# Evaluating Global Blockage engineering parametrizations with LES

G Centurelli1 L Vollmer2 J Schmidt2 M Dörenkämper2 M Schröder1,2 L J Lukassen1
and J Peinke1 1 ForWind, Carl von Ossietzky University Oldenburg, Küpkersweg
70, 26129 Oldenburg, Germany  2 Fraunhofer IWES, Küpkersweg 70, 26129
Oldenburg, Germany<EMAIL_ADDRESS>
###### Abstract

Under the term global blockage, the cumulative induction of wind turbines in a
wind farm has been recently suspected to be responsible for observed
overestimations of the energy yield in large-size wind farms. In this paper,
the practice of modeling this effect after linear superposition of single
turbine inductions, calculated with three of the most recent analytic models,
is compared to Large-Eddy-Simulations of wind farms. We compare the models
across two different farms, composed of 9 and 49 turbines, with two different
heights of the atmospheric boundary layer, 300 and 500 m. The results show
that the differences between the analytical models are negligible while they
substantially differ from the LES results. The linear superposition of
induction consistently underestimates the velocity deficit in front of the
farm with an error that increases as the wind farm size grows and the ABL
height decreases. Also, when calculating the power output at the turbines of
the farm, all the analytical models considered do not agree with the LES.
These comparisons reveal that the farm interactions with the atmospheric
boundary layer may highly outclass the turbine induction in determining the
extent of the global blockage effect. Therefore, we present a first
dimensional approach to the problem based on LES, aimed at simplifying its
characterization.

##  1 Introduction

Despite being already conceived in the very early theory of Betz, it is only
in recent years that the induction of a wind turbine was finally given a
representation in engineering models that aim at modelling the flow inside of
wind farms. There is, in fact, a growing awareness that upstream interactions
between wind and wind turbines should no longer be neglected [1]. Supporting
this theory, there is consistent evidence of wind speed deceleration upstream
of wind farms suspected to cause efficiency losses, especially of the first
turbines row [2]. This effect, discussed under the name of wind farm blockage
or global blockage, is at our day and age still debated in science and
industrial applications, and different postulates on its origin exist.
An early study on wind farm power losses caused by inflow wind speed reduction
can be found in [3], in there the wind farm is suspected to generate
atmospheric gravity waves, similarly to mountains, capable of reducing the
inflow speed through pressure perturbation. The same idea has been more
recently extensively investigated by Allaerts and Meyers in [4, 5, 6], where
by means of Large-Eddy Simulations (LES) the authors were able to better
describe the extent of the phenomena and to connect it to the stratification
of the atmospheric boundary layer (ABL). A more general discussion on the
existence of global blockage and its effect on power production can be found
in [1]. In this work, the authors compared the wind speed measurements
collected by different met-masts before and after a nearby wind farm was
installed. The wind speed was observed mostly to decrease in the second period
even when the met-masts measured 2$\div$ 10 rotor diameters upstream the wind
farm. Other, more specific, _in situ_ evidence of the phenomena can be found
in the measurements with remote sensing technique of Asimakopoulos et al. [7]
and especially Schneemann et al. [8]. In this novel work, the wind speed at
transition piece height ($\approx$25 m above mean sea level) is found to
decrease up to 4.5 % between 40 and 10 rotor diameters upstream of the _Global
Tech 1_ wind farm when stable stratification characterizes the ABL.
As the pool of evidence for global blockage is growing larger, the discussion
on whether and how to consider its effect in wind farm energy yield assessment
is still quite open.
An important contribution in this direction is given by the work of Segalini
and Dahlberg [9]. They derived an empirical correlation describing velocity
deficit for the first row at varying farm layout, from a series of wind tunnel
measurements. A different approach can be found in [10, 11]. In both
publications, the authors propose to model the global blockage effect by
linear superposition of the induction effect of the single turbines, in a
similar fashion to what is currently done for wake modeling.

Nygaard et al. [11] also attempted a validation of the method. In there, the
power output of the first row at the UK-based _Gunfleet Sands_ wind farm is
simulated for two particular wind direction bins at a fixed velocity. By
comparing the results with averaged Supervisory control and data acquisition
(SCADA) data, the authors observe that “in its present form the model is
likely missing some element of the relevant physics”. This important
conclusion cannot, unfortunately, be considered definitive. The wind farm
chosen is surrounded either by the coast or by nearby farms, in this condition
the blockage effect is not the only factor determining wind velocity
distributions, and phenomena such as external wind farm wakes or coastal wind
speed gradients may even play a greater role. Within this work, we want to
contribute in assessing how much the linear superposition of single turbine
induction should be expected to suffice in the modeling of wind speed
deceleration and hence wind farm power reduction connected to the effect of
global blockage. As reference for the comparison we use LES capable of
describing the development of the atmospheric boundary layer under physical
sound conditions. With the numerical environment we remove parasite
perturbations that may compromise the fairness of the comparison in
measurement-based validation approaches.
After a brief description of the induction models, their implementation in the
Fraunhofer IWES in-house software _flappy_ (Farm Layout Program in Python),
and the LES set-up in section 2, we present results for the comparison
models/LES on wind velocity and power production in section 3. As a closing
argument, in section 4, we propose a simple dimensional study to characterize
the wind velocity deficit the LES predict in front of the farm, seeing the
necessity of improving the current models.

##  2 Models Description and set-up

Within the last years a good number of engineering models describing the
velocity deficit in the induction zone of a wind turbine have been proposed.
Branlard et al. [12] review most of them, offering a comparison of their
performance against an actuator disk (AD) Reynolds-Averaged Navier-Stokes
(RANS) simulation. The Vortex Cylinder (VC) model of [10] appears as the one
offering the best accuracy. In this manuscript, we consider the VC model and
two additional models with lower computational effort: the _Self-Similar_
model of Troldborg et al. [13] and the potential flow model of Gribben and
Hawkes [14], here as _Rankine Half-Body_.

###  2.1 Vortex Model (VC)

Inspired by the cylinder model for impellers of Joukowski, the Vortex Cylinder
model [10] computes the perturbation velocity components by linear
superposition of the effects from the three main vortexes generated in the
interaction between wind and turbine blades: the straight root vortex, the
straight bound lifting lines at the blades, and helical vortex trailed at the
blades tip. As an infinite number of blades is assumed, the lifting lines at
the rotor disk merge into a bound vortex sheet and the helical vortex can be
decomposed in line vortex filaments and a circular component responsible for a
tangential circulation, $\gamma_{t}$ [15].
The computation of velocity perturbation happens thanks to the Biot-Savart law
integration applied to any of the vortex systems defined. Multiple concentric
cylinders supporting $\gamma_{t}$ may be considered to describe the wake
expansion, but, as done in [10, 12], only the single cylinder matching the
rotor dimension is considered here.
Under the previous assumptions, only the tangential circulation is the capable
of providing the axial induction. Therefore it is possible to connect
$\gamma_{t}$ to the induction factor, $\bar{a}$, thanks to the relation
$\gamma_{t}=-2\,U_{ref}\,\bar{a}$, where $U_{ref}$ is the velocity of the free
stream approaching the turbine, as demonstrated in [15]. In the present work,
$\bar{a}$ for the model is computed according to the polynomial expansion on
the global thrust coefficient $C_{t}$,

$\bar{a}=0.169C_{t}+0.400C^{2}_{t}-0.482C^{3}_{t}+0.396C^{4}_{t}$ (1)

, found in [10] to provide a better agreement with an AD RANS throughout the
inflow region $x\in[-2.5\,$D,$-0.5\,$D], with D the rotor diameter, with
respect to the one-dimension momentum theory. The VC model has been introduced
in our wind farm modeling tool implementing the approach followed for FLORIS
[16] available at https://github.com/ebranlard/wiz.

###  2.2 Self-Similar model (SS)

The axial velocity deficit at hub height, $u_{x}(x,r=0)$, in the VC model
takes a very simplified formulation that reads as,

$u_{x}(x,r=0)\equiv
u_{x,0}=\bar{a}U_{ref}\left(\frac{x}{\sqrt{R^{2}+x^{2}}}\right)$ (2)

with $x$ and $r$ being the coordinates of a cylindrical frame or reference
centered at the rotor hub height ($z_{HH}$). Troldborg et al., [13], have
observed the induction profiles for different turbines simulated with RANS to
exhibit self-similar behaviour when $x>R$. They hence proposed to compute the
velocity deficit using (2) and describing the radial decay as

$\displaystyle\centering
f(\epsilon)=\frac{1}{cosh^{\alpha}(\beta\epsilon)},\quad\quad\epsilon=\frac{r}{r_{1/2}(x)},\quad\quad
r_{1/2}(x)=\sqrt{\lambda\left[\eta-\left(\frac{x}{R}^{2}\right)\right]}\@add@centering$

with $\alpha=8/9,\,\beta=\sqrt{2},\,\lambda=0.587,\,\eta=1.32$. Differently to
what chosen for the VC model, the induction factor $\bar{a}$ is based here on
the 1 D momentum theory, $\bar{a}=0.5\,(1-\sqrt{1-\nu C_{t}})$ with the
constant $\nu=1.1$ providing better agreement with RANS simulations.

