# Constructing games on networks for controlling the inequalities in the
capital distribution

Jarosław Adam Miszczak  Institute of Theoretical and Applied Informatics,
Polish Academy of Sciences,
Bałtycka 5, 44-100 Gliwice, Poland<EMAIL_ADDRESS>[

(02/10/2021 (v. 0.30))

###### Abstract

The inequality in capital or resource distribution is among the important
phenomena observed in populations. The sources of inequality and methods for
controlling it are of practical interest. To study this phenomenon, we
introduce a model of interaction between agents in the network designed for
reducing the inequality in the distribution of capital. To achieve the effect
of inequality reduction, we interpret the outcome of the elementary game
played in the network such that the wining of the game is translated into the
reduction of the inequality. We study different interpretations of the
introduced scheme and their impact on the behaviour of agents in the terms of
the capital distribution, and we provide examples based on the capital
dependent Parrondo’s paradox. The results presented in this study provide
insight into the mechanics of the inequality formation in the society.

###### keywords:

social inequality , Matthew effect , Parrondo’s games , agent-based modeling

###### MSC:

[2010] 05C57, 62C86

††journal: Physica A: Statistical Mechanics and its Applications

url]https://orcid.org/0000-0001-8790-101X

##  1 Introduction

The inequality in the communities has been studied by various authors and
numerous methods for describing and measuring inequalities have been proposed
[1, 2]. However, it is not clear why this phenomenon emerges in communities,
and how to mitigate it. As the qualitative analysis of the populations
resembles in many aspects the analysis of interactions between agents or
particles, the natural approach for modelling the dynamics of inequalities is
to use the methods borrowed from statistical physics. The field of socio-
physics aims at applying the rules of statistical mechanics for exploring the
phenomena observed in social structures, including the human society [3].

One of the mechanisms appearing in many scenarios analysing the global
behaviour of different populations is the effect of cumulative advantage. The
mechanism of cumulative advantage has also been observed in the various
aspects of human society [4]. This interesting effect is observed in many
small communities, with the community of scientists being the first one
considered. In particular, in the sociology of science, the term Matthew
effect has been coined [5] to refer to the effect of eminent scientists
getting more credit than a less known researchers, even if their contribution
is similar. Matthew effect is obeyed by small-world networks, where it is
observed as the preferential attachment mechanism. In the context of network
formation this mechanism can be used to explain the formation of World Wide
Web network [6, 7, 8, 7] giving rise to the preferential attachment process.

Recently, Matthew effect has been studied in the context of sociological
competition. In particular in [9] a model capturing the acquisitions of power
– understood as a total energy – during the interaction between the agents has
been proposed. On the other hand, in [10] the authors investigated a model of
evolutionary game with rules evolving in each round so that the agent tends to
copy the strategy of the most successful agent from their neighbourhood.

Even if many models of describing the rise of inequality have been proposed,
it is equally interesting to ask to what degree one can control the rise of
inequalities. For these reasons, in this paper we propose a simple model
constructed for the purpose of mitigating the inequality of wealth
distribution in the population. It can be argued that providing such
mechanisms is important fom the purpose of social science. It is also worth
noting that inequalities and diversity in the population can be used to
explain the emergence of cooperation [11, 12, 13]. However, one should note
that the inequality in the distribution of resources is not unique for the
human society. On the contrary, a similar mechanisms can be used to explain
inequality in the technical systems. In such case the agents ca represent
software agents acting in the computer network or processes in the operating
system.

The main contribution of the presented paper is the study of the mechanisms
which can be used to mitigate inequality in the distributed system of
interacting agents. We introduce a simple scheme in which a selected two-
player game can be used for controlling the inequality of capital distribution
in the population. To this end we interpret the winning not as the increase in
the capital, but rather as the decrease in the inequality between the agents
participating in the game. We show that the Matthew effect can be defined as
an example of the policy for interpreting the results of the game. To study
the effects of such policies we analyse the behaviour of the inequalities
using alternative polices.

As a special example of the method used for controlling the inequality of
capital distribution we use the game constructed using capital dependant
Parrondo’s scheme. In particular, we study the interplay between the
parameters of the Parrondo’s scheme and the resulting inequality reduction. We
argue that the appearance of the paradoxical behaviour does not have a
straightforward interpretation in terms of the capital distribution control.

This paper is organized in the following manner. In Section 2 we introduced
basic concepts used in this work. In Section 3 we describe a framework of
building games on networks suitable for controlling the inequality in the
capital distribution. In Section 4 we provide examples of scenarios where the
inequality of capital distribution is controlled in various manners. In
particular, we study random policy selection and a policy based on Parrondo’s
scheme. Finally, in Section 5 we summarize the presented results, provide a
discussion of the presented results, and outline possible extensions and
applications.

##  2 Preliminaries

Let us start by introducing concepts and fixing the notation used in this
work. We assume that our population consists of $n$ agents. The agents are
free to interact which each of their neighbours and the neighbourhood is
defined by the structure of the graph. Additionally, the agent can move
according to the structure of the underlying graph.

Our goal is to analyse a simple scheme of mitigating the inequality in the
capital distribution in the population. We will focus on the capital of money
possesed by the individuals [14]. The scheme discussed in this work assumes
that capital obeys the conservation law and was inspired by the approach
applied in [14, 15, 16]. Moreover, for the sake of simplicity we do not
consider debt. One can also note that the described model can be used to
describe any kind of resources which can be exchanged by the individuals as
long as they obey the conservation law and can be represented by non-negative
numbers.

The amount of money possessed by individuals is one of the ingredients of the
wealth and the distribution of money can be analysed using the statistical
mechanics approach [14, 15]. In the case where only the conservation law is
assumed, it can be shown that in the equilibrium the distribution of money
obeys Boltzmann-Gibbs law, with probability mass function

$P(m_{i})\sim\exp^{-m_{i}/T},$ (1)

where the effective temperature is given by the average amount of money per
agent, $T=\frac{\sum_{i=1}^{n}m_{i}}{n}$. Here $m_{i}\in\\{1,2,\dots,M\\}$,
where $M$ is the total amount of money distributed among the agents. Such
equilibrium distribution of money can be derived under the assumption that
money obey the conservation law [14].

The main goal of the presented work is to analyse the inequalities in the
capital distributions between the agents on the networks. For the purpose of
quantifying the effects of the introduced strategies for mitigating the
inequalities we use commonly used metrics used to measure the inequality in
the population. One of the most popular metrics used for the purpose of
measuring the inequality of the capital distribution is _Gini index_. This
quantity is commonly adopted as a measure of statistical dispersion used to
describe the economic inequality [17]. For the vectors of positive numbers
$\textbf{x}=(x_{1},x_{2},\dots,x_{n})$, Gini index is defined as an average of
the absolute differences for all pairs of elements,

$\mathcal{G}(x)=\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}|x_{i}-x_{j}|}{n\sum_{i=1}^{n}x_{i}}.$
(2)

The Gini index can range from zero, meaning complete equality in the
population, to one, which indicates complete inequality. In what follows, we
use the Gini index to measure the inequality of the capital in the population
of agents on the network. The no-debt assumption is crucial in this case, as
it is not possible to use the Gini directly index in the case of negative
values.

