Textbook Renewable Energy Forecasting and Risk Management Paris France June 7 9 2017 Philippe Drobinski Ebook All Chapter PDF
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Springer Proceedings in Mathematics & Statistics
Philippe Drobinski
Mathilde Mougeot
Dominique Picard
Riwal Plougonven
Peter Tankov Editors
Renewable
Energy:
Forecasting and
Risk Management
Paris, France, June 7–9, 2017
Springer Proceedings in Mathematics & Statistics
Volume 254
Springer Proceedings in Mathematics & Statistics
This book series features volumes composed of selected contributions from
workshops and conferences in all areas of current research in mathematics and
statistics, including operation research and optimization. In addition to an overall
evaluation of the interest, scientific quality, and timeliness of each proposal at the
hands of the publisher, individual contributions are all refereed to the high quality
standards of leading journals in the field. Thus, this series provides the research
community with well-edited, authoritative reports on developments in the most
exciting areas of mathematical and statistical research today.
Peter Tankov
Editors
Renewable Energy:
Forecasting and Risk
Management
Paris, France, June 7–9, 2017
123
Editors
Philippe Drobinski Riwal Plougonven
Laboratoire de Méteorologie Laboratoire de Météorologie
Dynamique Dynamique
CNRS Ecole Polytechnique
Palaiseau, France Palaiseau, France
Dominique Picard
UFR de Mathématiques
Université Paris Diderot
Paris, France
This Springer imprint is published by the registered company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
v
vi Preface
vii
Contents
ix
x Contents
xi
xii Contributors
Abstract We propose a dynamical model for the wind speed which is a Markov
diffusion process with Weibull marginal distribution. It presents several advantages,
namely nice modeling features both in terms of marginal probability density function
and temporal correlation. The characteristics can be interpreted in terms of shape and
scale parameters of a Weibull law which is convenient for practitioners to analyze
the results. We calibrate the parameters with the maximum quasi-likelihood method
and use the model to generate and forecast the wind speed. We have tested the model
on wind-speed datasets provided by the National Renewable Energy Laboratory.
The model fits well the data and we obtain a very good performance in point and
probabilistic forecasting in the short-term in comparison to the benchmark.
1.1 Introduction
The two-parameter Weibull probability density function has become widely used to
fit wind speed datasets in the literature of wind energy (see [5, 9] and the reference
therein). It has been included in regulations concerning wind energy and in most
popular software on wind modeling like HOMER and WAsP.1
and http://www.wasp.dk/.
A. Bensoussan
International Center for Risk and Decision Analysis, Jindal School of Management,
The University of Texas, Dallas, TX, USA
e-mail: alain.bensoussan@utdallas.edu
A. Bensoussan
Department SEEM, City University of Hong Kong, Hong Kong, People’s Republic of China
A. Brouste (B)
Institut du Risque et de l’Assurance du Mans, Laboratoire Manceau de Mathématiques,
Le Mans Université, Le Mans, France
e-mail: alexandre.brouste@univ-lemans.fr
The energy production of a wind farm is related in particular to the wind speed on
the site through the power transfer function of wind turbines. Forecasting production
on timescales from minutes to days improves the operational management of a wind
farm. From seconds to minutes, this forecast allows an accurate indication to achieve
to control the turbine and to smooth the production on the electricity grid by storage.
From hours to half a day, it can be used to decide whether or not to store for efficient
trading on electricity markets. Finally, forecasting production on timescales of a day
to several days helps in scheduling maintenance operations. These problems belong
to the class of stochastic optimization problems due to the intermittent nature of the
production (or the wind) and the solutions depend mainly on the underlying model
used for the production (or the wind speed).
Several dynamical models such as classical time series (ARMA, FARIMA, . . .),
Markov chains [6, 13, 18, 20], semi-Markov chains [7, 8] and neural networks [17]
have been considered in the literature for both modeling and short-term forecasting.
Short-term forecasting corresponds in this paper to timescales of half an hour to half
a day and can be distinguished for long-term forecasting where numerical weather
prediction models are involved in addition to statistical methods.
We have considered the Cox–Ingersoll–Ross (CIR) diffusion in [2] to model the
square of the wind speed. Diffusion processes in general provide efficient point and
probabilistic forecasts obtained from transition probability density functions. For
instance, the CIR model outperforms the persistence benchmark (i.e. last measured
value as forecast value) in terms of one-step forecast mean square error (MSE).
Moreover, stochastic optimization problems with such continuous Markov processes
are handled conveniently.
In this paper, we propose a diffusion process for the wind speed whose marginal
law is Weibull. Temporal correlation structure of the stochastic process is also
parametrized. We show that the model is interesting as a wind generator model.
Indeed, the fact that characteristics are understandable in terms of shape and scale
parameters of a Weibull law helps practitioners analyzing the results. This model
also shows good performances in short-term forecasting for a large class of datasets
provided by the National Renewable Energy Laboratory (NREL). The model is com-
pared to the persistence benchmark and the Ornstein–Uhlenbeck diffusion process
both in terms of MSE (for point forecasts) and continuous ranked probability score
(CRPS, for probabilistic forecasts) for short-term lead times (see also [12] for defini-
tions of other score and other benchmark for one-step ahead forecasting). Contrary
to the CIR process, the transition probability density function is no longer available
in closed form. Consequently, new calibration and forecast methods are presented in
this paper which differ from our previous work [2].