###  2.3 Rankine Half-Body (RHB)

Gribben and Hawks in [14] suggest that the flow around a wind turbine
resembles the response of a uniform flow field to a constant momentum source.
In this typical problem of potential flow theory, the main flow is inviscidly
deflected around a Rankine Half-Body, the imaginary surface described by the
main stagnation streamline, described in a spherical frame of coordinates
($r,\theta$) centered at turbine hub height as

$cos(\theta)-\frac{2\pi U_{ref}}{\sigma}\,r^{2}\,sin^{2}(\theta)=-1$ (3)

with $\sigma=2\,\bar{a}\,A_{rotor}\,U_{ref}$ being the momentum source, and
$\bar{a}$ again derived from the one-dimensional momentum theory. Any of the
cartesian velocity perturbation components is given by

$\centering
u_{k}=\frac{\sigma}{4\pi}\frac{k}{\left(\sum_{k}{k^{2}}\right)^{3/2}}\quad\quad\quad
k=x,y,z.\@add@centering$ (4)

###  2.4 Empirical wind speed reduction by Segalini and Dahlberg (Seg)

As additional model for comparison, we consider the empirical wind speed
reduction derived by Segalini and Dahlberg [9] (_Seg_)

$u_{x_{0}}=U_{ref}\left\\{0.097\left(\frac{\Delta_{x}\Delta_{y}}{D^{2}}\right)^{-0.9}[1-\exp(0.88-0.88\,N_{rows})]\right\\}.$
(5)

Here, $u_{x_{0}}$ is the wind speed deficit at the front-most row middle
turbine, $\Delta_{x}$ and $\Delta_{y}$ the streamwise and spanwise distances
between the turbines and $N_{rows}$ the number of streamwise rows of the wind
farm. While not explicitly stated in [9], we use here $u_{x_{0}}$ as
initialization velocity at any turbine of the farm.

###  2.5 Case study setup and LES

The framework of our comparison is to perform simulations for the very same
wind farm layout either in AD LES and in the Farm Layout Program in Python111
_flappy_ v0.4.3 (_flappy_) developed at Fraunhofer IWES [17], the successor of
_flapFOAM_ (Farm Layout Program coupled to OpenFOAM) [18].
The wind turbines considered in the simulations have a constant $C_{t}$ of
0.86 and a constant power coefficient. Thus, they are characterized by a
rather high, but not unusual, induction factor. The turbine’s geometry is
typical for offshore application with a rotor diameter (D) of 160 m, resulting
in a rated power of approximately 8 MW, and a hub height of 110 m. Two basic
wind farm setups were simulated, one with three turbines in streamwise and
spanwise direction, respectively (3 $\times$ 3), and the other with seven
turbines in each direction (7 $\times$ 7). The distance between the turbines
in the streamwise and in spanwise directions is 7 D and 5 D, respectively.
Additional simulations with a single turbine and just the first turbine row
were conducted to serve as references.
The LES data was created with the Parallelized Large Eddy Simulations Model
(PALM) [19]. Further details about the used model setup can be found in [20].
The wind profile and turbulence of the simulations are developed during a
model spin-up period of 35 hours. The stratification of the simulated boundary
layer at the start of the spin-up simulation is neutral up to a capping
inversion. For simplification, we set the inversion layer to have a strong
potential temperature gradient of 8 K/100 m and a thickness of 100 m, above
this region a stable stratification of 1 K/100 m finds place up to the upper
boundary of the domain. Thus, any flow above the inversion layer is basically
decoupled from the boundary layer flow.
Two different wind fields are created for the wind farm simulations in the
LES, differentiating in the height of the inversion layer, with the inversion
in the “ABL 300”-simulation starting at 300 m height and the “ABL
500”-simulation at 500 m, thus resulting in four sets of LES reference wind
farm simulations: (3$\times$3_500, 3$\times$3_300, 7$\times$7_500 and
7$\times$7_300). The horizontal domain sizes of the simulations are adapted to
the size of the wind farm. The domains extend over approximately 13 $L_{x}$ in
streamwise and 13 $L_{y}$ in spanwise and over 3500 m in vertical direction,
with $L_{x}$ and $L_{y}$ the length and width of the wind farm, respectively.
The average wind speed profiles and hub height turbulence intensity at the
inflow boundary serve as input to the engineering model simulations with
_flappy_.
In _flappy_ the final flow field is derived in an iterative approach. Any
velocity perturbation computed by induction or wake models are linearly
superimposed. No induction model is allowed to provide a contribution in the
projected cylinder at the back of the rotor at which the induction is
computed. The presence of the ground is mimicked by also modelling a second
turbine mirrored at the ground surface. Wake effects are modelled with a
modified version of the Bastankhah wake model as described in [21]. However,
as the wake modelling only has a marginal influence on the modification of the
flow field by the induction models, we will not explore the representation of
the wakes further.

##  3 Results

Figure 1: Contours of upstream mean velocity deficit averaged in the span-wise
direction for the rotor frontal cross section. LES values are the dashed
lines. Single turbine with ABL 500 (top), middle turbine of 7$\times$7 farm
first row with ABL 500 (bottom)

In this section, the predictions of the engineering models are compared
against the LES results. First we focus mostly on the axial velocity deficit
that we define as

$U_{def}=(U_{x}(x,z)-U_{x,ref}(z))/U_{x,ref}(z=z_{HH})\times 100$ (6)

, the reference velocity is taken as the spanwise averaged value from the LES
at the inlet of the domain. Later, we look at the models effect when applied
to wind farm power prediction. In this section (3.2), the empirical
correlation of Segalini and Dahlberg [9] will be discussed as well.
It is important to mention that the induction models have no dependence on ABL
height. Thus, the comparison with the LES of different ABL height serves as an
illustration of an effect that is not reproducible with the implemented
models.

###  3.1 Velocity field prediction

Figure 1 presents the distribution of the velocity deficit in the vertical
plane upstream of a wind turbine. We observe a good agreement between models
and LES when the single turbine is simulated in isolation, Fig. 1(top),
confirming the results of Branlard et al. [12] in their comparison with AD-
RANS simulations. Qualitatively, the trend of the velocity deficit match up
quite well in the region $x>-2\,D$, especially for the VC model. For further
upstream distances the models tend to overestimate the deficit slightly.

However, repeating the same comparison for the middle turbine of the front row
of the 7$\times$7_500 case, Fig. 1(bottom), the previous difference are
exacerbated, with significant underestimations of the wind speed reduction for
all three models. While the wind farm in the LES induces wind speed reductions
of up to 2 % at distances of 5 D upstream, the reductions in the models do not
exceed 1 %. Interestingly, a good agreement remains for the near-rotor region
in the VC model simulation. Hence we deduce that the linear superposition
method fails to describe the cumulative effect of the wind turbines on the
flow.

Figure 2: Mean Axial velocity components at hub height upstream the wind
farms. The value is averaged on the span-wise direction on a window centered
on the farm and $5\,D$ larger of its cross section.
Figure 3: Mean axial velocity deficit averaged spanwise at cross planes 1
rotor diameter upstream the first, middle and last row in the farm. Horizontal
dashed lines represent the rotor edges.

Figure 2 provides a different visualisation of the upstream velocity reduction
with a display of the mean axial wind speed at hub height. By comparison to
the LES, the models clearly underestimate the velocity deficit throughout the
considered domain and compared to the LES results the model difference become
marginal. From the 3$\times$3 to the 7$\times$7 wind farm, the velocity
deficit at $x=-10\,D$ grows of 3 times larger in the LES simulations, while
the models predict only a 70 % increase. Similarly, varying the height of the
boundary layer strongly affects the inflow velocity deficit at both the farms
in the LES. The complete lack of sensitivity on this parameter proves to be
critical for the model.

Given the general good agreement observed for the single turbine case, the
results obtained until here point out that there are significant overhead
effects contributing to wind speed decelerations in front of the farm that
turbine induction cannot justify.

It is by looking at the vertical distribution of the velocity deficit, Figure
3, that it becomes even more evident that the inductions effects have too
local traits and they cannot fully motivate the global blockage effect. The
LES shows the velocity perturbation upstream of the first row to not only be
more intense with respect to the models, but also to interest the whole
vertical dimension of the ABL.

Figure 4: Contours of normalized wind speed at hub height for the
7$\times$7_500 farm. The induction model selected for comparison is the VC
model.