One should note that a plethora of inequality measures is used in the
literature [1, 2]. Another quantity commonly used to measure the inequality in
the income distribution is Hoover index, also know as Robin Hood111Legendary
heroic outlaw originally depicted in English folklore. index. Hoover index is
defined as the total amount of capital which have to be redistributed to
ensure that every agent posses equal capital,

$\mathcal{H}(x)=\frac{1}{2}\frac{\sum_{i=1}^{n}|x_{i}-\frac{1}{n}\sum_{i=1}^{n}x_{i}|}{\sum_{i=1}^{n}x_{i}}.$
(3)

As in the case of Gini index, Hoover index ranges from zero, in which case all
agents posses the same amount of the capital, to one, which is the case of
maximal inequality. Comparison og Gini and Hoover index for two different case
of capital distribution is presented in Fig. (1). However, in this paper we
focus on the Gini index.

(a) constant increase of the capital

(b) equilibrium distribution of the capital

Figure 1: Initial value of the inequality indices – Gini index and Hoover
index – obtained for the capital initialization 1(a) based on the constant
increase of the capital give by Eq. (4), and 1(b) equilibrium distribution
given for Boltzmann-Gibbs distribution from Eq. (5). In the second case,
$10^{5}$ initial samples were used for each number of agents
$n=20,40,\dots,200$, and each sample represents a possible initial state used
in the numerical experiments starting in the equilibrium.

As we aim at analysing the scheme the inequality, we need to consider various
initial distributions of the capital. For the purpose of this work we will
consider two scenarios for the initial capital distribution. In the first
scenario the initial value of the wealth assigned to the agents was set to

$m_{i}=W+i,$ (4)

where $i=1,2,\dots,n$ is the number of agents, and $W$ is the base value of
the capital. Using these initialization values we get the initial behaviour of
Gini index as depicted in Fig. (1).

In the second scenario, in order to check the behaviour of the model in the
situation when we start from the equilibrium, we will use Boltzmann-Gibbs
distribution of the initial capital. We assume that the initial amount of
money is the same as in the previous initialization strategy. The probability
mass function is given by

$P(w)\sim\exp(-\frac{wn}{M}),$ (5)

for $w=0,1,\ldots,M$, where $M=2n+\frac{n^{2}}{2}$ and the effective
temperature $T=2+\frac{n}{2}$. This leads to the minimum and maximum values of
the Gini index as depicted in Fig. (1(b)).

One should note that Gini index tends to be small for small populations [18].
This is visible in the case of initialization with a constant increase of
capital depicted in Fig. (1(a)). At the same time, in the case of Boltzmann-
Gibbs distribution, for small populations one can observe larger difference
between the minimal and the maximal value of the initial value of Gini
coefficient (cf. Fig. (1(b))). For this reason, one can start from the smaller
initial value of Gini index when a small population is considered.

As an particular example of the method of choosing the elementary games played
by the agents in the network we consider the example of the Parrondo’s effect
applied in the proposed scheme. _Parrondo’s paradox_ or _Parrondo’s game_ is a
scheme used to combine two games with negative gain in order to obtain a
scheme with a positive payoff [19, 20]. The scheme can be used in various
scenarios and the paradoxical effect has been observed in various setups [21,
22, 23, 24]. For the case of games on networks, which is most relevant for the
purpose of this work, it was demonstrated in [25, 26] that the combination of
two loosing games can lead to positive gain.

For more details one can consult recent results on Parrondo’s games on graphs
and complex networks [25, 26, 27]. For a comprehensive review of the recent
investigations concerning the application of the Parrondo’s scheme in the
context of social dynamics see [28].

In the context of this work we are interested in the connections between the
policy selection based on Parrondo’s scheme and the resulting inequality
reduction. Thus, we utilize the scheme used to obtain the paradoxical effect
observed by Parrondo, and we use it as the elementary process for controlling
the redistribution of capital in the population.

##  3 Network game for inequality control

Let us consider a population of $n$ agents on the network. Initially each of
the agents has some amount of capital $m_{i}$, $i=1,2,\dots,n$. At each time-
step an agent selects randomly one mate from the current node. Next, the agent
plays a game against the selected mate. Finally, the result of the game is
used to alter the capital of both players.

In the simplest cast, the game played by the agents is a coin-flip, which can
be possibly biased. The modification of the capital is based on the result. In
Game 1, such scheme on a graph $G$ with $n$ agents and an initial wealth
assigned to each agent equal to $m_{i}$ is presented. We assume that the coin
flip game played by the agents can have a non-zero bias,
$\epsilon\in[0,\frac{1}{2}]$, toward loosing.

0: coin-flip game A with game bias $\epsilon\in[-\frac{1}{2},0]$, graph $G$,
$m_{i}\geq 0,i=1,2\dots,n$,

for $i=1$ to $n$ do

$j$ $\leftarrow$ another agent from the current node such that $m_{j}>1$
{Avoid the debt}

$r$ $\leftarrow$ result of the A played by agent $i$ {Positive outcome means
win for agent $i$}

$\Delta\leftarrow\operatorname{sgn}{(m_{i}-m_{j})}$

if $r>0$ then

$(m_{i},m_{j})\leftarrow(m_{i}-\Delta,m_{j}+\Delta)$ {Decrease the difference
between $m_{i}$ and $m_{j}$}

else

$(m_{i},m_{j})\leftarrow(m_{i}+\Delta,m_{j}-\Delta)$ {Increase the difference
between $m_{i}$ and $m_{j}$}

end if

select one of the neighbour nodes for the current node occupied by $i$

move agent $i$ to the new position

end for

Game 1 Scheme for the direct interpretation of the results of the coin-flip
game A played on the network.

The crucial element distinguishing the proposed scheme from the standard
method of defining the game on graph is the interpretation of the result. In
the proposed case the difference in the value of capital $m_{i}$ is used in
the course of the scheme. Instead of altering their capital according to the
outcome of the game, agent interpret the outcome of the game in such manner
that winning leads to the action reducing the inequality in the capital
distribution. In particular, if agent $i$ wins against agent $j$, they both
agree to increase the capital of the agent with the smaller capital, and
reduce the capital of the agent with larger capital. On the other hand, if the
agent loses, the difference is amplified. Thus, the scheme is constructed with
the goal of controlling the distribution of capital in the population of
agents.

