve from 2002 to 2013.

Figure 9.Installed electrical power curve from 2002 to 2013.

Figure 10. Average electrical power curve from 2002 to 2013.

Figure 11. Theoretical energy produced annually from 2002 to 2013.

These simulations model the energy capacity of the planned hydroelectric power plant at the Sounda Gorges on the Kouilou-Niari river (Republic of Congo), as a function of the mean annual flow recorded over the period 2002-2013. Each curve represents an energy variable as a function of flow rate, thereby revealing the sensitivity of power generation to hydrological variations.

The curve on Figure 5 shows a near-perfect linear relationship between the total basin flow (Qriv, ranging from 630 to 1.180 m³/s) and the gross maximum power (ranging from 480 to 870 MW). The steep and regular slope indicates that the MGP is directly and proportionally linked to the available flow. This confirms that the gross potential of the plant is entirely dependent on the river’s hydrological inputs. Of the same linear nature, the curve on Figure 6 shows the MAP varying from 330 to 615 MW for the same flow rates. The parallelism between curves of Figure 5 and Figure 6 confirms the internal consistency of the model.

The Figure 7 demonstrates that the nominal gross potential of the plant is entirely determined by the mobilizable flow. The Normal Available Power on Figure 8, the power actually exploitable after losses ranges from 2.3 to 4.4 10² MW. The curve is also linear and parallel to the Gross Nominal Power.

The Installed Electrical Power on Figure 9, which corresponds to the power actually installed on the machines, also follows a strictly linear law with flow (from 380 to 690 MW). Very similar to Figure 9, the curve of average installed electrical Power (AIEP) on Figure 10 confirms the stability of the model. The AIEP values (350 - 700 MW) represent an estimate of the average exploitable power over the year, independent of seasonal peaks.

The Annual Theoretical Energy Production (ATEP) curve on Figure 11 is the most information-rich and most significant curve for plant sizing. Unlike the others, it presents a non-linear growth and a greater scatter. Indeed, the ATEP rises from 0.3 to 3.2 × 10³ GWh as flow increases from 560 to 680 m³/s, representing a very rapid increase and the data points no longer align on a straight line, reflecting the influence of additional factors (operating hours, seasonal variability, reservoir management).

3.2.2. Calculation Parameters

The calculation of Hydraulic Power and Electrical Potential are realized with the parameters defined in Table 2 below:

Table 2. Calculation parameters.

3.2.3. Hydraulic Power and Electrical Potential during Flood Period

The general formula: P = ρ ⋅ g ⋅ H n ⋅ Q [ W ]

With ρ = 1000 kg/m³, g = 9.81 m/s²,

Which gives the specific power coefficient k = ρ ⋅ g ⋅ H n ⋅ η / 10 6 [ MW / ( m 3 / s ) ] .

Then, for Hn = 68.5 m, η = 0.85: k = 1000 × 9.81 × 68.5 × 0.85 / 10 6 = 0.5713   MW / ( m 3 / s )

For Hn = 65 m: k= 0.5423 MW/(m³/s).

In addition, Q t = Q r i v − Q r e s

Thus, we calculate the hydraulic power and electrical potential during flood period

Knowing the flows rate: Qriv_flood = 1129.50 m³/s; Qriv_low = 526.60 m³/s; Q_ann = 928.53 m³/s.

And consider Hn = 68.5 m, η = 0.85, Hb = 75 m,

The results of calculation are reported in Table 3 [3][10][12]:

Table 3. Hydraulic power and electrical potential during flood period.

The flood period allows mobilization of 592.15 MW, representing approximately 124% of average annual power.

3.2.4. Hydraulic Power and Electrical Potential during Low-Water Period

Table 4 below shows the Hydraulic Power and Electrical Potential during the low-water period (June to September), the average river flow rate is 526.60 m³/s and the net turbine flow rate is 433.80 m³/s.

Table 4. Hydraulic power and electrical potential during low-water period.

The low-water period generates 247.78 MW, representing 41.8% of flood period power. Despite shorter duration (4 months), it contributes 713.61 GWh to annual production.

3.2.5. Annual Energy Balance

The effective energy of 3580.19 GWh/year accounts for 86.8 % availability (7500 hours out of 8760 annual hours), reflecting scheduled maintenance and unscheduled outages.

3.3.1. Turbine Selection Logic: Count, Rated Flow, and Operational Feasibility

Turbine count selection rationale:

The number of units N is determined by two competing constraints:

Each unit must be large enough to achieve high hydraulic efficiency (unit power > 100 MW for Francis turbines at this head range, per IEC 60193 [14]);The plant must remain operational during low-water periods without extreme part-load penalties.

Rated flow per unit and constraint analysis:

For a given net head Hn and turbine technology, the specific speed Ns governs the optimal operating point.

The rated flow per unit Qt_unit = Qt_total/N must satisfy:

Minimum: the plant must turbine at least Qt_min = 0.40 × Qt_unit × N during low-water to stay above cavitation and efficiency cliff.

Four hydroelectric development sizing variants are proposed, covering different optimization strategies based on the following criteria: high power, optimal performance, maximum flexibility, and investment economy [15]-[19].

3.3.2. Variant 1: High Power Configuration (6 Francis Turbines, Hn = 68.5 m)

This configuration prioritizes high installed capacity with six Francis turbines, offering large production capacity such as described in Table 5 [18].

Table 5. High power configuration (6 Francis turbines, Hn = 68.5 m).

3.3.3. Variant 2: Optimal Configuration (4 Francis Turbines, Hn = 68.5 m)

Table 6 below describes a variant which offers the best compromise between investment cost, energy performance, and capacity factor [17][20].

Table 6. Optimal configuration (4 Francis turbines, Hn = 68.5 m).

3.3.4. Variant 3: Mixed Configuration (3 Francis + 2 Kaplan Turbines, Hn = 68.5 m)

An hybrid configuration combines Francis turbines (high water) and Kaplan turbines (low water) to maximize operational flexibility is described in Table 7 [18][20].

Table 7. Mixed configuration (3 Francis + 2 Kaplan turbines, Hn = 68.5 m).

3.3.5. Variant 4: Economical Configuration (4 Francis Turbines, Hn = 65 m)

The variant which represents an economical alternative with reduced net head, offering a good balance between initial investment and energy production is presented in Table 8 [20][21].

Table 8. Economical configuration (5 Francis turbines, Hn = 65 m).

A comparative analysis of the four variants is presented in Table 9 below:

Table 9. Comparative analysis of the four variants.

After a comparative analysis of the different variants, we recommend variant 2 as the optimal solution (64.7% capacity factor, 4352.45 GWh/year) for the Sounda Gorges power plant construction project. However, variant 4 constitutes a viable economic alternative (3913.66 GWh/year).

Sensitivity Analysis

The three dominant parameters are: net head Hn (nominal: 68.5 m), overall plant efficiency η (nominal: 85%), and head losses Δh (nominal: 6.5 m from Hb = 75 m).

The annual energy production in the base configuration is:

Table 10 shows sensitivity variation parameters on ATEP.

Variant ranking robustness

The relative ranking (V2 > V3 > V1 > V4 in energy; V2 > V4 > V1 > V3 in capacity factor) is preserved across all single-parameter sensitivity tests. This confirms that V2 is the robustly optimal variant over the plausible parameter range.

Table 10. Sensitivity variation.