Dynamical Control Model of the Cascaded Kainji-Jebba
Hydropower Operating Head

1 2 3 3 3

- Olalekan Ogunbiyi, Cornelius T. Thomas, Isaac A. O. Omeiza, Jimoh Akanni and Benjamen J. Olufeagba

1Department of Electrical and Computer Engineering, Kwara State University, Malete, Nigeria
Department of Electrical and Information Engineering, Achievers University, Owo, Nigeria

2Department of Electrical and Information Engineering, Achievers University, Owo, Nigeria
3Department of Electrical and Electronics Engineering, University of Ilorin, Nigeria

3Department of Electrical and Electronics Engineering, University of Ilorin, Nigeria
{biyikan\|corneliustthomas}@gmail.com\| [isaacavazi@yahoo.com](mailto:isaacavazi@yahoo.com) \|{jimaka2005\|benjabezolufsenx}@gmail.com

{biyikan\|corneliustthomas}@gmail.com\| [isaacavazi@yahoo.com](mailto:isaacavazi@yahoo.com) \|{jimaka2005\|benjabezolufsenx}@gmail.com

Abstract— Operation and design of control system for the cascaded Kainji-Jebba hydropower system poses a great challenge to
researchers and engineers. The difficulties arose from the fact that the system is affected by several nonlinear interacting factors such as
variations in inflows, stochastic factors that are weather related, availability of the turbo-alternators, and numerous other constraints that are
influenced by the system dynamics. All these makes the mathematical representation of the system difficult. This paper presents the
development of a dynamical model for the operation and optimal control of the operating heads of the cascaded system. The mathematical
models were developed from energy conversion equation and Bernoulli’s equation. The model was calibrated and tuned using measured
data. Upon validation by comparing the response of the model with measured head, a deviation within ±2% was observed, making it a
good prediction of the system response and appropriate for control system design.

Keywords— Control model, Discharge, inflows, Operating head, Turbo-alternators.
—————————— ◆ ——————————

1 INTRODUCTION
H

ydropower generation in Nigeria is currently
Hprovided at three major locations: the Kainji
hydroelectric power station (KHEPS), the Jebba
hydroelectric power station (JHEPS) and the Shiroro
hydroelectric power station. KHEPS and JHEPS are
located on the River Niger about 103 km from each other
with the latter downstream while SHEPS is built on
Kaduna River in Niger State, Nigeria (Sambo, Garba,
Zarma, & Gaji, 2012).

0 ′ ′′ 0 ′ ′′
The KHEPS located at 09 5145 𝑁, 04 3648 𝐸 has an
installed capacity of installed capacity of 760 MW from 8
units of turboalternators, units 1G5 to 1G12. Units 5 and
6 are rated 120MW each, units 7, 8, 9 and 10 are rated 80
MW while units 11 and 12 are 100MW each. The JHEPS
on the other hand is located 103 km downstream of the
0 ′ ′′ 0 ′ ′′
KHEPS at 09 0808 𝑁, 04 4716 𝐸. It was
commissioned on April 13, 1985 with an installed
capacity of 578.4 MW. It has has six fixed blade propeller
type turboalternators, each rated at 96.4MW (Omeiza et
al., 2019; Salami, 2007)

09008^{\\prime}08^{\\prime\ 1}N,;04^{\\prime}47^{\\prime}16^{\\prime\\prime}E

2 OPERATIONAL CHARACTERISTICS OF THE
KHEPS AND JHEPS
An aspect of the problems affecting the operation of

KHEPS AND JHEPS
An aspect of the problems affecting the operation of
KHEPS and JHEPS in the cascade is the reliance on
intuitive (trial and error) water release rules instead of
scientifically motivated policies(Jimoh, 2008; Salami,
2007). KHEPS turbo-alternators have variable blades and
internal governors; this allows the reservoir head to vary
between 24 m and 42 m, depending on the time of the
year. Clearly, under such conditions, fixed blade turbine
cannot operate.

JHEPS, employs fixed bladed turbines and as a result,
the head has to be maintained strictly between the range
of 99 m and 103 m. Achieving this objective pose some
very serious challenges to the operator since the two
reservoirs are in cascade separated by 103 km and the
discharge from KHEPS depends on the condition of the
machine, inflow into Kainji and the rainfall in between.