We assume that agents are randomly distributed on the network. Moreover, each
agent has ability to move to the nearest nodes. In the case considered in this
work we limit our considerations to the case where the underlying network is a
2D grid with periodic boundary conditions. Preliminary simulation results
suggest that the structure of the network has no clear impact of the reduction
of inequalities. However, only basic experiments were performed.

Capital possessed by each agent is governed by two processes. At each step,
each agent plays a game. Subsequently, the result of the game is interpreted
in the process of capital update. In summary, each step consists of the
following actions.

  * 1.

If there are any other agents at the current node, play a selected game
against a randomly selected opponent.

  * 2.

Using the outcome of the game, update the capital of both players such that,
in the case of the win for the current player, the difference will decrease.

  * 3.

Choose one of the neighbouring nodes and move to a new location.

The pseudocode for the above scheme is presented in Game 1. One should note
that the agent will play only if there are other agent at the node it
currently occupies. Thus one should expect faster changes in the Gini index
for the networks with more connections and for the situation when there are
more agents in the network.

Moreover, one can consider a case where the model included additional bias
parameters introduced for the consideration of _unfair_ interpretations of the
results of the game A. Such scheme is provided in Game 2, where the additional
parameters $[b_{+},b_{-}]$ are considered in the process of updating the
capital of players engaged in each elementary game. One should note that in
this case we deal with the biased game played at each step and the biased
interpretations of the results. However, we do not consider the probabilistic
interpretation of the results and the selection of bias $[b_{+},b_{-}]$ is
made at the beginning of the game.

0: game A, game bias $\epsilon\in[-\frac{1}{2},0]$, graph $G$, $m_{i}\geq
0,i=1,2\dots,n$, and policy bias $b=[b_{+},b_{-}]$

for $i=1$ to $n$ do

$j$ $\leftarrow$ choose another agent from the node occupied by agent $i$

$r$ $\leftarrow$ result of the game A played by agent $i$

$\Delta\leftarrow\operatorname{sgn}{(m_{i}-m_{j})}$

if $r>0$ then

$(m_{i},m_{j})\leftarrow(m_{i}-b_{+}\Delta,m_{j}+b_{+}\Delta)$

else

$(m_{i},m_{j})\leftarrow(m_{i}+b_{-}\Delta,m_{j}-b_{-}\Delta)$

end if

Select one of the neighbour nodes for the current node occupied by $i$.

Move agent $i$ to the new position.

end for

Game 2 Scheme for the interpretation of the results of game A, with policy
bias.

In comparison to the procedure Game 1, procedure provided in Game 2 includes
additional pair of parameters, $b$, which influences the interpretation of the
wining and loosing. Thanks to this, the scheme defined in Game 2 can be used
to control the interpretation of the winning in the game A used in the scheme.
In particular, one can consider the following policies.

  * 1.

_Janosik 222Janosik was a Slovak folk hero taking from the rich and giving to
the poor, main character of many legends taking place in Tatra mountains on
Polish-Slovak border. policy_ or _anti-Matthew policy_ ,
$[b_{+},b_{-}]=[1,1]$, leading to the decrease of the difference of capital
between agents if the game outcome is positive.

  * 2.

_Mathhew policy_ , $[b_{+},b_{-}]=[-1,-1]$, leading to the interpretation of
the positive outcome as a increase in the difference between the agents.

  * 3.

_Strong Janosik policy_ or _strong anti-Matthew policy_ ,
$[b_{+},b_{-}]=[2,1]$, with the increase in the equality preferred over the
increase in the inequality, as in this case the reduction of the capital of
the player with larger capital is bigger than the increase of his capital in
the case of negative outcome.

  * 4.

_Strong Matthew policy_ , $[b_{+},b_{-}]=[-1,-2]$, with the increase in the
inequality preferred over the decrease in the inequality, with the
interpretation of the outcomes symmetric to the interpretations in the case of
strong anti-Matthew policy.

In particular, in the case $b=[1,1]$, the procedure Game 2 is reduced to Game
1. Thus, in the case of Janosik policy, the wining in the game is interpreted
toward the reduction of inequality in the population.

##  4 Policy selection

The behaviour of the scheme proposed in the previous section is controlled by
the choice of the elementary games played by the agents. This choice can be
crucial for the final results of the inequality reduction procedure. In this
section we focus on two particular examples of elementary games used in the
proposed scheme.

The most straightforward method of choosing the policy selection game is to
use a bit flip game. In this case the policy is fair in the sense that on
average only the difference of capitals will affect the exchange of capital
between the players.

Another possible scenario is the selection policy governed by the Parrondo’s
scheme. In this case one can study the interplay between the personal average
gain of the capital and the gain of the population expressed in terms of
inequality reduction.

In this section we provide the analysis of the numerical experiments conducted
using the introduced schemes and using two proposed policy selection methods.
The presented results were obtained using the following setup. Agents are
distributed on a graph, in this case a regular 2D grid with $N\times N$ nodes,
with periodic boundary conditions. The size of the grid is fixed to $N=10$.
The graph is used to select agent which takes part in the elementary games. At
each step, the agent is activated, and plays an elementary game against one of
the agents occupying the same node.

In the presented numerical experiments, the activity of the agents is governed
by _random activation policy_ [29]. According to this policy, each agent is
activated once per step, and the activation is in random order.

For the purpose of the presented results we have utilized Mesa programming
library [30, 31]. For the sake of reproducibility [32, 33] source code for the
presented numerical experiments can be obtained from publicly accessible
repository [34].

###  4.1 Random policy selection

Figure 2: Inequality in the population for different policies and the bias
parameter $\epsilon$ used in Game 2 for constant difference in the
distribution of the initial capital. For each policy the maximal and the
minimal values of the Gini index are marked. Plots are based on the 250 steps
of the procedure Game 2 repeated 50 times for each number of agents,
$n=20,40,\dots,140$. Black dash-dotted line denotes the initial value of the
Gini index for the initialization of capital given in Eq. (4).

In the simplest scenario the selection of policy is governed by the bit flip
game. In this case one can consider the relation between the bias of the coin
and the dynamics of the inequality. Numerical results obtained from the
realization of the procedure described in Game 1 and Game 2 lead to the
results presented in Fig. (2) and Fig. (3).

Parameter $\epsilon$ controls the bias of the game played by agents towards
loosing, and its value corresponds to the average gain of the agent, which is
given by $-2\epsilon$. In particular, value $\epsilon=0$ results in fair coin
flip, and value $\epsilon=-0.5$ results in the situation where the agent
always wins.

As one can observe in Fig. (2), for the case of a fair coin used during the
elementary game played by the agents ($\epsilon=0$), both unbiased policies –
Matthew and anti-Matthew – result in the preservation of the inequality. This
is expected as this case is symmetric. Similarly, both biased policies –
strong Matthew and strong anti-Matthew – lead to a significant reduction of
the inequality in the capital distribution.