Figure 3 also displays properties of the flow within the wind farm. The
vertical profiles of the velocity deficit at the middle and last row in the
LES reveal that the initial differences between the two ABL simulated are
recovered within the wind farm, and above the rotors the wind speed may even
exceed the velocity of the free flow far upstream. This effect is completely
outside the analytical models’ capability, even the one describing the speed-
ups around the rotors.

To our understanding, the flow patterns observed up to now can only be
motivated by pressure perturbations in the lower boundary layer originating,
as explained by Allaerts and Meyers in [4], as a consequence of a displacement
of air masses with different density aloft. Clearly, such an effect cannot be
replicated with the engineering model approaches where, to derive the
perturbation of the velocity field, pressure is considered to be in
equilibrium.

As a conclusion to our analysis of the velocity field representation, Fig. 4
visualises the horizontal velocity contours at hub height for the
7$\times$7_500 case. We observe that the use of an induction model enhances
the similarity with the results of the LES, especially if the model accounts
for lateral speed-ups like the VC model used in the comparison. This
underlines once more that the models are capable of reproducing the local
turbine blockage, but their linear superposition cannot adequately represent
the global blockage effect, as the wind farm interactions with the atmospheric
boundary layer makes the phenomenon strongly non-linear.

Figure 5: Reduction of the average power production of the first row and the
whole wind farm caused by modeling the induction effect, for farm 3$\times$3
a) and 7$\times$7 b). The references are the same simulations without
modelling any induction effects. For the LES, the reference of the first row
power is a simulation of the isolated first row. For the whole farm power no
reference is possible for the LES case. c) The reduction of the average power
production of the first row and the whole wind farm for the two different wind
farms in the simulation setups with different boundary layer height. The
reference case is the setup ABL 500.

###  3.2 Wind Farm power

In the previous section, the comparison with the LES has shown the linear
superposition of induction models to be far from sufficient to describe the
velocity deficit caused by global blockage. The enhancement of the flow
visualization may not be a sufficient reason justifying the method deployment
in context such as Annual Energy Production (AEP) assessment or wind farm
optimization algorithms. The many degrees of freedom involved in the solution
of these problems lead to iterative procedures that require fast calculation
of the farm output at a given state. Any computational cost added to this task
must be compensated by clear improvements in the accuracy of the prediction.
In the following section, we debate the benefits of considering an induction
model in the assessment of wind farm production.

In Fig. 5(a,b) we show the percentage of power reduction at the first row and
to the whole farm, by the application of the induction models and the global
wind speed reduction at any of the farm’s turbine, according to Eq. 5. Because
induction in the LES cannot be switched off, we define its reference as the
results for power output of a simulation where a single row of the farm is
considered. This practice is identical to the reference taken for the
analytical models, as removing induction in this models is equivalent to
removing the effects of the downwind turbines. Such a normalization for the
LES is also beneficial to remove row effects that lead to higher power
production as shown in [22]. Unfortunately, it is not possible to define a
similar reference for the whole farm in the LES.

The trend of velocity reduction observed in Fig. 2, can also be observed in
the first row power predicted by the LES, that decreases evidently as the farm
size grows and the ABL height decreases. From the 3$\times$3 to the 7$\times$7
the power reduction grows of 14 times in the LES at ABL 500, while in the
models it merely double. A further evidence of a strongly non linear
phenomena. The models also consistently underestimate the power reduction at
first row for all the studied cases but the 3$\times$3_500, in which the
effect of flow blockage are minimal. This result shows that the inlet velocity
deficit may be partially recovered, another sign of the importance of the
pressure effects.

The power production of the whole farm is less affected by the global blockage
effect than the first row in all the simulations with the induction models.
This result is also confirmed by the LES. The relative power reduction across
the two ABL simulated is much greater for the first row than for the whole
farm, Fig. 5c, indicating as in Fig. 2 that the inflow deficit is recovered at
the downwind turbines. We therefore urge caution with use of approaches based
on reducing the velocity at all the turbines in the farm according to a
correction measured upstream, e.g. Eq. 5. Despite we could not prove it here,
we expect the methods to be biased towards underestimating the whole farm
power.

To conclude the results’ section, we finally discuss the impact induction
models have on the whole farm power production. In our simulations, the use of
neither the VC model nor the RHB model significantly affect this prediction.
To our understanding, this is connected to the speed up around the rotor
described in these models, that despite being subtle it is sufficient to
balance the negative effect of induction on power. To this extent, only the
_Self-Similar_ model provides a prediction of the order of magnitude typically
assumed for the global blockage effect. More research in this direction is
still necessary but for the other two models, the significant computational
cost we experienced in our simulations, especially with the VC, and the
marginal influence on power production are results that do not encourage their
use in AEP calculation.

##  4 A simple dimensional analysis of global blockage effect

According to our analysis, we find that wind farm size, as well as atmospheric
boundary layer height, have a significant impact on the extent of velocity
reduction in front of the wind farm, see Fig. 2. This motivates us to perform
a simple dimensional analysis to find out whether a universal behaviour of the
velocity deficit can be identified.
Our LES simulations suggest that the wind speed reduction upstream the farm,
here defined as $u_{x,loss}=U_{x,ref}-U_{x,LES}$, can be represented as a firt
approach by the functional relation

$u_{x,loss}=f(x,U_{x,ref},L_{farm},H_{abl}).$ (7)

In Eq. 7, $x$ denotes the axial coordinate with $x=0$ at the farm first row,
$U_{x,ref}$ the undisturbed velocity, $L_{farm}$ the length of the farm, and
$H_{abl}$ the height of the ABL. We decide to neglect the width of the farm,
as having fixed the aspect ratio in the two farms simulated, $L_{farm}$ is
sufficient to describe their geometry change.

Casting the problem in the context of the Buckingham-Vaschy $\pi$\- theorem
[23, 24], there are five total variables, and two fundamental dimensions,
length and time, therefore three $\pi$ parameters must be defined.

$\displaystyle\centering\pi_{u}=\frac{u_{x,loss}}{U_{x,ref}}\quad\quad\pi_{x}=\frac{x}{L_{farm}}\quad\quad\pi_{h}=\frac{H_{abl}-D}{L_{farm}}\@add@centering$

Figure 6: Profiles of normalized loss velocity $\pi_{u}=u_{x,loss}/U_{ref}$ at
hub height in spanwise average, along the full LES domain $x$, here normalized
as $\pi_{x}=x/L_{farm}$. No rescaling for $\pi_{u}$ a), rescaling with
$(\frac{H_{abl}-D}{L_{farm}})^{-0.5}b)$

The choice of the first two groups appears as the most convenient, the one for
the last group deserves a further comment. From our understanding, the wind
can adapt to the presence of the farm by deflecting upward, downward, and to
the sides of the area occupied by the rotors. As any deflection to the sides
comes virtually with no consequence, the deflection up and down can cause
extra shear as the flow is bounded by the ground and the capping inversion.
The parameter $\pi_{h}$ is therefore representing the height of the channels
upwards and downwards the wind farm rotors, whose cumulative height is
$H_{abl}-D$. In terms of the $\pi$\- theorem, we have incorporated the
variable $D$ (the rotor diameter) only here to avoid an additional $\pi$\-
term. This choice, taken for sake of simplicity, is also justified by the fact
that $D$ is constant across the LES, therefore we could have not investigated
the dependence on this parameter.

According to the new non-dimensional $\pi$\- terms, Eq. (7) can be rewritten
as,

$\pi_{u}=F\left(\pi_{x},\,\pi_{h}\right)$ (8)

Mathematically, we now propose that $F$ factorises to
$F\left(\pi_{x},\,\pi_{h}\right)=G\left(\pi_{x}\right)g(\pi_{h})$. As the LES
results provide us the trend of $\pi_{u}(\pi_{x})$, we search for the
functional relation $g(\pi_{h})$, so that

$\frac{\pi_{u}}{g(\pi_{h})}=G\left(\pi_{x}\right)$ (9)

with $G(\pi_{x})$ being the universal curve describing the velocity deficit.
As demonstrated in the right figure of (6), choosing

$g(\pi_{h})=\pi_{h}^{-0.5}$ (10)

gains the four curves predicted by the LES, Fig. 6 (left), to collapse to one
in the inflow region.

It must be underlined that this is only a first and early attempt at
characterising the interactions of wind farm and ABL we believe to originate
the GBE. It is interesting that this simple application of the $\pi$\- theorem
provided a rather promising result. Lacking a more substantial set of
observations to test the model on, we avoid trying to interpret to much out of
it. But we believe the whole modelling of global blockage would benefit from
more work in this direction.