The availability of the turbo-alternators also has a direct
effect on the reservoir head. Whenever a unit fails at
KHEPS, it reduces the inflow into JHEPS. Operators of
JHEPS may be forced to reduce the number of operating
machines which in turn reduces the energy on the grid.
The head at KHEPS keeps increasing and a significant
percentage of the water is lost to evaporation. A reverse
situation occurs when machines fail at JHEPS while
KHEPS keeps releasing water, the excess water spill
away instead of being utilised at KHEPS. A review of the
daily report of operations from the two stations shows
that there are several days when the units are shut down
due to the inability to manage resources appropriately

\*Corresponding Author

* * *

(TCN-NCC, 2017).

To solve the optimal control problem that will maximise
the energy generation in the two stations, the problem
must be properly posed in a standard form (Robert &
Michaud, 2011; Zheng, Fu, & Wei, 2013).

There have been various researches to model the system
such as to come up with a scientifically motivated
operational policy. Some are based on the statistical
model of inflow observation (Using regression analysis)
and the use of dynamic programming for optimal policy
formulation (Aribisala, 2007; Jimoh, 2008; Salami, 2007).
Such models are not suitable for dynamic system.

A time series model has also been suggested but the
model formulated using this method is non-causal,
hence cannot be used in control design (Ale et al., 2011;
Aribisala, 2007; Nwobi-Okoye & Igboanugo, 2012).

The artificial neural network model was presented in
(Abdulkadir et al., 2013; Igboanugo & Nwobi-Okoye,
2013; Salami et al., 2015), similarly the model does not
leads to a real time control system design.

A more appropriate model is those involving the system
dynamics. Nevertheless there have been focus on the
turbine dynamic and its effect on system stability (Lu &
Hogg, 2000; Nanaware, Sawant, & Jadhav, 2012). A more
appropriate model should consider the reservoir
dynamics, turbine dynamics and availability of the
turboalternators.

The literature review carried out during this work could
not find a related work that attempts to develop a
control model for either or the two hydropower stations.
This research, therefore, focuses on developing a control
model for the two stations which can be used to
determine the optimal control policies for the release of
water from KHEPS such that the reservoir head at
JHEPS remains relatively constant.

The operating parameters are also defined as listed
below: ℎ represents the water head (𝑚), 𝑄 is the Inflow
into the reservoir (𝑚3/𝑠), 𝑞 is the inflow into the
penstock (𝑚3/𝑠), QL represents the losses (m3/s), 𝑄𝑠
stands for the discharge through spill way (𝑚3/𝑠), 𝐴1 is
the effective surface area (𝑚2), 𝐴2 is the area of the inlet
to the penstock (𝑚2) and 𝑈1 represents the turboalternator units.

({\\bf m}^{3}/s{\\bf})

3 MATHEMATICAL MODELLING OF THE KHEPS
AND JHEPS
As demonstrated in this Figure 1 presents the schematic

(m3/s)

K - Kainji
J - Jebba

J - Jebba
1 – KHEPS Reservoir

1 – KHEPS Reservoir
2 – KHEPS Turbo-alternator

2 – KHEPS Turbo-alternator

4 – JHEPS Reservoir
5 – JKEPS Turbo-alternator

5 – JKEPS Turbo-alternator
6 – JHEPS Penstock

6 – JHEPS Penstock
7- River Channel

7- River Channel
8 – Discharge for JHEPS

8 – Discharge for JHEPS

(k\ {\\check{g}}/{check{m}}^{3})

JHEPS
Given that 𝑃𝑒is the electrical power developed in from a
hydropower plant, 𝜂 represents the energy conversion
3
efficiency, 𝜌 is the density of water in (𝑘𝑔⁄𝑚 ), 𝑔 is the
2
acceleration due to gravity (𝑚⁄𝑠 ), ℎ is the operating
head of the reservoir (𝑚); and 𝑞 is the flow rate in
(𝑚3⁄𝑠).