Figure 3: Inequality in the population for different policies and the bias
parameter $\epsilon$ used in Game 2 with equilibrium distribution of the
initial capital given by Eq. 5. Black dash-dotted line denotes the mean Gini
index averaged over $10^{4}$ initial configurations. Total capital in the
population is set to $20n$, where $n$ is the number of agents. Other
parameters are the same as in the case of Fig. 2.

It is interesting to notice that even a relatively small change in the value
of game bias parameter leads to a visible change in the effects of the
unbiased policies. In the presented case, for $\epsilon=-0.025$, anti-Matthew
(Janosik) policy leads to the desired goal and the final value of the Gini
index is decreased.

One should also note that strong Matthew policy leads to the decrease of the
inequality for the values of the game bias parameter
$\epsilon\in\\{0,-0.025,-0.05,-0.1,-0.15\\}$. Only for $\varepsilon>-0.5$, the
strong Matthew policy leads to the increase of the inequality. In this
situation one can make the following observations:

  * 1.

Anti-Matthew (Janosik) policy is not effective for the case $\epsilon=0$,
which means that decrease in the Gini index is compensated by its increase.
However, the application of strong anti-Matthew (strong Janosik) policy leads
to the expected reduction of the Gini index.

  * 2.

The effect of Matthew policy is the increase of inequality, and the effect of
anti-Matthew policy in the decrease of inequality is visible even for small
non-zero values of $\epsilon$.

  * 3.

Surprisingly, for the case $\epsilon=0$, strong Matthew policy leads to the
elimination of inequalities. The decrease of inequality obtained as the effect
of strong-Matthew policy is visible for values up to $\epsilon=-0.125$.

Results of the numerical simulations for the case of equilibrium
initialization of capital distribution are provided in Fig. (3). One can
observer that the initial distribution has a significant impact on the
behaviour of all strategies.

In particular, in the case of equilibrium initialization of the capital, one
can see that strong-Matthew policy is more effective. Already for
$\epsilon=-0.1$ such policy can lead to the increase in the inequality. Also
in the symmetric interpretation expressed as Matthew policy, one can observe
the increase in the inequality even for the unbiased case. This effect is more
prominent that in the case of uniform difference in the initial capitals (cf.
Fig. (2)).

###  4.2 Parrondo’s scheme policy selection

In the schemes considered in procedures Game 1 and Game 2 we assumed that the
inequality regulation policy is applied using a simple coin flip game. This
simplifies the model, but it is interesting to see how does the form of the
elementary game influences the values of capitals.

As an alternative example of the elementary game played by the agents we use
Parrondo’s scheme. In this case we consider two elementary game, game A and
game B. The choice of the particular game used at each step depends on the
strategy of the player. Additionally, the average gain for the selected game
can depend on the current capital possessed by the player. Such construction
leads to the procedure described in Game 3.

0: games A and B defined by $\delta$ and $K$, game selection policy
$\mathcal{P}$, graph $G$, $m_{i}\geq 0,i=1,2\dots,n$, and interpretation
policy bias $b=[b_{+},b_{-}]$,

for $i=1$ to $n$ do

$j$ $\leftarrow$ choose another agent from the node occupied by agent $i$

$r$ $\leftarrow$ result of the Parrondo’s game played by agent $i$ using
policy $\mathcal{P}$

$\Delta\leftarrow\operatorname{sgn}{(m_{i}-m_{j})}$

if $r>0$ then

$(m_{i},m_{j})\leftarrow(m_{i}-b_{+}\Delta,m_{j}+b_{+}\Delta)$

else

$(m_{i},m_{j})\leftarrow(m_{i}+b_{-}\Delta,m_{j}-b_{-}\Delta)$

end if

move agent $i$ to the new position

end for

Game 3 Scheme for the biased interpretation of the results with policy bias
and policy for choosing games according to the Parrondo’s scheme.

In this scenario each step consists of the following actions.

  * 1.

If there are any other agents at the current node, choose a game which will be
played during the step. The choice of the played game depends on the game
selection policy.

  * 2.

Play the selected game from the Parrondo’s scheme using the value of the
capital of the agent.

  * 3.

Alter the value of the capital for both players depending on the outcome of
the game and taking into account the inequality controlling policy.

Values of the bias parameter used in the Parrondo’s scheme was selected from
to $\delta=0.005$ to $\delta=0.1$. The parameter governing the behaviour of
game B was set to $K=3$. Combination $\delta=0.005$ and $K=3$ leads to the
example of Parrondo’s paradox in the capital dependant scenario for using
randomly selected game [20, 21].

The results for the numerical experiments conducted in the case of Game 3 are
presented in Fig. (4). From the obtained results one can conclude that anti-
Matthew policy, as well as Matthew policy, are not effective in this case.
They do not lead to the expected results for the value of parameter
$\delta>0.05$. Surprisingly, strong Matthew policy is almost not affected by
changes in the game parameters. Moreover, it leads to the significant decrease
in the value of the Gini index in all considered cases. Additionally, anti-
Matthew policy is effective only for $\delta<0.01$. It can be observed that
for $\delta=0.005$ and the uniform selection of elementary games in the
Parrondo’s scheme, the anti-Matthew policy leads to the expected reduction of
inequality in the capital distribution. However, as this effect is visible
also for the B and AABB selection of elementary games, it cannot be attributed
to the paradoxical effect which is observed in the case of the uniform
selection of elementary games in the Parrondo’s scheme. On the contrary, for
$\delta=0.01$, and the elementary game AABB, the effect of inequality
reduction is less prominent that in the case of using B only. This suggest
that the paradoxical effect observed in the Parrondo’s scheme in terms of
average gain, is not observed in terms of inequality reduction. Moreover,
using losing game only – in this case game B – can be beneficial from the
point of view of inequality reduction.

Figure 4: Influence of the parameters used in the Parrondo’s scheme in Game 3
on the changes in the Gini index. In this case four policies for constructing
Parrondo’s scheme were used, namely: always play game A, always play game B,
play games A and B in sequence AABB, and choose which game to play randomly
with equal probability. Parameter $\delta\in\\{0.1,0.05,0.01,0.005\\}$ is used
in games A and B used in the Parrondo’ scheme. Plots are based on the 250
steps of the procedure Game 2 repeated 50 times for each number of agents,
$n=20,60,\dots,180$. Black dash-dotted line denotes the initial value of the
Gini index for the initialization of capital with constant difference (cf. Eq.
(4)).

##  5 Final remarks

The goal of this work was to introduce a scheme for reducing the inequality in
the capital distribution among the agents on the network. To this end we
proposed an interpretation of the game on network in terms of inequality
reduction, and we considered symmetric and asymmetric interpretation policies.