##  5 Conclusion

By means of comparison with LES, the analytic induction models are shown to
correctly describe the induction generated by a single wind turbine in
isolation and to improve the flow representation within the wind farm,
especially if the model accounts for lateral speed-ups. Under the conditions
simulated, these results do not carry over when these models are linearly
superimposed to describe the wind farm inflow deceleration due to global
blockage. We observe a substantial underestimation in the model predictions
that increases as the wind farm dimension grows and the ABL height decreases.
These results show the poor sensitivity, or complete lack thereof, to these
parameters to generate large inaccuracies in the models predictions.

We deduce that the deceleration of the inflow is mainly determined by the
interactions and the physics of the ABL rather than by the turbines’
induction. Compelling evidence at support of this thesis are also found in the
production data. We observe the inflow deficit to be partially recovered at
first row and especially at the row downstream in the farm. Suggesting the
global blockage effect to be not correlated to the turbines induction but
rather to exogenous pressure perturbations, likely due to the displacement of
air masses with different density aloft [4], capable of redistributing the
available energy. We conclude that methods to predict the GBE at wind farm
based on reducing the whole farm speed velocity according to the measured
inflow deficit, e.g. as in Eq. (5), are not advisable. We also believe the
linear superposition of induction models to be inadequate to the modeling of
GBE in its present form where no dynamics of the ABL are considered.

Eventually, we perform a dimensional analysis of the inlet velocity deficit
predicted by the LES aimed at simplifying the description of the interactions
between wind farm and atmospheric boundary layer. Despite being very early,
the results are promising and they could give birth to a new approach for the
modeling of the GBE.

Future work is necessary to more broadly characterize the extent of the GBE at
varying conditions of the ABL, and the wind farm size and shape. This will
gain a better testing ground for the presented dimensional model, and nourish
further development of the approach. In parallel, we foresee the modeling of
the lower atmosphere’s vertical structure, such the one presented in [6], as a
possible step forward of the current status.

RANS

    Reynolds-Averaged Navier-Stokes
LES

    Large-Eddy Simulation
AD

    actuator disk
WT

    wind turbine
VC

    Vortex Cylinder
RHB

    Rankine Half-Body
SS

    Self-Similar
ABL

    atmospheric boundary layer
GBE

    global blockage effect
AEP

    Annual Energy Production
EM

    engineering models
SCADA

    Supervisory control and data acquisition

The work presented in this paper was developed in the framework of the X-Wakes
project (FKZ 03EE3008D,A) which is funded by the German Federal Ministry for
Economic Affairs and Energy (Bundesministerium für Wirtschaft und Energie -
BMWi) due to a decision of the German Bundestag. The simulations were partly
performed at the HPC Cluster EDDY, located at the University of Oldenburg
(Germany) and funded by BMWi (FKZ 0324005). Additional computer resources have
been provided by the North German Supercomputing Alliance (HLRN).

## References

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  *[ABL]: atmospheric boundary layer
  *[SCADA]: Supervisory control and data acquisition
  *[AD]: actuator disk
  *[RANS]: Reynolds-Averaged Navier-Stokes
  *[VC]: Vortex Cylinder
  *[AEP]: Annual Energy Production


# Integrated and Differentiated Sequence Spaces

Murat Kirişci  Department of Mathematical Education, Hasan Ali Yücel Education
Faculty, Istanbul University, Vefa, 34470, Fatih, Istanbul, Turkey
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we investigate integrated and differentiated sequence spaces
which emerge from the concept of the space $bv$ of sequences of bounded
variation.The integrated and differentiated sequence spaces which was
initiated by Goes and Goes [4]. The main propose of the present paper, we
study of matrix domains and some properties of the integrated and
differentiated sequence spaces. In section 3, we compute the alpha-, beta- and
gamma duals of these spaces. Afterward, we characterize the matrix classes of
these spaces with well-known sequence spaces.

###### Key words and phrases:

Matrix transformations, sequence spaces, $BK$-space, dual spaces, Schauder
basis, $AK$-property

###### 2010 Mathematics Subject Classification:

Primary 46A45; Secondary 46A35, 46B15.

This work was supported by Scientific Projects Coordination Unit of Istanbul
University. Project number 34465.

##  1\. Introduction

The set of all sequences denotes with
$\omega:=\mathbb{C}^{\mathbb{N}}:=\\{x=(x_{k}):x:\mathbb{N}\rightarrow\mathbb{C},k\rightarrow
x_{k}:=x(k)\\}$ where $\mathbb{C}$ denotes the complex field and
$\mathbb{N}=\\{0,1,2,\ldots\\}$. Each linear subspace of $\omega$ (with the
induced addition and scalar multiplication) is called a _sequence space_. The
following subsets of $\omega$ are obviously sequence
spaces:$\ell_{\infty}=\\{x=(x_{k})\in\omega:\sup_{k}|x_{k}|<\infty\\}$,
$c=\\{x=(x_{k})\in\omega:\lim_{k}x_{k}~{}\textrm{ exists}~{}\\}$,
$c_{0}=\\{x=(x_{k})\in\omega:\lim_{k}x_{k}=0\\}$,
$bs=\\{x=(x_{k})\in\omega:\sup_{n}|\sum_{k=1}^{n}x_{k}|<\infty\\}$,
$cs=\\{x=(x_{k})\in\omega:(\sum_{k=1}^{n}x_{k})\in c\\}$ and
$\ell_{p}=\\{x=(x_{k})\in\omega:\sum_{k}|x_{k}|^{p}<\infty,\quad 1\leq
p<\infty\\}$. These sequence spaces are Banach space with the following norms;
$\|x\|_{\ell_{\infty}}=\sup_{k}|x_{k}|$,
$\|x\|_{bs}=\|x\|_{cs}=\sup_{n}|\sum_{k=1}^{n}x_{k}|$ and
$\|x\|_{\ell_{p}}=\left(\sum_{k}|x_{k}|^{p}\right)^{1/p}$ as usual,
respectively. And also the concept of integrated and differentiated sequence
spaces was employed as $\int X=\left\\{x=(x_{k})\in\omega:(kx_{k})\in
X\right\\}$ and $d(X)=\left\\{x=(x_{k})\in\omega:(k^{-1}x_{k})\in X\right\\}$
in [4].

A sequence, whose $k-th$ term is $x_{k}$, is denoted by $x$ or $(x_{k})$. _A
coordinate space_ (or _$K-$ space_) is a vector space of numerical sequences,
where addition and scalar multiplication are defined pointwise. That is, a
sequence space $X$ with a linear topology is called a $K$-space provided each
of the maps $p_{i}:X\rightarrow\mathbb{C}$ defined by $p_{i}(x)=x_{i}$ is
continuous for all $i\in\mathbb{N}$. A $BK-$space is a $K-$space, which is
also a Banach space with continuous coordinate functionals $f_{k}(x)=x_{k}$,
$(k=1,2,...)$. A $K-$space $K$ is called an _$FK-$ space_ provided $\lambda$
is a complete linear metric space. An _$FK-$ space_ whose topology is normable
is called a _$BK-$ space_. If a normed sequence space $X$ contains a sequence
$(b_{n})$ with the property that for every $x\in X$ there is unique sequence
of scalars $(\alpha_{n})$ such that

$\displaystyle\lim_{n\rightarrow\infty}\|x-(\alpha_{0}b_{0}+\alpha_{1}b_{1}+...+\alpha_{n}b_{n})\|=0$

then $(b_{n})$ is called _Schauder basis_ (or briefly basis) for $X$. The
series $\sum\alpha_{k}b_{k}$ which has the sum $x$ is then called the
expansion of $x$ with respect to $(b_{n})$, and written as
$x=\sum\alpha_{k}b_{k}$. An _$FK-$ space_ $X$ is said to have $AK$ property,
if $\phi\subset X$ and $\\{e^{k}\\}$ is a basis for $X$, where $e^{k}$ is a
sequence whose only non-zero term is a $1$ in $k^{th}$ place for each
$k\in\mathbb{N}$ and $\phi=span\\{e^{k}\\}$, the set of all finitely non-zero
sequences.

Let $X$ and $Y$ be two sequence spaces, and $A=(a_{nk})$ be an infinite matrix
of complex numbers $a_{nk}$, where $k,n\in\mathbb{N}$. Then, we say that $A$
defines a matrix mapping from $X$ into $Y$, and we denote it by writing
$A:X\rightarrow Y$ if for every sequence $x=(x_{k})\in X$. The sequence
$Ax=\\{(Ax)_{n}\\}$, the $A$-transform of $x$, is in $Y$; where

(1.1) $\displaystyle(Ax)_{n}=\sum_{k}a_{nk}x_{k}~{}\textrm{ for each
}~{}n\in\mathbb{N}.$

For simplicity in notation, here and in what follows, the summation without
limits runs from $1$ to $\infty$. By $(X:Y)$, we denote the class of all
matrices $A$ such that $A:X\rightarrow Y$. Thus, $A\in(X:Y)$ if and only if
the series on the right side of (1.1) converges for each $n\in\mathbb{N}$ and
each $x\in X$ and we have $Ax=\\{(Ax)_{n}\\}_{n\in\mathbb{N}}\in Y$ for all
$x\in X$. A sequence $x$ is said to be $A$-summable to $l$ if $Ax$ converges
to $l$ which is called the $A$-limit of $x$.