:P\_{\\epsilon}

below: ℎ represents the water head (𝑚), 𝑄 is the Inflow
3.1 ELECTRIC POWER GENERATED AT KHEPS AND
is the inflow into the
JHEPS
/s), 𝑄𝑠
Given that 𝑃 is the electrical power developed in from a

(m/s^{2})

(m^{3}/s)

* * *

From Figure 1, the flow rate q is related to the discharge
velocity by;

q

q=A\_{2}v\_{2}

(2)

𝐴2 is the cross-sectional area of the penstock intake (𝑚)
and 𝑣 is the velocity of water (𝑚⁄𝑠).

A\_{2}

P\_{e}=,\\eta\\rho g h A\_{2}v

(3)

The velocity can be express as a function of head by
applying the Bernoulli’s energy equation (4) to the input
and output of the reservoir (White, 2015).
1⁄

(4)

Let 𝑣1 be the velocity at the intake and 𝑣2
discharge, 𝑃1 𝑎𝑛𝑑 𝑃2 are atmospheric pressure values at
the surface and the outlet. In practice 𝑃1
approximately equal for Kaplan low head schemes. ℎ2

be the velocity at the intake and 𝑣2 at the
are atmospheric pressure values at
the surface and the outlet. In practice 𝑃1 𝑎𝑛𝑑 𝑃2 are
approximately equal for Kaplan low head schemes. ℎ2 is

approximately equal for Kaplan low head schemes. ℎ2 is
the head at the outlet which equals zero. Since the
velocity of water at the head of the reservoir is much less
than the velocity at the penstock outlet, 𝑣2 is far greater

than the velocity at the penstock outlet, 𝑣2
than 𝑣1 and equation (4) can be reduced to the form in

than the velocity at the penstock outlet, 𝑣2 is far greater
and equation (4) can be reduced to the form in

than 𝑣1 and equation (4) can be reduced to the form in
equation (5) (Guo et al., 2009; Kyung et al., 2010):

equation (5) (Guo et al., 2009; Kyung et al., 2010):

equation (5) (Guo et al., 2009; Kyung et al., 2010):

\\stackrel{1}{\\slash!{}}\\rho v{v\_1}^{2}+\\rho g h\_{1}+P\_{1}=\\stackrel{1}{\\slash{}}!{\ /{}}\ \\stackrel{2}{\\rho{v\_{2}}^{2}}+\\rho g h\_{2}+P\_{2}~\ (4)

v\_{1}

v\_{2}

P\_{1}

P\_{1}

P\_{2}

v\_{2}

v\_{1}

\\rho g h\_{1}=\\stackrel{1}{!}/ _{2}\\rho\\ensuremath{v_{2}}^{2}

(5)

v\_{2}=\\sqrt{2g h\_{1}}

(6)

q=A\_{2}\\sqrt{2g h}

(7)

(17)

(8)

\ {cal P} _{e}=\\beta\\eta{\ h h}^{3/{}_{2}}

(9)

\ \ mathrm w w h r r e\\quad\\beta=\\sqrt{2},\\rho A\_{2},g^{3}/\_{2}

The three groups of units at KHEPS can be represented
as follows: 𝑛1𝐾stands for 4 sets of 80 MW units, 𝑛2𝐾
stands for the 2 sets of 100 MW, while 𝑛3𝐾represents the
two sets of 120 MW. Hence the electric power generated
at KHEPS is represented by equation (10).

(18)

n\_{1K}

n\_{3K}

(10)

(14)

\\begin{array}{r}{P\_{K}=\\left(\\sqrt{,2},,n\_{1K},\\eta\_{1K},\\rho,A\_{2K},g^{3/ _{{}2}}}\\right){_{{K}}^{3/ _{{}_{2}}}\ +}\ {\\left(\\sqrt{,2},,n\_{2K},\\eta\_{2K},\\rho,A\_{2K},g^{3/ _{{}_{2}}}\\right){h\_{K}}^{3/ _{{}_{2}}}\ +}\ {\\left(\\sqrt{,2},,n\_{3K},\\eta\_{3K},\\rho,A\_{2K},g^{3/ _{{}_{2}}}\\right){h\_{K}}^{3/ _{{}_{2}}}\ +}\\end{array}

P\_{J}=\\left(\\sqrt{2},,n\_{J},\\eta\_{J},\\rho,A\_{2J}g^{3}/ _{2}\\right){h h_{J}}^{3/\_{2}}

n\_{K}

JHEPS has 6 identical units and they are designated
by 𝑛𝐽.