In Sect. 1.2, the dynamic marginal Weibull diffusion model is presented. Point
forecast and probabilistic forecast methods are developed in Sect. 1.3. In Sect. 1.4,
the estimation method of the parameters of the aforementioned model is presented
and the dynamic model is calibrated on NREL dataset. Forecasting performances of
the different models are also summarized.
1 Marginal Weibull Diffusion Model … 5
In this section, we describe the model mentioned in the introduction. The three-
parameter marginal Weibull diffusion process (Zt , t ≥ 0) is the solution of the
stochastic differential equation (sde)
t t
Zt = z0 + v0 (Zs , ϑ)ds + v1 (Zs , ϑ)d βs , t ≥ 0, (1.2)
0 0
where (βt , t ≥ 0) is a standard Wiener process. Moreover the drift coefficient v0 (·, ·)
and the diffusion coefficient v1 (·, ·) are known functions that are described below.
The stochastic process (Zt , t ≥ 0) is an homogeneous Markov diffusion process
which can be characterized by its transition probability density function p(t, ·; x, ϑ)
representing the conditional probability density function of Zs+t given Zs = x (for
more information on Markov diffusion processes, see [1] or [11]). The long term
law of Zt (as t → ∞) is called the stationary distribution or invariant distribution.
It is denoted p(·; ϑ). If the initial condition z0 is random and follows the invariant
distribution, then the law of Zt is still the invariant distribution for all t ≥ 0. In this
setting, the stationary distribution of our model is Weibull with scale parameter ϑ2
and shape parameter ϑ3 (see Eq. (1.1)).
In this model, we fix the drift term equal to
1
v0 (z, ϑ) = ϑ1 ϑ2 Γ 1 + −z (1.3)
ϑ3
where Γ is the gamma function. Here, the parameter ϑ1 is the temporal correlation
parameter of the process. It can be shown, if the initial condition is the invariant
distribution, that the correlation structure is given by
6 A. Bensoussan and A. Brouste
For more information, see Appendix 1. Also, the computation of the proper diffusion
coefficient v1 (z, ϑ) to obtain Weibull probability density function as marginal distri-
bution given the previously fixed correlation structure is postponed to Appendix 2.
Namely, the diffusion coefficient is given by
⎛ ⎞
2ϑ1 ϑ2 Γ 1 + ϑ1
z ϑ3 z ϑ3
1+ ϑ1
v12 (z, ϑ) = 3 ⎝ 1 − e − ϑ2 −
1 ϑ2
u 3
−1 −u
e du⎠ .
f (z, ϑ) Γ 1 + ϑ1 0
3
(1.5)
For the marginal Weibull diffusion model the transition probability density func-
tion p(t, y; x, ϑ) cannot be obtained in closed form. Fortunately, it is the solution of
the Fokker–Planck (FP) equation
∂ ∂ 1 ∂2
p(t, y; x, ϑ) = − (v0 (y, ϑ)p(t, y; x, ϑ)) + v1 (y, ϑ)2 p(t, y; x, ϑ)
∂t ∂y 2 ∂y2
(1.6)
with initial condition p(0, y; x, ϑ) = δx (y) where δx (y) represents a Dirac distribution
at point x.
The operator of a wind farm is interested in forecasting the production in the next
hours to provide the information to the entity in charge of the electric grid and trade
on electricity markets. From the transfer function of the wind turbine, the production
is directly related to the wind speed. So the problem boils down to the short-term
forecasting of the wind speed.
1.3.1 Definitions
Suppose that we fix the present time at t = 0 and that the initial observed wind speed
is Z̃0 . Let us denote Z̃t the true (random and unknown) value of the wind speed at
time t > 0 and ψ̃(t, ·) its (unknown) probability density function (see Fig. 1.1).
In this paper, a point forecast is an estimator of the wind speed Z̃t given the
knowledge of Z̃0 . It is denoted by π(ZtZ̃0 ). We call persistence benchmark the current
knowledge
πper (ZtZ̃0 ) = Z̃0 . (1.7)
1 Marginal Weibull Diffusion Model … 7
πd (ZtZ̃0 ) = Eϑ (Zt )
where Zt is the solution of (1.2) with initial condition z0 = Z̃0 . In the particular case
of diffusion processes with a linear drift v0 (z, ϑ) = ϑ1 (α − z), it can be shown (see
Appendix 1) that the point forecast has the following closed-form
πd (ZtZ̃0 ) = α + Z̃0 − α e−ϑ1 t . (1.9)
We recall that the parameter ϑ1 parametrizes the correlation structure and stands for
the mean-reverting speed. For the marginal Weibull diffusion model
1
α = ϑ2 Γ 1+ .
ϑ3
have also linear drift (see also [2]) and consequently generate similar point forecasts.
This indicator makes it possible to compare persistence model and diffusion model
point forecasts. For the persistence model defined in (1.7),
2
MSEper (t) = E Z̃0 − Z̃t .