Let $X$ is a sequence space and $A$ is an infinite matrix. The sequence space

(1.2) $\displaystyle X_{A}=\\{x=(x_{k})\in\omega:Ax\in X\\}$

is called the matrix domain of $X$ which is a sequence space(for several
examples of matrix domains, see [2] p. 49-176). By $\mathcal{F}$, we will
denote the collection of all finite subsets on $\mathbb{N}$. In [3], Başar and
Altay have defined the sequence space $bv_{p}$ which consists of all sequences
such that $\Delta$-transforms of them are in $\ell_{p}$ where $\Delta$ denotes
the matrix $\Delta=(\delta_{nk})$

$\displaystyle\delta_{nk}=\left\\{\begin{array}[]{ccl}(-1)^{n-k}&,&\quad(n-1\leq
k\leq n)\\\ 0&,&\quad(0\leq k<n-1~{}\textrm{ or }~{}k>n)\end{array}\right.$

for all $k,n\in\mathbb{N}$. And also we define the matrices
$\Gamma=(\gamma_{nk})$ and $\Sigma=(\sigma_{nk})$ by

(1.7) $\displaystyle\gamma_{nk}=\left\\{\begin{array}[]{ccl}k&,&\quad(n=k)\\\
-k&,&\quad(n-1=k)\\\ 0&,&\quad(other)\end{array}\right.$

(1.11)
$\displaystyle\sigma_{nk}=\left\\{\begin{array}[]{ccl}\frac{1}{k}&,&\quad(n=k)\\\
-\frac{1}{k}&,&\quad(n-1=k)\\\ 0&,&\quad(other)\end{array}\right.$

The integrated and differentiated sequence spaces which was initiated by Goes
and Goes [4]. In the present paper, we study of matrix domains and some
properties of the integrated and differentiated sequence spaces. In section 3,
we compute the alpha-, beta- and gamma duals of these spaces. Afterward, we
characterize matrix classes of these spaces with well-known sequence spaces.

##  2\. The Sequence Spaces $\int bv$ and $d(bv)$

The integrated spaces defined by

$\displaystyle\int\ell_{1}$ $\displaystyle=$
$\displaystyle\left\\{x=(x_{k})\in\omega:\sum_{k}|k.x_{k}|<\infty\right\\}$
$\displaystyle\int bv$ $\displaystyle=$
$\displaystyle\left\\{x=(x_{k})\in\omega:\sum_{k}|k.x_{k}-(k-1).x_{k-1}|<\infty\right\\}$

and the differentiated spaces defined by

$\displaystyle d(\ell_{1})$ $\displaystyle=$
$\displaystyle\\{x=(x_{k})\in\omega:\sum_{k}|k^{-1}.x_{k}|<\infty\\}$
$\displaystyle d(bv)$ $\displaystyle=$
$\displaystyle\\{x=(x_{k})\in\omega:\sum_{k}|k^{-1}.x_{k}-(k-1)^{-1}.x_{k-1}|<\infty\\}.$

Consider the notation (1.2) and the matrices (1.7), (1.11). From here, we can
re-define the spaces $\int bv$ and $d(bv)$ by

(2.1) $\displaystyle\left(\int\ell_{1}\right)_{\Delta}=\int bv~{}\textrm{ or
}~{}\left(\ell_{1}\right)_{\Gamma}=\int bv$

and

(2.2) $\displaystyle\left[d(\ell_{1})\right]_{\Delta}=d(bv)~{}\textrm{ or
}~{}\left(\ell_{1}\right)_{\Sigma}=d(bv).$

Let $x=(x_{k})\in\int bv$ and $\Delta x_{k}=x_{k}-x_{k-1}$. The
$\Gamma-$transform of a sequence $x=(x_{k})$ is defined by

(2.5) $\displaystyle y_{k}=(\Gamma
x)_{k}=\left\\{\begin{array}[]{ccl}x_{1}&,&\quad k=1\\\ \Delta(kx_{k})&,&\quad
k\geq 2\end{array}\right.$

where $\Gamma$ is defined by (1.7). Let $x=(x_{k})\in d(bv)$ and $\Delta
x_{k}=x_{k}-x_{k-1}$. The $\Sigma-$transform of a sequence $x=(x_{k})$ is
defined by

(2.8) $\displaystyle y_{k}=(\Sigma
x)_{k}=\left\\{\begin{array}[]{ccl}x_{2}/2&,&\quad k=2\\\
\Delta(k^{-1}x_{k})&,&\quad k\geq 3\end{array}\right.$

where $\Sigma$ is defined by (1.11).

######  Theorem 2.1.

The spaces $\int\ell_{1}$ and $d(\ell_{1})$ are $BK-$spaces with the norms
$\|x\|_{\int\ell_{1}}=\sum_{k}|kx_{k}|$ and
$\|x\|_{d(\ell_{1})}=\sum_{k}|k^{-1}x_{k}|$, respectively.

###### Proof.

Let $x=(x_{k})\in\int\ell_{1}$. We define $f_{k}(x)=x_{k}$ for all
$k\in\mathbb{N}$. Then, we have

$\displaystyle\|x\|_{\int\ell_{1}}=1.|x_{1}|+2.|x_{2}|+3.|x_{3}|+\cdots+k.|x_{k}|+\cdots$

Hence $k.|x_{k}|\leq\|x\|_{\int\ell_{1}}\Rightarrow|x_{k}|\leq
K.\|x\|_{\int\ell_{1}}\Rightarrow|f_{k}(x)|\leq K.\|x\|_{\int\ell_{1}}$. Then,
$f_{k}$ is continuous linear functional for each $k$. Thus $\int\ell_{1}$ is a
$BK-$space.

In a similar way, we can prove that the space $d(\ell_{1})$ is a $BK-$spaces.
∎

######  Lemma 2.2.

[4] The space $\int bv$ is a $BK-$space with the norm $\|x\|_{\int
bv}=\sum_{k}|\Delta(kx_{k})|$.

######  Theorem 2.3.

The space $d(bv)$ is a $BK-$space with the norm
$\|x\|_{d(bv)}=\sum_{k}|\Delta(k^{-1}x_{k})|$.

###### Proof.

Since $d(bv)=\left[d(\ell_{1})\right]_{\Delta}$ holds, $d(\ell_{1})$ is a
$BK-$space with the norm $\|x\|_{d(\ell_{1})}$ and the matrix $\Delta$ is a
triangle matrix, then Theorem 4.3.2 of Wilansky[6] gives the fact that the
space $d(bv)$ is a $BK-$space. ∎

######  Theorem 2.4.

  * (i).

The spaces $\int\ell_{1}$ and $d(\ell_{1})$ have $AK-$property.

  * (ii).

The spaces $\int bv$ and $d(bv)$ have $AK-$property.

###### Proof.

The fact that of the space $\int bv$ has $AK-$property was given by Goes and
Goes[4]. Then, we will only prove that the space $d(bv)$ has $AK-$property in
(ii).

Let $x=(x_{k})\in d(bv)$ and
$x^{[n]}=\\{x_{1},x_{2},\cdots,x_{n},0,0,\cdots\\}$. Hence,

$\displaystyle
x-x^{[n]}=\\{0,0,\cdots,0,x_{n+1},x_{n+2},\cdots\\}\Rightarrow\|x-x^{[n]}\|_{d(bv)}=\|0,0,\cdots,0,x_{n+1},x_{n+2},\cdots\|$

and since $x\in d(bv)$,

$\displaystyle\|x-x^{[n]}\|_{d(bv)}=\sum_{k\geq
n+1}|\Delta(k^{-1}x_{k})|\rightarrow 0~{}\textrm{ as}~{}\ n\rightarrow\infty$
$\displaystyle\Rightarrow\lim_{n\rightarrow\infty}\|x-x^{[n]}\|_{d(bv)}=0\Rightarrow
x^{[n]}\rightarrow\infty~{}\textrm{ as}~{}\ n\rightarrow\infty~{}\textrm{
in}~{}\ d(bv).$

Then the space $d(bv)$ has $AK-$property. ∎

######  Theorem 2.5.

The spaces $\int bv$ and $d(bv)$ are norm isomorphic to $\ell_{1}$.

###### Proof.

We must show that a linear bijection between the spaces $\int bv$ and
$\ell_{1}$ exists. Consider the transformation $T$ defined, with the notation
(2.5), from $\int bv$ to $\ell_{1}$ by $x\mapsto y=Tx$. The linearity of $T$
is clear. Also, it is trivial that $x=\theta$ whenever $Tx=\theta$ and
therefore, $T$ is injective.