(P\_{T})

Equation (15) represents the power generated, indicating
that it depends on the reservoir head and number of
operating units. Consequently, the dynamical model of
hydropower station is determined by the dynamical
consideration of both the alternator and reservoir
operating head.

(19)

\\Psi\_{J}=\\beta\_{J},n\_{J},\\eta\_{J}

P\_{K}=\\beta\_{K}n\_{1K}\\eta\_{1K}h\_{K}^{^{3/2}}+\\beta\_{K}n\_{2K}\\eta\_{2K}h\_{K}^{^{3/2}}+\\beta\_{K}n\_{3K}\\eta\_{3K}h\_{K}^{^{3/2}}\ (

P\_{T}=P\_{K}+P\_{J}

P\_{J}=\\beta\_{J},n\_{J},\\eta\_{J},h\_{J}^{\ /\_{2}}

3.2 DYNAMICAL HEAD EQUATIONS FOR THE KHEPS AND
JHEPS RESERVOIRS

Consider the KHEPS reservoir as represented in Figure 1

Consider the KHEPS reservoir as represented in Figure 1
with Q as the inflow and q as the outflow;

\\begin{array}{r}{A\_{1K}\\cfrac{d h\_{K}}{d t}=Q\_{K}-Q\_{L K}-Q\_{S K}-q\_{2K}=Q\_{K}-Q\_{L K}-Q\_{S K}-}\ {A\_{2K}v\_{2K}}\\end{array}

Equation (16) can be combined with equation (6) to
give:

n\_{J}

\\begin{array}{r}{A\_{1K}\\c{frac{d h\_{K}}{d t}}=Q\_{K}-Q\_{L K}-Q\_{S K}-\\sqrt{2g h\_{K}\ A A\_{2K}}}\\end{array}

Hence, the dynamical model for KHEPS with 𝑛𝐾
number of units is expressed as:

\\begin{array}{r}{\\frac{d h\_{K}(t)}{d t}=-,n\_{K}\\propto\_{K}h\_{K}^{1/2}(t)+\\mu\_{K}(Q\_{K}(t)-Q\_{L K}(t)-Q\_{s K}(t))}\\end{array}

\\propto\_{K}!=\\sqrt{2g},{A\_{1K}}^{-1}A{ _{2K}};;\\mathrm{a n d};;\\mu_{K}={A\_{1K}}^{-1}

The dynamical model for JHEPS with 𝑛𝐽number of
units and 𝑄𝐶𝐽(𝑡) (the inflow from the catchment area inbetween the two reservoirs) can be expressed as:

\\begin{array}{r}{\\frac{d h\_{J}(t)}{d t}=-,n\_{J}\\propto\_{J}h\_{J}^{^{1/2} _{2}}(t);+;\\mu_{J}(Q\_{J}(t)-Q\_{L J}(t)-Q\_{s J}(t))}\\end{array}

Q\_{c I}(t)

\\mathrm{W h e r e},,,Q\_{J}=q\_{k}+Q\_{s k}+Q\_{C J}

Combining equations (19) and (20) gives the JHEPS
model as presented in (21).

\\begin{array}{r l}&{\\frac{d h\_{J}(t)}{d t}=-n\_{J}\\propto\_{J}h\_{J}^{1/2}(t),+,\\mu\_{J}((q\_{k}(t)+Q\_{s k}(t)),+}\ &{\\quad\\quad\\quad\\quad Q\_{C J}(t)-Q\_{L J}(t)-Q\_{s J}(t))}\ &{

\\mathrm{w h e r e}\\propto\_{J}=\\sqrt{2g},{A\_{1J}}^{-1}{A\_{2J}},,\\mathrm{a n d},,,\\mu\_{J}={A\_{1J}}^{-1}.

3.3 ESTIMATION OF MODEL PARAMETERS
If observations of the inflow and head are studied such

If observations of the inflow and head are studied such

Q that a section of time where the behavior of the system is
almost linear is selected. If 𝑄(𝑡)is the net inflow
in (𝑚3⁄𝑠), 𝑞(𝑡)is the discharge in (𝑚3⁄𝑠) and ∆ℎ is the
change in head between time 𝑡1 and 𝑡𝑛, then;

Q t

(m^{3}/s),,q\_{(t)}

(m^{3}/s)

t\_{1}

\\Delta h

t\_{n}

(23)

Equation (21) was applied to a measured data to obtain:

A\_{1k}=883\ ,208,,571.43,m^{2},\ \ {tt n n d}

A\_{1j}=287\ .775\ 00\ m^{2}.