Diffusion models (1.2) have the property that the well-specified MSE (namely MSE
(1.12) where
Zt = Zt ) can be written as
1 Marginal Weibull Diffusion Model … 9
MSE(t) = Eϑ Zt2 − (Eϑ (Zt ))2 = u(t, Z0 ) − (Eϑ (Zt ))2
∂u ∂u v12 (x, ϑ) ∂ 2 u
= v0 (x, ϑ) +
∂t ∂x 2 ∂x2
with
u(0, x) = x2 .
For instance, the MSE can be obtained in closed form for the Ornstein–Uhlenbeck
process and the Cox–Ingersoll–Ross process (see [2]). Direct computations lead to
σ 2 (1 − e−2ϑ1 t )
MSE(t) =
2ϑ1
σ2
lim MSE(t) = and MSE(t) ∼0 σ 2 t,
t→∞ 2ϑ1
and, comparatively,
ασ 2
lim MSE(t) = and MSE(t) ∼ σ 2 Z0 t as t → 0.
t→∞ 2ϑ1
In the marginal Weibull setting, no closed form are available but numerical compu-
tations can be performed for the MSE with MSE(t) ∼ v12 (Z0 , ϑ)t as t → 0.
For a diffusion model in general, the probabilistic forecast is given by its transition
probability density function
For the Gaussian distribution, the previous function is explicitly computable, namely
1 x−μ
x−μ x−μ
CRP(N (μ, σ ), x) = −σ
2
√ − 2ϕ − 2Φ −1
π σ σ σ
(1.16)
where ϕ and Φ are respectively the probability distribution function and the cumu-
lative probability distribution function of a standard Gaussian random variable. For
a deterministic forecast,
2
CRP(δy , x) = 1{u≥y} − 1{u≥x} du = |y − x|. (1.17)
R
It is worth mentioning that for deterministic forecast, thanks to (1.17) the CRPS is
reduced to the mean absolute error (MAE) defined by
MAE(t) = E π(ZtZ̃0 ) − Z̃t . (1.18)
Conversely to the CIR model [2], transition (forecast) probability density func-
tions are not in closed form. Consequently, computations of the CRPS requires using
the Fokker–Planck finite element numerical solving scheme which is sophisticated
(see Sect. 1.3.3.3). In the following, we propose probabilistic forecasts methods based
on the approximation of the transition probability density function p(t, ·; x, ϑ).
1 Marginal Weibull Diffusion Model … 11
For a very short lead time t > 0, the Gaussian approximation for transition proba-
bility density function is valid. Consequently, we use the Gaussian distribution as
the probabilistic forecast with the first order or second order Itò-Taylor expansion
characteristics. For the first order expansion, we recall that the mean is given by
mx = x + v0 (x, ϑ)t
and variance by
Here v0 and v1 are given respectively by (1.3) and (1.5). For the second order expan-
sion, the mean is given by
2
∂ 1 2 ∂2 t
mx = x + v0 (x, ϑ)t + v0 (x, ϑ) v0 (x, ϑ) + v1 (x, ϑ) 2 v0 (x, ϑ)
∂x 2 ∂x 2
For the marginal Weibull diffusion model and linear drift diffusion models,
∂2
v0 (x, ϑ) = 0
∂x2
and corresponding terms in the previous equation disappear.
12 A. Bensoussan and A. Brouste
Let Θ be a bounded open subset of (R+ ∗ ) . We present in this section the calibration
3
n
L (ϑ, Z (n) ) = log p(Δn , Zti ; Zti−1 , ϑ) (1.19)
i=1
where the transition probability density p(t, y; x, ϑ) is given by the solution of the
Fokker–Planck equation (1.6).
1 Marginal Weibull Diffusion Model … 13
ϑn = max L (ϑ, Z (n) )
ϑ∈Θ
can be computed numerically. But, for the three-parameter marginal Weibull diffu-
sion process, this is not the case anymore.
Several methods are available in this context depending on the discretization time
scheme. For a large observation horizon tn and large mesh size Δn , one can use an
approximation of the likelihood function (numerical approximation of the Fokker–
Planck equation in [15], Monte-Carlo simulation approximation [16], . . .). A recent
review of possible methods in this setting is proposed in [19].
When the mesh is relatively small (thereafter Δn will be equal to 1 hour), which is
the case we are considering, it is possible to use the quasi-likelihood approach. For
small Δn , under proper assumptions, it is possible to approximate the transition prob-
ability density function (and consequently the likelihood) by a Gaussian probability
density function with the same mean and variance.
The simplest approximation is the Euler method,2 in which the transition proba-
bility density p(Δn , y; x, ϑ) is approximated by a Gaussian with mean
and variance
σx2 = v12 (x, ϑ)Δn . (1.21)
n 2
n 1 Zti − Zti−1 + v0 (Zti−1 , ϑ)Δn
L ∗ (ϑ, Z (n) ) = − log 2π v12 (Zti−1 , ϑ)Δn − .
2 2
i=1
v2 (Zti−1 , ϑ)Δn
1
(1.22)
ϑn∗ = max L ∗ (ϑ, Z (n) ).