Let $y\in\ell_{1}$ and define the sequence $x=(x_{k})$ by
$x_{k}=\frac{1}{k}.\sum_{j=1}^{k}y_{j}$. Then

$\displaystyle\|x\|_{\int bv}$ $\displaystyle=$
$\displaystyle\sum_{k}|\Delta(kx_{k})|=\sum_{k}\left|k.\frac{1}{k}\sum_{j=1}^{k}y_{j}-(k-1).\frac{1}{k-1}\sum_{j=1}^{k-1}y_{j}\right|=\sum_{k}|y_{k}|=\|y\|_{\ell_{1}}<\infty.$

Then, we have that $x\in\int bv$. So, $T$ is surjective and norm preserving.
Hence $T$ is a linear bijection. It shown us that the space $\int bv$ is norm
isomorphic to $\ell_{1}$.

As similar, using the notation (2.8), we can define the transformation $S$
from $d(bv)$ and $\ell_{1}$ by $x\mapsto y=Sx$. And also, if we choose the
sequence $x=(x_{k})$ by $x_{k}=k.\sum_{j=2}^{k}y_{j}$ while $y\in\ell_{1}$,
then we obtain the space $d(bv)$ is norm isomorphic to $\ell_{1}$ with the
norm $\|x\|_{d(bv)}$. ∎

######  Theorem 2.6.

The spaces $\int bv$ and $d(bv)$ have monotone norm.

###### Proof.

Let $x=(x_{k})\in\int bv$. We define the norms $\|x\|_{\int
bv}=\sum_{k}|\Delta(kx_{k})|$ and $\|x^{[n]}\|_{\int
bv}=\sum_{k=1}^{n}|\Delta(kx_{k})|$, for all $x\in\int bv$. For $n<m$,

$\displaystyle\|x^{[n]}\|=\sum_{k=1}^{n}|\Delta(kx_{k})|\leq\sum_{k=1}^{m}|\Delta(kx_{k})=\|x^{[m]}\|,$

that is,

(2.9) $\displaystyle\|x^{[m]}\|\geq\|x^{[n]}\|.$

The sequence $\|x^{[n]}\|$ is monotonically increasing sequence and bounded
above.

(2.10)
$\displaystyle\sup\|x^{[n]}\|=\sup\left(\sum_{k=1}^{n}|\Delta(kx_{k})|\right)=\left(\sum_{k=1}^{n}|\Delta(kx_{k})|\right)=\|x\|.$

From (2.9) and (2.10), it follows that the space $\int bv$ has the monotone
norm.

In similar way, we can obtain to the space $d(bv)$ has the monotone norm. ∎

Because of the isomorphisms $T$ and $S$, defined in the proof of Theorem 2.5,
are onto the inverse image of the basis $\\{e^{(k)}\\}_{k\in\mathbb{N}}$ of
the space $\ell_{1}$ is the basis of the spaces $\int bv$ and $d(bv)$.
Therefore, we have the following:

######  Theorem 2.7.

  * (i).

Define a sequence $t^{(k)}=\\{t_{n}^{(k)}\\}_{n\in\mathbb{N}}$ of elements of
the space $\int bv$ for every fixed $k\in\mathbb{N}$ by

$\displaystyle t_{n}^{(k)}=\left\\{\begin{array}[]{ccl}1/k&,&\quad(n\geq k)\\\
0&,&\quad(n<k)\end{array}\right.$

Therefore, the sequence $\\{t^{(k)}\\}_{k\in\mathbb{N}}$ is a basis for the
space $\int bv$ and if we choose $E_{k}=(Ax)_{k}$ for all $k\in\mathbb{N}$,
where the matrix $A$ defined by (1.7), then any $x\in\int bv$ has a unique
representation of the form

$\displaystyle x=\sum_{k}E_{k}b^{(k)}.$

  * (ii).

Define a sequence $s^{(k)}=\\{s_{n}^{(k)}\\}_{n\in\mathbb{N}}$ of elements of
the space $d(bv)$ for every fixed $k\in\mathbb{N}$ by

$\displaystyle s_{n}^{(k)}=\left\\{\begin{array}[]{ccl}k&,&\quad(n\geq k)\\\
0&,&\quad(n<k)\end{array}\right.$

Therefore, the sequence $\\{s^{(k)}\\}_{k\in\mathbb{N}}$ is a basis for the
space $d(bv)$ and if we choose $F_{k}=(Bx)_{k}$ for all $k\in\mathbb{N}$,
where the matrix $B$ defined by (1.11), then any $x\in d(bv)$ has a unique
representation of the form

$\displaystyle x=\sum_{k}F_{k}b^{(k)}.$

The result follows from fact that if a space has a Schauder basis, then it is
separable. Hence, we can give following corollary:

######  Corollary 2.8.

The spaces $\int bv$ and $d(bv)$ are separable.

##  3\. The $\alpha-$, $\beta-$ and $\gamma-$ Duals of the spaces $\int bv$
and $d(bv)$

In this section, we state and prove the theorems determining the $\alpha$-,
$\beta$\- and $\gamma$-duals of the sequence spaces $\int bv$ and $d(bv)$.

Let $x$ and $y$ be sequences, $X$ and $Y$ be subsets of $\omega$ and
$A=(a_{nk})_{n,k=0}^{\infty}$ be an infinite matrix of complex numbers. We
write $xy=(x_{k}y_{k})_{k=0}^{\infty}$, $x^{-1}*Y=\\{a\in\omega:ax\in Y\\}$
and $M(X,Y)=\bigcap_{x\in X}x^{-1}*Y=\\{a\in\omega:ax\in Y~{}\textrm{ for all
}~{}x\in X\\}$ for the _multiplier space_ of $X$ and $Y$. In the special cases
of $Y=\\{\ell_{1},cs,bs\\}$, we write $x^{\alpha}=x^{-1}*\ell_{1}$,
$x^{\beta}=x^{-1}*cs$, $x^{\gamma}=x^{-1}*bs$ and $X^{\alpha}=M(X,\ell_{1})$,
$X^{\beta}=M(X,cs)$, $X^{\gamma}=M(X,bs)$ for the $\alpha-$dual, $\beta-$dual,
$\gamma-$dual of $X$. By $A_{n}=(a_{nk})_{k=0}^{\infty}$ we denote the
sequence in the $n-$th row of $A$, and we write
$A_{n}(x)=\sum_{k=0}^{\infty}a_{nk}x_{k}$ $n=(0,1,...)$ and
$A(x)=(A_{n}(x))_{n=0}^{\infty}$, provided $A_{n}\in x^{\beta}$ for all $n$.

######  Lemma 3.1.

[1, Theorem 2.1] Let $\lambda,\mu$ be the BK-spaces and $B_{\mu}^{U}=(b_{nk})$
be defined via the sequence $\alpha=(\alpha_{k})\in\mu$ and triangle matrix
$U=(u_{nk})$ by

$\displaystyle b_{nk}=\sum_{j=k}^{n}\alpha_{j}u_{nj}v_{jk}$

for all $k,n\in\mathbb{N}$. Then, the inclusion
$\mu\lambda_{U}\subset\lambda_{U}$ holds if and only if the matrix
$B_{\mu}^{U}=UD_{\alpha}U^{-1}$ is in the classes $(\lambda:\lambda)$, where
$D_{\alpha}$ is the diagonal matrix defined by $[D_{\alpha}]_{nn}=\alpha_{n}$
for all $n\in\mathbb{N}$.

######  Lemma 3.2.

[1, Theorem 3.1] $B_{\mu}^{U}=(b_{nk})$ be defined via a sequence
$a=(a_{k})\in\omega$ and inverse of the triangle matrix $U=(u_{nk})$ by

$\displaystyle b_{nk}=\sum_{j=k}^{n}a_{j}v_{jk}$

for all $k,n\in\mathbb{N}$. Then,

$\displaystyle\lambda_{U}^{\beta}=\\{a=(a_{k})\in\omega:B^{U}\in(\lambda:c)\\}.$

and

$\displaystyle\lambda_{U}^{\gamma}=\\{a=(a_{k})\in\omega:B^{U}\in(\lambda:\ell_{\infty})\\}.$

######  Lemma 3.3.

Let $A=(a_{nk})$ be an infinite matrix. Then, the following statements hold:

  * (i)

$A\in(\ell_{1}:\ell_{\infty})$ if and only if

(3.1) $\displaystyle\sup_{k,n\in\mathbb{N}}|a_{nk}|<\infty.$

  * (ii)

$A\in(\ell_{1}:c)$ if and only if (3.1) holds, and there are
$\alpha_{k},\in\mathbb{C}$ such that

(3.2) $\displaystyle\lim_{n\rightarrow\infty}a_{nk}=\alpha_{k}~{}\textrm{ for
each }~{}k\in\mathbb{N}.$