The effective area of the scroll casing can also be
estimated from the observation. This is motivated by the
fact that

(24)

Hence given the observation for a whole year, the
median value of 𝐴2𝑖was used as 𝐴2 in the model. Where
𝑛𝑖represents the number of operating units on day 𝑖, 𝑞𝑖
is the total station discharge on day 𝑖.

A\_{2}

A\_{2i}

n\_{i}

q\_{i}

In estimating the effective area of the scroll casing (A2)
for KHEPS and JHEPS, the area was calculated per day
using equation (24) from 1stof Jan. to 31stof Dec. 2013.
The calculated values were modelled and the median in
each case was estimated as:

A\_{2k}=8.55005;m^{2};\\mathrm{a n d};;A\_{2j}=\ 13.53289;m^{2}

The evaporation loss used in this model was
estimated from observations between 1974 and 2009 for
KHEPS and 1985-2010 for JHEPS. Monthly maximum,
minimum and average evaporation loss were plotted in
each case. The average value was mathematically
modelled and presented as:
𝑄 = 0.0003𝑡6 − 0.0126𝑡5 + 0.0845𝑡4 + 1.8323𝑡3 −

\\begin{array}{r}{Q\_{e v p,K}=0.0003t^{6}-;0.0126t^{5}+;0.0845t^{4}+1.8323t^{3}-}\ {25.699t^{2}+88.123t+7.267\\qquad\\qquad\\qquad\\qquad\\qquad2((25)}\\end{array}

(25)

\\pm2%

\\begin{array}{r}{Q\_{e v p,~J}=0.0003t^{6}-0.0126t^{5}+0.0845t^{4}+1.8323t^{3}-}\ {25.699t^{2}+88.123t+7.267}\ \ \ \ \ \ \ \ \ \ \ \ \ \ (26)}\\end{array}

4 MODEL VALIDATION OF THE CASCADED
KHEPS AND JHEPS USING OBSERVATIONS
The model was validated subject to the fact that the two

The model was validated subject to the fact that the two
stations are connected through the channel such that the
inflow into JHEPS is equal to the sum of the discharge from
KHEPS, spill from KHEPS and the flow from tributaries
along the connected channel. A comparison between the
measured head and the computed head for year 2013 is as
presented in Figure 2 (a) to (d). The inflows and the number
of operating machines per day were passed into the model
in addition to the system parameters: effective surface area,
effective scroll casing area and the evaporation. The results
show a good agreement between the measured and
computed head in each case. An error within ±2% was
observed, making the model a good prediction of the
system response and appropriate for control system design.
The slight deviation is as a result of the estimated value
used in the system parameters and the approximation
errors from the numerical solution to the model.

(26)

(a) Jan.- Mar.

\[Image: Image76\]

\[Image: Image76\]
(b) April. - Jun.
\[Image: Image78\]

(d) Oct.- Dec.
Fig. 2: Comparison of Observed Head with Computed Head for the
Cascade KHEPS and JHEPS in 2013

A model for nonlinear control system design and
analysis for the cascaded KHEPS and JHEPS has been
presented in this paper. The dynamical models were
developed from flow continuity conditions. The
developed model, together with the optimal control
algorithms will work well to provide a more dependable scheme for the operators. This will boost the generation
potential of the cascaded power system and ensure safe
operation such that the operating heads are kept within
limits.
For future research, we recommend that the turbine

For future research, we recommend that the turbine
dynamics should be embedded in the equation to
replace the conversion efficiency. Also, it is
recommended that a similar work should be carried out
on the Shiroro hydropower station, to aid research in
ensuring the operation of the station throughout the
year.

ACKNOWLEDGMENT
We wish to appreciate the management of Mainstream

We wish to appreciate the management of Mainstream
Energy Solution for allowing access to their facilities and
the Transmission company of Nigeria, National Control
Center Oshogbo for providing the needed data.

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