ϑ∈Θ
Although the estimator is not in a closed form, it can be computed numerically. This
approximation is valid for a “rapidly increasing experimental design” which means
Δn → 0, nΔn → ∞ and nΔ2n → 0 (see [14] for details).
2 The Gaussian approximation of the conditional density function p(Δ, y; x, ϑ) proposed in [14]
uses the higher order Itò-Taylor expansion to approximate the mean and the variance. It is worth
emphasizing that the Euler method is the one order Itò-Taylor expansion.
14 A. Bensoussan and A. Brouste
with respect to the calibrated marginal distribution F (Weibull and Gaussian respec-
tively) and (3) selection of the distribution corresponding to the smallest value of the
computed statistic.
Stationary distribution for the marginal Weibull diffusion model and the Ornstein–
Uhlenbeck process (respectively Weibull and Gaussian) are illustrated in Fig. 1.2 for
the ID 24310 wind turbine in 2011. Correlation structure for the marginal Weibull
diffusion model is also illustrated in Fig. 1.2.
The result of the selection for all wind turbines and all the years are given in the
following Table 1.1.
The previous table shows for instance that the marginal Weibull diffusion model
outperforms the (Gaussian) Ornstein–Uhlenbeck process in terms of wind-speed
generation in 2011 in Fig. 1.2. This is merely confirming common practice of using
the Weibull distribution rather than using a Gaussian to describe the distribution of
the wind speed.
0.15
0.10
Density
0.05
0.00
0 5 10 15 20 25
Wind speed (m/s)
1.0
0.8
autocorrelation
0.6
0.4
0.2
0.0
Fig. 1.2 On the top, histogram of the training NREL wind speed dataset (see Sect. 1.4). Stationary
distribution for the marginal Weibull diffusion model (plain line) and Gaussian Ornstein–Uhlenbeck
process (dashed line) fitted on the training dataset are superposed. On the bottom, empirical auto-
correlation of the training NREL wind speed dataset (plain black line) and calibrated correlation
structure of the marginal Weibull diffusion model (plain gray line)
t ,z
π Ztjj+τj
the forecast value knowing that Z̃tj = zj . The empirical mean square error (eMSE) is
given by
16 A. Bensoussan and A. Brouste
−1−k
N 2
t ,z
π Ztjj+τj − zj+k
j=0
eMSE(τ ) = .
N −k
We compute the empirical MSE for all models described above. Namely,
t ,z
1. For the persistence benchmark, π Ztjj+τj = zj ;
2. For the dynamic 3 parameters marginal Weibull diffusion model with linear drift,
the point forecast is given by (1.9) or
t ,z 1 1
π Ztjj+τj = ϑ2 Γ 1 + + zj − ϑ2 Γ 1 + e−ϑ1 τ
ϑ3 ϑ3
Table 1.2 Bias and RMSE in parenthesis (all measures in m/s) for wind-speed forecasts for lead
times of relevance
τ = 1h τ = 2h τ = 3h τ = 6h
Weibull diffusion 0.00 (1.97) 0.00 (2.59) 0.00 (2.97) 0.00 (3.58)
Persistence 0.00 (2.04) 0.00 (2.73) 0.00 (3.19) 0.00 (4.04)
benchmark
1 Marginal Weibull Diffusion Model … 17
Table 1.3 CRPS and MAE in brackets for wind-speed forecasts for lead times of relevance
τ = 1h τ = 2h τ = 3h τ = 6h
FEM 0.25 [0.34] 0.36 [0.49] 0.41 [0.56] 0.51 [0.71]
1st order TI 0.25 [0.34] 0.36 [0.49] 0.44 [0.57] 0.61 [0.71]
Gaussian
Persistence [0.33] [0.50] [0.58] [0.78]
benchmark
Table 1.4 Bias, RMSE in parenthesis, RMSE in % of the nominal power for production forecasts
(all measures in kWh) for lead times of relevance
τ = 1h τ = 2h τ = 3h τ = 6h
Weibull diffusion −10 (171) [9%] −13 (229) [11%] −20 (261) [13%] −29 (310) [16%]
Ornstein– −23 (192) [10%] −31 (253) [12%] −51 (289) [14%] −75 (339) [17%]
Uhlenbeck
In this section, we compare the production forecast given, on the one hand, by the
three-parameter marginal Weibull diffusion model calibrated on the wind speed and
the computation of the corresponding production through the manufacturer power
function and, on the other hand, by an Ornstein–Uhlenbeck directly calibrated on
the production dataset. Production point forecasting (in kWh) performance is given
in terms of RSME in the following Table 1.4. The computation has been done on a
subset of the dataset (every 25 hourly measurements).
It is worth emphasizing that, considering no errors on the manufacturer transfer
function and a transfer function only depending on the wind speed, forecasting using
the wind speed outperforms the Ornstein–Uhlenbeck model on production.
1.5 Conclusion
Yt = f (t)Zt + g(t)
where f (·) and g(·) are periodic functions (see for instance [3] for a related work
where the diurnal cycle is taken into account). The characteristics of the marginal
Weibull diffusion model are understandable in terms of shape and scale parameter
of a Weibull law that is appreciated by practitioners to analyze the results.