  * (iii)

$A\in(\ell_{1}:\ell_{1})$ if and only if

(3.3) $\displaystyle\sup_{k\in\mathbb{N}}\sum_{n}|a_{nk}|<\infty.$

######  Theorem 3.4.

$\left[\int bv\right]^{\alpha}=d(\ell_{1})$

###### Proof.

We take the matrix $\Gamma$ as defined by (1.7) and $\Gamma_{n}$ denotes the
sequences in the $n$th rows of the matrices $\Gamma$. We define the matrix $C$
whose rows are the product of the rows of the matrix $\Gamma^{-1}$ and the
sequence $a=(a_{n})$, i.e., $C_{n}=(\Gamma^{-1})_{n}a$. From the relation
(2.5), we obtain

(3.4) $\displaystyle
a_{n}x_{n}=\sum_{k=1}^{n}\frac{a_{n}}{n}y_{k}=(Cy)_{n}\quad\quad(n\in\mathbb{N}).$

It follows from (3.4) that $ax=(a_{n}x_{n})\in\ell_{1}$ whenever
$x=(x_{k})\in\int bv$ if and only if $Cy\in\ell_{1}$ whenever $y\in\ell_{1}$.
By using Lemma 3.3 (iii), we obtain that $\left[\int
bv\right]^{\alpha}=d(\ell_{1})$. ∎

######  Theorem 3.5.

$[d(bv)]^{\alpha}=\int\ell_{1}$

###### Proof.

As similar way in proof of Theorem 3.4, if we take the matrix $\Sigma$ as
defined by (1.11) and define the matrix $D=(d_{nk})$ with
$a_{n}x_{n}=\sum_{k=2}^{n}n.a_{n}.y_{k}=(Dy)_{n}$ for all $n\in\mathbb{N}$,
using by the relation (2.8), this gives us that
$[d(bv)]^{\alpha}=\int\ell_{1}$. ∎

######  Theorem 3.6.

$\left[\int bv\right]^{\beta}=d(bs)$

###### Proof.

Consider the equation

(3.5)
$\displaystyle\sum_{k=1}^{n}a_{k}x_{k}=\sum_{k=1}^{n}a_{k}\left(k^{-1}\sum_{j=1}^{k}y_{j}\right)=\sum_{k=1}^{n}\left(\sum_{j=k}^{n}\frac{a_{j}}{j}\right)y_{k}=(Ey)_{n}$

where $E=(e_{nk})$ is defined by

(3.8) $\displaystyle
e_{nk}=\left\\{\begin{array}[]{ccl}\sum_{j=k}^{n}j^{-1}a_{j}&,&\quad(0\leq
k\leq n)\\\ 0&,&\quad(k>n)\end{array}\right.$

for all $n,k\in\mathbb{N}$. Then we deduce from Lemma 3.3 (ii) with (3.5) that
$ax=(a_{k}x_{k})\in cs$ whenever $x=(x_{k})\in\int bv$ if and only if $Ey\in
c$ whenever $y=(y_{k})\in\ell_{1}$. Thus, $(a_{k})\in cs$ and $(a_{k})\in
d(bs)$ by (3.1) and (3.2), respectively. Since the inclusion $d(bs)\subset cs$
holds, then, we have $(a_{k})\in d(bs)$, whence $\left[\int
bv\right]^{\beta}=d(bs)$. ∎

######  Lemma 3.7.

[4] $(cs)^{\beta}=bv\Rightarrow[d(cs)]^{\beta}=\int bv$

From Theorem 3.6 and Lemma 3.7, we have,

######  Theorem 3.8.

$(bv)^{\beta}=cs\Rightarrow[d(bv)]^{\beta}=\int cs$.

######  Theorem 3.9.

$\left[\int bv\right]^{\gamma}=d(bs)$

###### Proof.

This can be obtained by analogy with the proof of Theorem 3.6 with Lemma 3.3
(i) instead of Lemma 3.3 (ii). So we omit the details. ∎

######  Theorem 3.10.

$[d(bs)]^{\gamma}=\int bv$

##  4\. Matrix Mappings on the spaces $\int bv$ and $d(bv)$

In this section, we characterize some matrix transformations on the spaces
$\int bv$ and $d(bv)$.

We shall write throughout for brevity that

$\displaystyle\overline{a}_{nk}=k^{-1}\sum_{j=k}^{\infty}a_{nj},\quad\quad\widetilde{a}_{nk}=k\sum_{j=k}^{\infty}a_{nj},$
$\displaystyle\widehat{a}_{nk}=n.a_{nk}-(n-1).a_{n-1,k},\quad\quad\overrightarrow{a}_{nk}=n^{-1}.a_{nk}-(n-1)^{-1}.a_{n-1,k}$

for all $k,n\in\mathbb{N}$.

######  Lemma 4.1.

[1] Let $X,Y$ be any two sequence spaces, $A$ be an infinite matrix and $U$ a
triangle matrix matrix.Then, $A\in(X:Y_{U})$ if and only if $UA\in(X:Y)$.

######  Theorem 4.2.

Suppose that the entries of the infinite matrices $A=(a_{nk})$ and
$F=(f_{nk})$ are connected with the relation

(4.1) $\displaystyle f_{nk}=\overline{a}_{nk}$

for all $k,n\in\mathbb{N}$ and $Y$ be any given sequence space. Then,
$A\in(\int bv:Y)$ if and only if $\\{a_{nk}\\}_{k\in\mathbb{N}}\in[\int
bv]^{\beta}$ for all $n\in\mathbb{N}$ and $F\in(\ell_{1}:Y)$.

###### Proof.

Let $Y$ be any given sequence space. Suppose that (4.1) holds between
$A=(a_{nk})$ and $F=(f_{nk})$, and take into account that the spaces $\int bv$
and $\ell_{1}$ are norm isomorphic.

Let $A\in(\int bv:Y)$ and take any $y=(y_{k})\in\ell_{1}$. Then $\Gamma F$
exists and $\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ which
yields that (4.1) is necessary and
$\\{f_{nk}\\}_{k\in\mathbb{N}}\in(\ell_{1})^{\beta}$ for each
$n\in\mathbb{N}$. Hence, $Fy$ exists for each $y\in\ell_{1}$ and thus

$\displaystyle\sum_{k}f_{nk}y_{k}=\sum_{k}a_{nk}x_{k}~{}\textrm{ for all
}~{}n\in\mathbb{N},$

we obtain that $Fy=Ax$ which leads us to the consequence $F\in(\ell_{1}:Y)$.

Conversely, let $\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ for
each $n\in\mathbb{N}$ and $F\in(\ell_{1}:Y)$ hold, and take any
$x=(x_{k})\in\int bv$. Then, $Ax$ exists. Therefore, we obtain from the
equality

$\displaystyle\sum_{k=1}^{m}a_{nk}x_{k}=\sum_{k=1}^{m}\left[k^{-1}\sum_{j=k}^{m}a_{nj}\right]y_{k}~{}\textrm{
for all }~{}m,n\in\mathbb{N}$

as $m\rightarrow\infty$ that $Ax=Fy$ and this shows that $F\in(\ell_{1}:Y)$.
This completes the proof.

∎

######  Theorem 4.3.

Suppose that the entries of the infinite matrices $A=(a_{nk})$ and
$G=(g_{nk})$ are connected with the relation

(4.2) $\displaystyle g_{nk}=\widetilde{a}_{nk}$

for all $k,n\in\mathbb{N}$ and $Y$ be any given sequence space. Then,
$A\in(d(bv):Y)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in[d(bv)]^{\beta}$ for all $n\in\mathbb{N}$ and
$G\in(\ell_{1}:Y)$.

######  Theorem 4.4.

Suppose that the entries of the infinite matrices $A=(a_{nk})$ and
$H=(h_{nk})$ are connected with the relation

(4.3) $\displaystyle h_{nk}=\widehat{a}_{nk}$

for all $k,n\in\mathbb{N}$ and $Y$ be any given sequence space. Then,
$A\in(Y:\int bv)$ if and only if $M\in(Y:\ell_{1})$.

###### Proof.

Let $z=(z_{k})\in Y$ and consider the following equality

$\displaystyle\sum_{k=0}^{m}\widehat{a}_{nk}z_{k}=\sum_{k=0}^{m}(n.a_{nk}-(n-1)a_{n-1,k})z_{k}\quad~{}\textrm{
for all, }~{}m,n\in\mathbb{N}$

which yields that as $m\rightarrow\infty$ that $(Hz)_{n}=\\{\Gamma(Az)\\}_{n}$
for all $n\in\mathbb{N}$. Therefore, one can observe from here that $Az\in\int
bv$ whenever $z\in Y$ if and only if $Hz\in\ell_{1}$ whenever $z\in Y$. ∎

######  Theorem 4.5.

Suppose that the entries of the infinite matrices $A=(a_{nk})$ and
$M=(m_{nk})$ are connected with the relation

(4.4) $\displaystyle m_{nk}=\overrightarrow{a}_{nk}$

for all $k,n\in\mathbb{N}$ and $Y$ be any given sequence space. Then,
$A\in(Y:d(bv))$ if and only if $F\in(Y:\ell_{1})$.