We also studied two particular examples for controlling the elementary game
selection. In the first scenario, we used a biased coin to mimic the random
selection of the policy for reducing the inequality. We demonstrated how the
value of the bias influences the resulting Gini index.

In the second scenario, we investigated the relation between the Parrondo’s
effect and the capital inequality control. Following the scheme used in the
Parrondo’s paradox, we provided a construction of a scheme composed of two
losing games. We used the result of the game to introduce the behaviour of
suppressing the inequality of wealth distribution in the group of agents in a
network. We also provided arguments that the interpretation of the Parrondo
paradox in terms of the capital cannot be translated into the interpretation
in terms of the capital reduction.

One can note that the schemes studied in the presented work do not include
mechanics taking into account the information about the global inequality. In
the presence of such global governing mechanism such scenarios, the
information about the value of the inequality index could be incorporated into
the inequality regulation policy. In the schemes introduced in the presented
paper, the agents were not aware of the global inequalities in the capital
distribution. The interpretation of the local game was based on the difference
between the players only. It is natural to ask if the global mechanics for
distributing information about the inequalities, expressed by the Gini index
or by other inequality indicators, can lead to a different behaviour of the
model.

Another possible extension of the presented work is the study of the proposed
schemes in the case where the interactions between agents take place on the
complex network. Such extension could be useful to study the possibility of
managing the wealth interpreted not as capital, but as other forms of social
factors.

It is also worth stressing that the presented study is limited by the
applicability of the Gibbs distribution. In scenarios observed in economy and
social science one cannot assume that the system is in the equilibrium. In
such systems it is necessary to take into account the permanent dynamics of
the processes governing the population.

Finally, other possible methods of capital initialization could be
investigated. In particular, one could mimic the wealth distribution obtained
in some communities or countries, and thus test the proposed model in the
real-world context. Such study could be particularity interesting if the
historical data and the past decisions could be described and analysed in the
context of social inequality reduction.

## Acknowledgments

Author would like to thank Nenggang Xie, Ye Ye and Wang Meng for motivating
the presented study and for their hospitality during the visit at the Anhui
University of Technology.

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# Hierarchical Robust Adaptive Control for Wind Turbines with Actuator Fault

Sina Ameli and Olugbenga Moses Anubi∗ Department of Electrical Engineering
Florida State University
Tallahassee, Florida 32310
Email<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

This paper solves the problem of regulating the rotor speed tracking error for
wind turbines in the full-load region by an effective robust-adaptive control
strategy. The developed controller compensates for the uncertainty in the
control input effectiveness caused by a pitch actuator fault, unmeasurable
wind disturbance, and nonlinearity in the model. Wind turbines have multi-
layer structures such that the high-level structure is nonlinearly coupled
through an aggregation of the low-level control authorities. Hence, the
control design is divided into two stages. First, an $\mathcal{L}_{2}$
controller is designed to attenuate the influence of wind disturbance
fluctuations on the rotor speed. Then, in the low-level layer, a controller is
designed using a proposed adaptation mechanism to compensate for actuator
faults. The theoretical results show that the closed-loop equilibrium point of
the regulated rotor speed tracking error dynamics in the high level is finite-
gain $\mathcal{L}_{2}$ stable, and the closed-loop error dynamics in the low
level is globally asymptotically stable. Simulation results show that the
developed controller significantly reduces the root mean square of the rotor
speed error compared to some well-known works, despite the largely fluctuating
wind disturbance, and the time-varying uncertainty in the control input
effectiveness.

##  1 Introduction

Wind energy is increasingly becoming a significant source of clean energy.
However, its market penetration has been affected adversely by high costs
related to manufacturing, operation, and maintenance, especially for megawatt
size  wind turbines (WTs) producing a great deal of electricity. To expand the
lifetime of such large WTs, deploying active load alleviation control is
critical. One efficient approach to decrease structural loading, and thereby
extend the lifetime of the WTs while maintaining the rated electrical power is
regulating the rotor speed tracking error via controlling pitch angles [1, 2,
3, 4]. In [2] a PI controller is designed for a WT to alleviate the mechanical
stress; however, it highly relies on gain selection based on the wind speed.
In [3] a linear quadratic regulator is designed to track the generator speed.
However, the closed loop system performance is downgraded in wider operating
range. In [4], the authors proposed a linear quadratic Gaussian control.
However, the controller suffers from lack of robustness to model
uncertainties.

Since WTs are aero-electro-mechanical systems, their dynamics are surpassingly
complicated. There is no explicit relation between the rotor speed and the
pitch angle for WTs in the region that the wind speed is above the rated value
(full-load region). They have nonlinear dynamics exposed to stochastic wind
signal changing the WTs’ operating points. Thus, these issues necessitate the
design of the nonlinear controllers for WTs. In [5] a gain-scheduling optimal
controller is proposed utilizing genetic algorithm for pitch control; however,
it can only deal with a limited types of wind signals. In [6] a nonlinear
model predictive control (MPC) is designed for a WT to reduce pitch actuation
while reducing the mechanical loads and regulating the rotor speed tracking
error. However, the controller is not robust against model uncertainty.

Moreover, it has been proven that robust and adaptive control strategies
tackle with the wind disturbance fluctuations, model uncertainty, and time-
varying actuator faults in WTs [7, 8, 9, 10, 11, 12, 13, 14]. In [7], an
$\mathcal{H}_{\infty}$-state-feedback controller is designed for a WT to
mitigate actuator faults. However, the controller’s performance is limited to
specific operating points. In [9] a disturbance observer and effective wind
speed estimator are designed for a WT with model uncertainty to maintain the
rated electrical power by controlling the pitch angle. However, the controller
design relies on the estimation of the wind speed. In [8] a robust MPC is
designed by solving an LMI for a WT while the pitch actuator is subjected to a
fault. However, the controller’s performance is dependent on the linearized
model of the WT. In [10] a moving horizon $\mathcal{H}_{\infty}$ controller is
designed for a WT to control the pitch angle. It shows that the closed-loop
system achieves the $\mathcal{L}_{2}$ disturbance attenuation from the wind
signal to the states in the framework of dissipation theory. However, the
pitch actuators are assumed to be healthy during their operations. In [11],
and [12] an online system identifier using adaptive directional forgetting
scheme is designed to extract the WT model and approximate it to a second or
third order system for pitch and torque channels. Then an adaptive PI
controller is designed using modified Ziegler-Nichols rules. Although the
controller can estimate the time-varying parameters, and regulates the rotor
speed error, it is not robust against the wind signal with high turbulence
intensity. In [13] a robust adaptive controller is designed using clustering-
type fuzzy neural network for a WT subjected to pitch actuator faults.
However, the fuzzy controller itself has several parameters (defuzzification,
inference, input and output membership functions, rulebases, etc) to be
determined. In [14] an adaptive $\mathcal{L}_{2}$-gain controller is designed
for a class of singular systems with Lipschitz nonlinearity, actuator
saturation, and actuator faults. Although the results show that the closed-
loop system achieves smaller upper bound on the $\mathcal{L}_{2}$-gain, the
actuator faults are assumed to be constant. The major contributions of this
paper are (i) a multi-layered control structure consistent with the nature of
the system, (ii) a detailed finite-gain analysis and design to compensate for
unmeasurable wind disturbance, nonlinearity and model uncertainty in the high-
level layer, and (iii) low-level adaptive control design to compensate for
time-varying incipient pitch actuator faults. The remaining of the paper is
organized as follows: Section 2, introduces notations and preliminary. Section
3, presents the system model. Section 4, illustrates the control development
for all layers. Section 5, shows the numerical simulation results. Section 6,
provides conclusion remarks, and finally the future work is discussed in
section 7.