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during my absence, only one kafila had arrived from Bornou,—the
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above his head at the same time with equal rapidity; he would then
rejoin the dance. In the centre of the ring there were two drummers,
the drums standing on the ground. They were made of a hollow
block of wood about three feet high, with a skin drawn tensely over
the top by means of braces. A great concourse of natives were
assembled to witness the exhibition.
June 5.—Morning cloudy. At six in the morning we left Girkwa,
and reposing ourselves during the heat of the day under some
tamarind trees among the villages of Nansarina, we encamped at
sunset in the woods. The inhabitants were now very busy in the
fields planting grain. Their mode of planting it is very simple. A man
with a hoe scrapes up a little mould at regular intervals, and is
followed by a woman carrying the seed, of which she throws a few
grains into each hole, and treads down the mould over them with her
feet.
June 6.—At noon we halted in the town of Sangeia, the governor
of which was at Kano; so I fortunately escaped the pain of hearing
his squeaking voice. We encamped for the night in the woods.
June 7.—At one in the afternoon we halted outside the town of
Katungwa. At sunset two horsemen arrived at full gallop, with the
news of the governor of Kano having taken a town, at a very short
distance to the north, from the rebel Duntungua.
June 8.—Every where the inhabitants were busily employed
clearing the ground, and burning the weeds and stubble, preparatory
to sowing grain. We sheltered ourselves from the mid-day heat
under the shade of a tamarind tree, in the province of Sherra, and
halted for the night outside the town of Boosuea. A son of the
governor of Sherra was here, attended by a number of horsemen,
and a band of music. He drank coffee with me, and I was in turn
regaled with music the greater part of the night. The instruments
were chiefly flutes and long wooden pipes, called by the natives
frum-frum.
June 9.—At sunset we arrived at the town of Dugwa.
June 10.—At daybreak we left Dugwa, and travelled through a
thickly wooded country. It rained all day, and we also had some
thunder and lightning. At seven in the evening we arrived at Murmur.
I heard, at Kano, that a kafila of Arabs, belonging to Augela, had
destroyed the clay wall around Dr. Oudney’s grave, and made a fire
over it, telling the inhabitants he was a Kafir. This report, to my great
regret, I found to be true.
June 11.—At sunrise I sent for the governor, to inquire who had
committed the outrage, when he protested it was the Arabs, and not
the people of the town. I felt so indignant at this wanton act of
barbarity, I could not refrain from applying my horsewhip across the
governor’s shoulders, and threatened to report him to his superior,
the governor of Katagum, and also to despatch a letter on the
subject to the sultan, unless the wall was immediately rebuilt: which,
with slavish submission, he promised faithfully to see done without
delay. During my halt at noon, near Katagum, I sent Dumbojee
forward to inform Duncawa, the governor, of my return. In the
afternoon I heard that he was on his way to meet me; and I had
scarcely left my resting-place before he made his appearance,
attended by about thirty horsemen, who, when they saw me, came
up at full gallop, brandishing their spears. I presented the governor
with a hundred Goora nuts, every one of which he distributed
amongst his people. He gave me many very hearty welcomes, and
made numerous inquiries about Bello, and his behaviour to me. He
and his people now galloped into the town, yelling and skirmishing;
and although the governor had been sick for some time past, he
appeared as lively and cheerful as any of them. On entering
Katagum I was lodged in my old quarters, and was immediately
visited by my old friend Hameda, the Tripoline merchant, who was
still here. I invited him to accompany me to Tripoli, as the late Dr.
Oudney had advised him; but he excused himself, on the plea of
being unable to collect his outstanding debts from his numerous
creditors, who were scattered all over the country.
June 12.—Warm and sultry. Duncawa remained with me all day,
and informed me, that he had the sultan’s orders to conduct me to
Kouka, in Bornou. This mark of respect I positively declined, both on
account of his recent illness, and also lest his presence might give
umbrage to the sheikh; but agreed to accept from him an escort
through the Bede territory. I assured him, when once in Bornou, that
I felt myself as safe as in his house. If he insisted, however, on
somebody accompanying me, he might, if he pleased, send one of
his principal people. I made a formal complaint of the insult
committed to Dr. Oudney’s grave,—enforcing, in the strongest terms,
the disgrace of disturbing the ashes of the dead, whose immortal
part was now beyond the power of malignant man. He frankly
acknowledged the enormity of the act, and faithfully promised to
have the wall rebuilt,—even offering to send for the governor of
Murmur, and have him punished; but, at the same time, begging me
not to acquaint the sultan of the occurrence. I expressed my reliance
on his assurances, but apprised him I must inform the gadado of the
affair. I afterwards spent the evening with Hameda.
June 13.—There was a fresh breeze in the morning; but it
afterwards began to rain. Duncawa being laid up from lameness, I
had a day’s rest, and again spent the evening with Hameda. The
conversation turning on the trustworthiness of slaves, he mentioned
to me, that his servants never knew in what apartment of his house
he slept; and that he even lay with a dagger, and loaded pistols,
under his pillow, lest he should be murdered by his female slaves.