######  Lemma 4.6.

  * (i)

$A\in(\ell_{1}:bs)$ if and only if

(4.5)
$\displaystyle\sup_{k,m\in\mathbb{N}}\left|\sum_{n=0}^{m}a_{nk}\right|<\infty.$

  * (ii)

$A\in(\ell_{1}:cs)$ if and only if (4.5) holds, and

(4.6) $\displaystyle\sum_{n}a_{nk}~{}\textrm{ convergent for each
}~{}k\in\mathbb{N}.$

  * (iii)

$A\in(\ell_{1}:c_{0}s)$ if and only if (4.5) holds, and

(4.7) $\displaystyle\sum_{n}a_{nk}=0~{}\textrm{ for each }~{}k\in\mathbb{N}.$

######  Lemma 4.7.

  * (i)

$A\in(\ell_{\infty}:\ell_{1})=(c:\ell_{1})=(c_{0}:\ell_{1})$ if and only if

(4.8) $\displaystyle\sup_{N,K\in\mathcal{F}}\left|\sum_{n\in N}\sum_{k\in
K}(a_{nk}-a_{n,k+1})\right|<\infty$

  * (ii)

$A\in(bs:\ell_{1})$ if and only if

(4.9) $\displaystyle\lim_{k}a_{nk}=0~{}\textrm{ for each }~{}n\in\mathbb{N}.$
(4.10) $\displaystyle\sup_{N,K\in\mathcal{F}}\left|\sum_{n\in N}\sum_{k\in
K}(a_{nk}-a_{n,k+1})\right|<\infty$

  * (iii)

$A\in(cs:\ell_{1})$ if and only if

(4.11) $\displaystyle\sup_{N,K\in\mathcal{F}}\left|\sum_{n\in N}\sum_{k\in
K}(a_{nk}-a_{n,k-1})\right|<\infty$

  * (iv)

$A\in(c_{0}s:\ell_{1})$ if and only if (4.10) holds.

Now, we can give the following results:

######  Corollary 4.8.

The following statements hold:

  * (i)

$A=(a_{nk})\in(\int bv:\ell_{\infty})$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ for all
$n\in\mathbb{N}$ and (3.1) holds with $\overline{a}_{nk}$ instead of
${a}_{nk}$.

  * (ii)

$A=(a_{nk})\in(\int bv:c)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ for all
$n\in\mathbb{N}$ and (3.1) and (3.2) hold with $\overline{a}_{nk}$ instead of
${a}_{nk}$.

  * (iii)

$A\in(\int bv:c_{0})$ if and only if $\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int
bv\\}^{\beta}$ for all $n\in\mathbb{N}$ and (3.1) and (3.2) hold with
$\alpha_{k}=0$ as $\overline{a}_{nk}$ instead of $a_{nk}$.

  * (iv)

$A=(a_{nk})\in(\int bv:bs)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ for all
$n\in\mathbb{N}$ and (4.5) holds with $\overline{a}_{nk}$ instead of
${a}_{nk}$.

  * (v)

$A=(a_{nk})\in(\int bv:cs)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ for all
$n\in\mathbb{N}$ and (4.5), (4.6) hold with $\overline{a}_{nk}$ instead of
${a}_{nk}$.

  * (vi)

$A=(a_{nk})\in(\int bv:c_{0}s)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ for all
$n\in\mathbb{N}$ and (4.5), (4.7) hold with $\overline{a}_{nk}$ instead of
${a}_{nk}$.

######  Corollary 4.9.

The following statements hold:

  * (i)

$A=(a_{nk})\in(d(bv):\ell_{\infty})$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{d(bv)\\}^{\beta}$ for all $n\in\mathbb{N}$
and (3.1) holds with $\widetilde{a}_{nk}$ instead of ${a}_{nk}$.

  * (ii)

$A=(a_{nk})\in(d(bv):c)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{d(bv)\\}^{\beta}$ for all $n\in\mathbb{N}$
and (3.1) and (3.2) hold with $\widetilde{a}_{nk}$ instead of ${a}_{nk}$.

  * (iii)

$A\in(d(bv):c_{0})$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{d(bv)\\}^{\beta}$ for all $n\in\mathbb{N}$
and (3.1) and (3.2) hold with $\alpha_{k}=0$ as $\widetilde{a}_{nk}$ instead
of $a_{nk}$.

  * (iv)

$A=(a_{nk})\in(d(bv):bs)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{d(bv)\\}^{\beta}$ for all $n\in\mathbb{N}$
and (4.5) holds with $\widetilde{a}_{nk}$ instead of ${a}_{nk}$.

  * (v)

$A=(a_{nk})\in(d(bv):cs)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{d(bv)\\}^{\beta}$ for all $n\in\mathbb{N}$
and (4.5), (4.6) hold with $\widetilde{a}_{nk}$ instead of ${a}_{nk}$.

  * (vi)

$A=(a_{nk})\in(d(bv):c_{0}s)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ for all
$n\in\mathbb{N}$ and (4.5), (4.7) hold with $\widetilde{a}_{nk}$ instead of
${a}_{nk}$.

######  Corollary 4.10.

We have:

  * (i)

$A=(a_{nk})\in(\ell_{\infty}:\int bv)=(c:\int bv)=(c_{0}:\int bv)$ if and only
if (4.8) hold with $\widehat{a}_{nk}$ instead of ${a}_{nk}$.

  * (ii)

$A=(a_{nk})\in(bs:\int bv)$ if and only if (4.9) and (4.10) hold with
$\widehat{a}_{nk}$ instead of ${a}_{nk}$.

  * (iii)

$A=(a_{nk})\in(cs:\int bv)$ if and only if (4.11) holds with
$\widehat{a}_{nk}$ instead of ${a}_{nk}$.

  * (iv)

$A=(a_{nk})\in(c_{0}s:\int bv)$ if and only if (4.10) holds with
$\widehat{a}_{nk}$ instead of ${a}_{nk}$.

######  Corollary 4.11.

We have:

  * (i)

$A=(a_{nk})\in(\ell_{\infty}:d(bv))=(c:d(bv))=(c_{0}:d(bv))$ if and only if
(4.8) hold with $\overrightarrow{a}_{nk}$ instead of ${a}_{nk}$.

  * (ii)

$A=(a_{nk})\in(bs:d(bv))$ if and only if (4.9) and (4.10) hold with
$\overrightarrow{a}_{nk}$ instead of ${a}_{nk}$.

  * (iii)

$A=(a_{nk})\in(cs:d(bv))$ if and only if (4.11) holds with
$\overrightarrow{a}_{nk}$ instead of ${a}_{nk}$.

  * (iv)

$A=(a_{nk})\in(c_{0}s:d(bv))$ if and only if (4.10) holds with
$\widehat{a}_{nk}$ instead of ${a}_{nk}$.

##  5\. Conclusion

Goes and Goes [4] introduced the integrated and differentiated sequence
spaces. Subramanian et.al. [5] gave the integrated rate space $\int\ell_{\pi}$
and studied some properties of this space. And they also characterized the
matrix classes $\left(\int\ell_{\pi}:Y\right)$, where
$Y=\\{\ell_{\infty},c,c_{0},\ell_{p},bv,bv_{0},bs,cs,\ell_{\rho},\ell_{\pi}\\}$.
There are no studies on differentiated sequence spaces.

In this paper, we studied some properties of integrated and differentiated
sequence spaces. We compute the alpha-, beta- and gamma-duals of these spaces.
For $Y=\\{\ell_{\infty},c,c_{0},bs,cs,c_{0}s\\}$, we characterize matrix
classes $(\int bv:Y),(d(bv):Y)$ and $(Y:\int bv),(Y:d(bv))$ in the last
section.

We should note from now on that the investigation of the domain of some
particular limitation matrices, namely Cesàro means of order one, Euler means
of order r, Riesz means, Nörlund means, the double band matrix $B(r,s)$, the
triple band matrix $B(r,st),$etc., in the spaces $\int bv$ and $d(bv)$ will
lead us to new results which are not comparable with the present results. If
we can choose different sequence spaces for the space $Y$, it can study new
matrix characterizations of $(\int bv:Y),(d(bv):Y)$ and $(Y:\int
bv),(Y:d(bv))$. Also the spaces $\int bv$ and $d(bv)$ can be defined by a
index $p$ and paranormed sequence spaces as $p=(p_{k})$ is a sequence of
strictly positive numbers.

## References

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