##  2 Notation and Preliminary

The following notions and conventions are utilized in the paper: $\mathbb{R}$,
and $\mathbb{R}^{n}$ denote the space of real numbers, and real vectors of
length $n$, respectively. $\mathbb{R}_{+}$ includes zero and positive real
numbers. Normal-face lower-case letters ($x\in\mathbb{R}$) are used to
represent real scalars, bold-face lower-case letter
($\mathbf{x}\in\mathbb{R}^{n}$) represents vectors. The euclidean balls
$\mathbb{B}_{r}(0)$ is defined for some $r>0$ as
$\mathbb{B}_{r}(0)\triangleq\left\\{\textbf{x}:\left\|\textbf{x}\right\|\leq
r\right\\}$. Moreover, $\mathcal{L}_{2}$ is the space of all piecewise
continuous, square-integrable functions
$\mathbf{f}:\mathbb{R}_{+}\mapsto\mathbb{R}^{n}$, and the $\mathcal{L}_{2}$
norm of $\mathbf{f}\in\mathcal{L}_{2}$ is defined by

$\displaystyle\left\|\textbf{f}\right\|_{2}\triangleq\left(\int_{0}^{\infty}{\|\textbf{f}(\tau)\|^{2}d\tau}\right)^{\frac{1}{2}}<\infty,$

and the extended space $\mathcal{L}_{2e}$ is the space of all measurable
functions $\mathbf{f}$ such that $\mathbf{f}_{T}\in\mathcal{L}_{2}$ where

$\displaystyle\mathbf{f}_{T}(t)\triangleq\left\\{\begin{array}[]{rl}\mathbf{f}(t),&0\leq
t<T\\\ \textbf{0},&t\geq T\end{array}\right.$

, for all $T\in[0,\infty)$.

###### Definition 1

($\gamma$-dissipativity)[15] Consider the nonlinear system

$\displaystyle\mathcal{G}:\hskip
14.22636pt\begin{array}[]{rl}\dot{\mathbf{z}}&=g(\mathbf{z},\bm{\nu})\\\
\bm{\omega}&=h(\mathbf{z})\end{array}$ (3)

where $\bm{\nu}(t)\in\mathcal{L}_{2e}^{q}$,
$\mathbf{z}(t)\in\mathcal{L}_{2e}^{n}$,
$\bm{\omega}(t)\in\mathcal{L}_{2e}^{s}$ are the input, the state variable, and
the output, respectively. The nonlinear system in Eqn. (3) is dissipative with
respect to the supply rate
$\ell:\mathcal{L}_{2e}^{q}\times\mathcal{L}_{2e}^{s}\mapsto\mathcal{L}_{2e}$,
if an energy function $V(\mathbf{z})\geq 0$ exists such that, for all
$t_{1}\geq t_{0}$,

$\displaystyle V(\mathbf{z}(t_{1}))\leq
V(\mathbf{z}(t_{0}))+\int_{t_{0}}^{t_{1}}\ell(\bm{\nu},\mathbf{\omega})d\tau\hskip
5.69054pt\text{for all}\hskip 5.69054pt\bm{\nu}\in\mathcal{L}_{2e}^{q}.$ (4)

In addition, for a $\gamma>0$, if the supply rate is chosen as
$\ell(\bm{\nu},\bm{\omega})=\gamma^{2}\left\|\bm{\nu}\right\|_{2}^{2}-\left\|\bm{\omega}\right\|_{2}^{2}$,
then (4) indicates a finite-gain $\mathcal{L}_{2}$ stability. As a result, the
dynamic is $\gamma$-dissipative and inequality (4) yields
$\dot{V}\leq\gamma^{2}\left\|\bm{\nu}\right\|_{2}^{2}-\left\|\bm{\omega}\right\|_{2}^{2}$.

###### Lemma 1

The following characteristics are satisfied[16]

  1. 1.

$\textsf{sat}(\chi\mathbf{y},y_{1},y_{2})=\chi\textsf{sat}(\mathbf{y},\frac{y_{1}}{\chi},\frac{y_{2}}{\chi}),\text{
for all }\chi>0$

  2. 2.

For $r>0$, there exists $\mu>0$ such that
$\mathbf{y}^{\top}\textsf{sat}(\mathbf{y},y_{1},y_{2})\geq\mu\left\|\mathbf{y}\right\|_{2}^{2},\text{
for all }\mathbf{y}\in\mathbb{B}_{r}(0)$

##  3 Problem Formulation

In general WTs are operational in two areas, namely partial-load and full-load
regions shown Figure 1. In the former the optimum electrical power is
generated via a torque control. However, in the latter, since the wind speed
is above the rated value, the rated electrical power can be generated but it
should not pass its rated value and meanwhile the mechanical stress should be
reduced via the pitch control.

###  3.1 Drive-train Dynamics

Figure 1: Electrical power generated by a typical WT at different regions

The rotor dynamics in WTs is provided by [17, 1, 18]

$\dot{\omega}_{r}=f(\nu,\omega_{r},\theta_{j}),\hskip 14.22636ptj=1,2,3$ (5)

,where $\omega_{r}\in\mathcal{L}_{2e}$, is the rotor speed,
$\theta_{j}\in\mathbb{R}_{+}$ is the $j$th pitch angle,
$\nu\in\mathcal{L}_{2e}$ is the wind disturbance, and

$\displaystyle
f(\nu,\omega_{r},\theta_{j})=g_{1}(\nu,\omega_{r})-g_{2}(\nu,\omega_{r})\left\|\bm{\theta}\right\|^{2},$

where $\bm{\theta}\in\mathbb{R}^{3}$, and

$\displaystyle g_{1}(\nu,\omega_{r})=$
$\displaystyle\frac{\kappa\nu^{3}}{J\omega_{r}}\left(\frac{\nu}{\omega_{r}}-p_{1}\right)\textsf{e}^{(-p_{2}\frac{\nu}{\omega_{r}})}-\frac{P_{0}}{J\omega_{r}},$
(6) $\displaystyle g_{2}(\nu,\omega_{r})=$
$\displaystyle\frac{\kappa\nu^{3}}{3J\omega_{r}}p_{3}\textsf{e}^{(-p_{2}\frac{\nu}{\omega_{r}})},$
(7)

where $\kappa$ is a positive constant, and the positive constants $p_{1}$,
$p_{2}$, and $p_{3}$ are obtained experimentally. $P_{0}$, and $J$ are the
rated mechanical power, and the total drive-train inertia, respectively.