He also acquainted me, that almost all the Arabs did the same; for it
was chiefly females whom they had reason to fear, the master being
often strangled at night by the women of his household.
June 14.—Duncawa visited me again, and made me a present of
two tobes, two sheep, and a large quantity of Guinea corn, and gave
a tobe to each of my servants. I presented him with six hundred
Goora nuts, having brought a large supply of them from Kano.
June 15.—I had every thing prepared for continuing my journey,
but Duncawa pressed me to spend another day with him, and I
availed myself of the delay to write to Bello and the gadado. I
returned my humble thanks to the former for his protection and
favour while I sojourned in his territories; and, in acknowledging the
uniform kindness of the latter, I did not fail to acquaint him of the
outrage committed on Dr. Oudney’s grave. I delivered these letters to
the charge of Dumbojee, who, having fulfilled his orders, took leave
of me here, having first made him a present of a couple of tobes and
forty dollars. My guide, Mahomed Dumbojee, had now become rich
and gay, having a numerous train of attendants; for at every town
where we halted, the governor was bound in courtesy to make him a
present, in token of respect for the sultan.
Having sent my camels forward, I went to bid farewell to
Duncawa, who was still confined to his house by illness. He made
me breakfast with him. Our breakfast consisted of a sheep’s head,
singed in the same manner as is practised in Scotland—a sheep’s
fry—and bread and milk. I was accompanied across the Yeou by my
friend Hameda, and Duncawa’s horsemen, who all wished to be
allowed to attend me to Sansan; but I excused myself from this
guard of honour, at once troublesome and expensive, by pretending
it was unlucky to go beyond the banks of a river with a friend.
Attended only by one of Duncawa’s principal men, I passed the thick
woods on the bank of the river, and, halting under a tamarind tree
during the heat of the day, I encamped towards evening at a village
called Mica. The inhabitants were all very busy in the fields sowing
gussub. They brought me, however, an abundant supply of milk, and
repeated inquiries were made after Bello’s health; for although they
recently belonged to Bornou, of which country they are natives, they
entertain, nevertheless, a great respect for their new sultan.
June 17.—I started at daylight, and, as the weather was cloudy
and rather windy, I did not halt before reaching Sansan. I was here
provided with very indifferent accommodation; but, on threatening I
would encamp outside the town, the governor received me into his
own house, according to Duncawa’s orders, and also made me a
present of a sheep. At night there was a violent storm, with thunder
and lightning. The poor lad Joseph, who had been hired at Kouka by
the late Dr. Oudney to tend the camels, was out all night with them.
Being a native of Fezzan, and half an idiot, he was here considered
a holy man, and I still retained him in my service out of charity. It was
he who gave me an account of the people of Bede, as he had been
a slave among them; and related his story with such artless
simplicity, that I implicitly rely on its correctness.
June 18.—Cool and cloudy. I heard to-day of a courier being
delayed on his route, by his camel’s being knocked up; and as
Duncawa was also preparing a present for the sheikh El Kanemy, I
postponed my departure yet another day.
June 19.—At eleven in the forenoon the courier arrived, bringing a
sabre as a present for the sultan Bello, and letters from Major
Denham, the consul at Tripoli, and the secretary of state. Accordingly
at mid-day I set off on my return to Katagum, in order to have the
sword forwarded to Bello by Duncawa.
At ten in the morning I entered Katagum, and immediately waited
on Duncawa to acquaint him with the cause of my return. I showed
him the sword, and explaining to him the manner of attaching the
belt, he expressed himself in terms of the highest admiration of both
sheath and sabre; and looking again and again at the ornaments, he
frequently asked, “Is not this all gold?” He sent instantly for the cadi,
who wrote a letter in my name to Bello, and a courier was
despatched with it and the sword. In the evening, another thunder-
storm, with much rain.
June 21.—At one in the afternoon I arrived again at Sansan.
June 22.—Clear and sultry. I was further detained on account of
the present for the sheikh not being ready.
June 23.—Morning cloudy. At seven in the morning I left Sansan,
attended by part of the escort which was to conduct me through the
Bede territory, and was obliged to stop about noon at the village of
Girkwa, by a violent attack of ague and bilious vomiting. Previous to
starting, I was joined by two merchants of Tripoli, who had been at
Kano, and begged to be allowed to place themselves under my
protection during this perilous part of the journey. In the afternoon
Hadje Fudor, the governor of Sansan, arrived with the remainder of
the escort, and also brought me a sheep, more in the expectation, I
think, of receiving some Goora nuts in return, than from any regard
for me. At midnight more rain, thunder, and lightning.
June 24.—Cool and cloudy. At ten in the morning halted at the
village of Boorum, to fill our water-skins, and afterwards travelled
through a thick wood, where we saw a number of karigums and
elephants: the karigum is a species of antelope, of the largest size,
as high as a full grown mule. At sunset we pitched our tents in the
woods. The night was extremely boisterous, with rain, thunder and
lightning, and violent squalls of wind; and my tent being blown down,
the baggage was drenched with water.
June 25.—Next morning we continued our route through a thick
wood, and halted at Joba during the heat of the day, when I had my
baggage dried in the sun. We still travelled through a thick wood,
and at seven in the evening encamped at a village called Gorbua.
Rain, thunder, and lightning, all night.
June 26.—Cloudy, with rain. At ten in the morning I left Gorbua, or
“the strong town,” as it is ironically called in the Bornouese language,
from being enclosed with matting. Our road, still winding and woody,
led through the Bede territory; and at sunset we reached Guba, a
small town on the south bank of the Yeou, within the dominions of
Bornou.
June 27.—The forenoon was rainy, which obliged us to remain at
Guba till one in the afternoon; when the weather clearing up, we
loaded the camels, and crossing to the north bank of the channel of
the river, which was now dry, we travelled east by south to the town
of Muznee, where we halted for the night.
June 28.—Cloudy, with rain. We travelled eastward along a
crooked path, full of holes, and overgrown with brushwood, and took
up our abode for the night at the town of Redwa. An officer of the
sultan of Bornou was here, collecting his master’s dues, and sent me
milk, onions, and six fowls; and I presented him, in return, with fifteen
Goora nuts.
June 29.—After travelling east by north, we halted at noon at
Kukabonee, or “wood and fish,” a large town on the south bank of
the Yeou. We next passed Magawin, and a number of other villages
and towns on the banks of the river, which we had not visited before,
when we accompanied the sheikh last year.
June 30.—Cool and cloudy. We halted at ten in the morning at
Dungamee, in consequence of heavy rain with thunder and lightning,
which continued without intermission all day.
July 1.—Clear. The weather was hot and sultry. At sunset we
arrived at Mugabee. I shot at a hippopotamus which was swimming
in a lake, of which there are many in this part of the country; I
seemed to hit it, but it quickly disappeared.
July 2.—Stopped for the day to allow the camels to have food and
rest.
July 3.—Between Gateramaran and Mugabee we met Malam
Fanamee, the governor of Munga, who had been to Kouka on a visit
to the sheikh. He was a dirty looking old man, preceded by a
drummer beating a drum, and attended by a parcel of ragged
followers, armed with bows and spears. We encamped at night in a
wood.
July 4.—At mid-day we halted on the banks of the Yeou: in the
afternoon there was thunder, lightning, and rain. A dealer in fish, who
had joined our party, solicited me in vain to pursue a route through a
town named Sucko, where he was going, promising me a sheep,
with plenty of milk, as an inducement. We passed another night in
the woods.
July 5.—Clear and cool. At ten in the morning we halted and filled
our water-skins, and I here shot a hare and two Guinea-fowls. About
an hour after starting we had heavy squalls of wind, with thunder and
rain: the storm was so violent that the camels lay down with their
burdens, and my horse would neither move forward, nor face the
storm in spite of all I could do. It was an hour before we were able to
resume our journey, and at eight in the evening we encamped in the
woods. The dangers of the road being past, my two fellow travellers,
the merchants before mentioned, left me at midnight on account of
the want of water.
July 6.—To-day I shot a fine male mohur, or beautiful red and
white antelope; a female only of which species I had once shot at
Woodie. At noon we took shelter under the walls of Borgee from
heavy squalls of wind and sand, but without rain. At sunset we
encamped near a well where there had been a great fall of rain, and
all the hollows were filled with water. To roast our mohur a large fire
was kindled in a hole made in the sand, on which it was placed, and
then covered over with hot embers; but, in the morning, to our great
disappointment, nothing remained of our prize but the naked
skeleton.
July 7.—At noon we halted at the wells of Barta, and encamped at
night at the wells and town of Calawawa.
July 8.—At eight in the morning I returned to Kouka: Major
Denham was absent on a journey round the east side of the Tchad.
Hillman, the naval carpenter, was busily employed in finishing a
covered cart, to be used as a carriage or conveyance for the
sheikh’s wives: the workmanship, considering his materials, reflected
the greatest credit on his ingenuity; the wheels were hooped with
iron, and it was extremely strong, though neither light nor handsome.
July 9.—In the afternoon I waited on the sheikh, who was very
kind in his inquiries after my health, and expressed much regret at
Dr. Oudney’s death.
July 10.—To-day the sheikh sent me three pairs of slippers, two
loaves of sugar, and a supply of coffee; and two days afterwards a
sheep, two bags of wheat, and a jar of honey.
APPENDIX.
TRANSLATIONS FROM THE ARABIC, OF VARIOUS LETTERS AND
DOCUMENTS, BROUGHT FROM BORNOU AND SOUDAN BY
MAJOR DENHAM AND CAPTAIN CLAPPERTON.
BY A. SALAME, ESQUIRE.
No. I.
“In the name of God, the merciful and the clement. May God bless
our favourite Prophet Mohammed, and those who follow his sound
doctrine.
“To the head of the Christian nation, the honoured and the
beloved among the English people, George the Fourth, King of Great
Britain;
“Praise be to God, who inspires, and peace be unto those who
follow, the right path:
“Your Majesty’s servant, Ra-yes-Abd-Allah, (Mr. Clapperton’s
travelling name,) came to us, and we found him a very intelligent and
wise man; representing in every respect your greatness, wisdom,
dignity, clemency, and penetration.