###  3.2 Pitch Systems

The pitch actuators include a hydraulic system whose dynamics are expressed as
follows [19]

$\displaystyle\ddot{\theta}_{j}=-2\zeta\omega_{n}\dot{\theta}_{j}-\omega^{2}_{n}\theta_{j}+\omega_{n}^{2}\theta_{rj},\hskip
28.45274ptj=1,2,3,$ (8)

where $\zeta$, is the damping ratio, and $\omega_{n}$ is the natural
frequency. $\theta_{j}$, and $\theta_{rj}$ are the pitch angle output and the
pitch reference (control input) for the $j$-th pitch actuator, respectively.

###### Assumption 1

The pitch angle $\theta_{j}$ is mechanically constrained such that it is
bounded as $0\leq\theta_{j}\leq\overline{\theta}$, where $\overline{\theta}$
is the maximum allowable pitch angel.

####  3.2.1 Fault Model

One type of fault that could happen in pitch actuators is the high-air content
fault, which is an incipient fault, making the oil is mixed with air. The
occurrence of this fault changes the characteristics of the second-order
dynamics in Eqn. (8). Thus, the uncertainty caused by this fault is parametric
modeled as follows

$\displaystyle\omega_{n}^{2}$ $\displaystyle=\delta\omega_{n_{0}}^{2},$ (9)
$\displaystyle\zeta\omega_{n}$ $\displaystyle=\rho\zeta_{0}\omega_{n0},$

where $\omega_{n_{0}}$, and $\zeta_{0}$ are the operating points (faultless
situation). Moreover, $\delta$ and $\rho$ are uncertain parameters because of
the fault but they are bounded as $0<\delta\leq 1$, $0<\rho\leq 1$ [19].

##  4 Control Development

In this section, the aim is to track the rotor speed operating point
$\omega_{r0}$ robustly. This is achieved via $\gamma$-dissipativity. Define
the rotor speed tracking error

$\displaystyle\widetilde{\omega}_{r}=\omega_{r}-\omega_{r0}.$ (10)

Denote the following filtered error

$\displaystyle\sigma=$
$\displaystyle\widetilde{\omega}_{r}+\psi\underbrace{\int_{0}^{t}\widetilde{\omega}_{r}d\tau}_{\text{$\widetilde{\omega}_{rI}$}},$
(11)

where $\psi>0$. Taking first time derivative of Eqn. (11), and then inserting
the time derivative of Eqn. (10) into it yields

$\displaystyle\dot{\sigma}=$
$\displaystyle\dot{\widetilde{\omega}}_{r}+\psi\dot{\widetilde{\omega}}_{rI}$
$\displaystyle=$ $\displaystyle
f(\nu,\omega_{r},\theta_{j})+\psi\widetilde{\omega}_{r}.$ (12)

Consider the equilibrium points $\omega_{r0}$, $\nu_{0}$, and $\theta_{0}$ in
Eqn. (5), then $f(\nu_{0},\omega_{r0},\theta_{0})=0$. Subtracting this
equation from Eqn. (12) yields

$\displaystyle\dot{\sigma}=f(\nu,\omega_{r},\theta_{j})-f(\nu_{0},\omega_{r0},\theta_{0})+\psi\widetilde{\omega}_{r}.$

Then, invoking the mean value theorem [20] results in

$\displaystyle\dot{\sigma}=\rho_{\nu}\widetilde{\nu}+\left(\rho_{\omega}+\psi\right)\widetilde{\omega}_{r}+\bm{\rho}_{\theta}^{\top}\bm{\widetilde{\theta}}$
(13)

where

$\displaystyle\rho_{\nu}=$ $\displaystyle\frac{\partial
f(\eta_{\nu},\eta_{\omega},\bm{\eta}_{\theta})}{\partial\nu},$
$\displaystyle\rho_{\omega}=$ $\displaystyle\frac{\partial
f(\eta_{\nu},\eta_{\omega},\bm{\eta}_{\theta})}{\partial\omega},$
$\displaystyle\bm{\rho}_{\theta}=$
$\displaystyle\nabla_{\theta}f(\eta_{\nu},\eta_{\omega},\bm{\eta}_{\theta}),$

with $\eta_{\nu}=\lambda\nu_{0}+(1-\lambda)\nu$,
$\eta_{\omega}=\lambda\omega_{r0}+(1-\lambda)\omega_{r}$,
$\bm{\eta}_{\theta}=\lambda\bm{\theta}_{0}+(1-\lambda)\bm{\theta}$ for some
$\lambda\in(0,1)$, and

$\displaystyle\widetilde{\bm{\theta}}$
$\displaystyle=\bm{\theta}-\bm{\theta}_{0},$ $\displaystyle\widetilde{\nu}$
$\displaystyle=\nu-\nu_{0}$

, where $\bm{\theta}_{0}$, and $\nu_{0}$ are the pitch angle vector, and the
wind speed at the operating point. These operating points are known, and they
are usually used in the component design and rating. So, it is reasonable to
assume that they are known. Moreover, the simulation results in section 5,
will show that the controller regulates the rotor speed tracking error well at
different operating points.

Here, the objective is to design an auxiliary control law
$\widetilde{\bm{\theta}}$ for Eqn. (13), such that the rotor speed tracking
error $\widetilde{\omega}_{r}$ is robustly regulated for all
$\widetilde{\nu}\in\mathcal{L}_{2e}$. This is then deployed for a desired
pitch angle for the low-level dynamics. The subsequent properties hold for the
open-loop dynamics in Eqn. (13).