“When the time of his departure came, he requested us to form a
friendly relation, and correspond with you, and to prohibit the
exportation of slaves by our merchants to Ata-gher, Dahomi, and
Ashantee. We agreed with him upon this, on account of the good
which will result from it, both to you and to us; and that a vessel of
yours is to come to the harbour of Racka with two cannons, and the
quantities of powder, shot, &c. which they require; as also, a number
of muskets. We will then send our officer to arrange and settle every
thing with your consul, and fix a certain period for the arrival of your
merchant ships; and when they come, they may traffic and deal with
our merchants.
“Then after their return, the consul may reside in that harbour (viz.
Racka), as protector, in company with our agent there, if God be
pleased.”
“Dated 1st of Rhamadan, 1239 of Hejra.” 18th April, 1824.
No. III.
“Praise be to God, and prayers be unto him who was the last of
the Prophets (Mohammed).
“To the learned and accomplished, the virtuous Iman, the jealous
and zealous defender of the Mohammedan faith, our true friend the
Sheikh Mohammed El Kanemy, Lord of the country of Barnooh[66],
and its dependencies, whom may God protect and dignify, and
prolong his life long in happiness and felicity. Peace be unto you,
and the mercy and blessings of God be upon you, as long as the
inhabitants of the world shall exist.
“It follows, my Lord, subsequent to the due inquiry we make after
your health, which may God preserve, that your esteemed letter has
reached us, and we became acquainted with its contents. You
informed us that our beloved son, Aba Bak’r Ben Khalloom, arrived
in your presence, in company with some persons of the English
nation, our friends; and that you received them with extreme
kindness, and showed them all the marvellous things that your
country contains, and made them see all the extraordinary rivers and
lakes that surround it; and that you behaved to them as becoming
your high station, and indicating your esteem and regard towards us.
May God reward you for all this kindness, and protect you from all
evils. This kind treatment was our sanguine expectation, and indeed
we were already sure of it, from what we knew of the true friendship
and amity established between us.
“What we have now to acquaint you with, is to request that you
will continue your protection and assistance to the said English
travellers (though we doubt not you do not need this additional
recommendation), and cause them to proceed to the country of
Soudan, to behold its marvellous things, and traverse the seas
(lakes or rivers), and deserts therein. This being the proper desire of
the great King of the English himself, we beg of you to use your
utmost endeavours, as far as lies in your power, in their safe arrival
at the country of Soudan, accompanied either by letters of
recommendation, or by troops and guards, in order that they may
obtain the accomplishment of their wishes, and return to us safe and
unhurt; and whatever kindness you may do to them, it is done to us.
Resolve therefore, and exert yourself, as we are confident of your
goodness, and let them see all the places which they wish to visit.
“At the end there will be a splendid present, befitting your high
rank, sent to you through us, consisting of various rare and elegant
articles of value; for the delivery of which, unto your hands, we
pledge ourselves.
“This is all that we have to say at present, and if any affair should
occur to you in this country, let us know. And peace be unto you.
“Your friend,
(Signed) “YOUSUF PASHA .”
(Dated) “28th of Sha-wal, 1238 of Hejra;”
corresponding to August, 1823.
FOOTNOTES:
A Letter from the before named Pasha of Tripoli to Aba Bak’r ben
Khalloom, at Bornou.
“We received your letter, and comprehended all that you stated to
us. We were glad to hear that you, and our friends, the English
travellers, with whom we sent you as guide and conductor, had
arrived at Barnooh in safety; and that you were kindly received by
our friend, my Lord, the Sheikh Mohammed El Kanemy, who
immediately allowed the travellers to inspect all the deserts, and
seas, lakes and rivers, that are in his country. May God reward him
for this act of kindness. We have written to thank him for his laudable
behaviour; and we pray to God to enable us to show him equal
kindness in return.
“With regard to the persons of the different tribes, who were
obstinate and disobedient to you on the road, they have been
apprehended, and taken and punished one by one.
“As long as the English travellers remain at Barnooh, you have to
attend, and be with them wherever they go, till they shall have
obtained their wishes, and accomplished their object; and when they
desire to return, you may accompany and come with them as you
went. If this letter should reach you before you leave Barnooh, you
must stay with them, as above stated; if it reach you while you are on
the road homewards, you must return to Barnooh immediately, and
only send us the slave you have with you; and if you should arrive at
Fezzan before this letter reaches you, you may then send your
brother to Barnooh, to stay with them instead of you; for we only sent
you on their account, for the purpose of facilitating their proceeding,
and all their affairs. It is, therefore, impossible that you should leave
or part with them, but in this manner; and we are sure that, to a
person like you, there is no need to add any stronger words,
especially as you know that they are in our honour, and under our
protection, both in their going and returning in safety; which is the
accomplishment of our wishes. And may you live in happiness and
peace.
(Signed) “YOUSUF PASHA .”
(Dated) “2d of Ze-el-ka’da, 1238;”
corresponding to August, 1823.
No. V.
FOOTNOTES:
No. VI.