###### Assumption 2

The high-level dynamics is sufficiently smooth. Hence, the uncertain
parameters $\rho_{\nu}$, and $\rho_{\omega}$ in Eqn. (13) are bounded as
follows

$\displaystyle\left|\rho_{\nu}\right|\leq\bar{\rho}_{\nu},$
$\displaystyle\left|\rho_{\omega}\right|\leq\bar{\rho}_{\omega},$

where $\bar{\rho}_{\nu}$, and $\bar{\rho}_{\omega}$ are positive constants.
Moreover, there exists $\bm{\rho}_{0}\in\mathbb{R}^{3}$, and a positive
constant $\varphi$ such that

$\displaystyle\bm{\rho}_{\theta}^{\top}\bm{\rho}_{0}\geq\varphi.$ (14)

Consequently, the design of the auxiliary control law is as follows

$\displaystyle\widetilde{\bm{\theta}}$
$\displaystyle=-\bm{\rho}_{0}\textsf{sat}\left(k\sigma,-\theta_{0},\overline{\theta}-\theta_{0}\right),$
(15)

where $k$ is a positive control gain, and $\bm{\rho}_{0}$ satisfies the conic
constraint in Eqn. (14). Then, the corresponding high-level closed-loop error
system is obtained as follows

$\displaystyle\dot{\sigma}=$
$\displaystyle\left(\rho_{\omega}+\psi\right)\widetilde{\omega}_{r}+\rho_{\nu}\widetilde{\nu}-\bm{\rho}_{\theta}^{\top}\bm{\rho}_{0}\textsf{sat}\left(k\sigma,-\theta_{0},\overline{\theta}-\theta_{0}\right)$
$\displaystyle=$
$\displaystyle\left(\rho_{\omega}+\psi\right)\left(\sigma-\psi\widetilde{\omega}_{rI}\right)+\rho_{\nu}\widetilde{\nu}$
(16)
$\displaystyle-\bm{\rho}_{\theta}^{\top}\bm{\rho}_{0}\textsf{sat}\left(k\sigma,-\theta_{0},\overline{\theta}-\theta_{0}\right)$

The following theorem provides the robust performance of the auxiliary control
law in the high-level layer.

###### Theorem 1

Given the high-level control law in Eqn. (15), and $\gamma>0$, if the control
gain is chosen such that the following sufficient condition is satisfied

$\displaystyle
k\geq\frac{1}{\mu\varphi}\left(1+\frac{\left(\bar{\rho}_{\omega}+2\psi\right)^{2}}{4\psi}+\frac{\bar{\rho}_{\nu}^{2}}{4\gamma^{2}}\right),$
(17)

then the corresponding closed-loop error dynamics in (16) is
$\mathcal{L}_{2}$-gain stable and the $\mathcal{L}_{2}$-gain from the
exogenous signal $\widetilde{\nu}$ to the regulated error
$\widetilde{\omega}_{r}$ is upper bounded by $\gamma$.

######  Proof 4.2.

Denote the positive definite energy function

$\displaystyle
V=\frac{1}{2}\sigma^{2}+\frac{\psi^{2}}{2}\widetilde{\omega}_{rI}^{2}.$ (18)

Since
$\left\|\widetilde{\omega}_{r}\right\|_{2}\leq\left\|\left(1+\frac{1}{s}\widetilde{\omega}_{r}\right)\right\|_{2}=\left\|\sigma\right\|_{2}$,
and considering Definition. (1), it is sufficient to show that
$\dot{V}\leq\gamma^{2}\widetilde{\nu}^{2}-\sigma^{2}$. Taking the time
derivative of $V$, and inserting the high-level closed-loop dynamics in Eqn.
(16) into it, leads to

$\displaystyle\dot{V}$
$\displaystyle=\sigma\dot{\sigma}+\psi^{2}\widetilde{\omega}_{rI}\dot{\widetilde{\omega}}_{rI}$
$\displaystyle=\sigma\left(\left(\rho_{\omega}+\psi\right)\sigma-\psi\left(\rho_{\omega}+\psi\right)\widetilde{\omega}_{rI}+\rho_{\nu}\widetilde{\nu}\right)$
$\displaystyle-\frac{1}{k}\bm{\rho}_{\theta}^{\top}\bm{\rho}_{0}(k\sigma)\textsf{sat}\left(k\sigma,-\theta_{0},\overline{\theta}-\theta_{0}\right)+\psi^{2}\widetilde{\omega}_{rI}\left(\sigma-\psi\widetilde{\omega}_{rI}\right)$

Adding, and subtracting
$\left(\gamma^{2}\widetilde{\nu}^{2}-\sigma^{2}\right)$, and finally applying
Assumption. (2), and Lemma. (1) yield

$\displaystyle\dot{V}$
$\displaystyle=\left(\rho_{\omega}+\psi+1\right)\sigma^{2}-\psi\rho_{\omega}\sigma\widetilde{\omega}_{rI}+\rho_{\nu}\sigma\widetilde{\nu}$
$\displaystyle-\frac{1}{k}\bm{\rho}_{\theta}^{\top}\bm{\rho}_{0}(k\sigma)\textsf{sat}\left(k\sigma,-\theta_{0},\overline{\theta}-\theta_{0}\right)$
$\displaystyle-\psi^{3}\widetilde{\omega}_{rI}^{2}-\gamma^{2}\widetilde{\nu}^{2}+\left(\gamma^{2}\widetilde{\nu}^{2}-\sigma^{2}\right)$
$\displaystyle=\left(\rho_{\omega}+\psi+1+\frac{\rho_{\omega}^{2}}{4\psi}+\frac{\rho_{\nu}^{2}}{4\gamma^{2}}\right)\sigma^{2}-\psi\left(\psi\widetilde{\omega}_{rI}+\frac{\rho_{\omega}}{2\psi}\sigma\right)^{2}$
$\displaystyle-\gamma^{2}\left(\widetilde{\nu}-\frac{\rho_{\nu}}{2\gamma^{2}}\sigma\right)^{2}-\frac{1}{k}\bm{\rho}_{\theta}^{\top}\bm{\rho}_{0}(k\sigma)\textsf{sat}\left(k\sigma,-\theta_{0},\overline{\theta}-\theta_{0}\right)$
$\displaystyle+\left(\gamma^{2}\widetilde{\nu}^{2}-\sigma^{2}\right)$
$\displaystyle\leq-\left(k\mu\varphi-1-\frac{\left(\bar{\rho}_{\omega}+2\psi\right)^{2}}{4\psi}-\frac{\bar{\rho}_{\nu}^{2}}{4\gamma^{2}}\right)\sigma^{2}+\left(\gamma^{2}\widetilde{\nu}^{2}-\sigma^{2}\right)$
$\displaystyle\leq\left(\gamma^{2}\widetilde{\nu}^{2}-\sigma^{2}\right)$

Next, the desired pitch angle vector $\bm{\theta}$ designed in the high-level
layer is directly translated to individual desired pitch angle in the low-
level control such that $\bm{\theta}_{d}=\bm{\theta}$. Without loss of
generality, the control development in the low-level is performed using a
single actuator since the actuator model provided in Eqn. (8) is the same for
all actuators.

Consider the low-level tracking error $\varepsilon=\theta-\theta_{d}$, where
$\theta_{d}$ is the desired pitch angle for one actuator. Since the pitch
actuator dynamics in Eqn. (8) is significantly faster than the rotor dynamics
given in Eqn. (5), then it is reasonable to assume that $\dot{\theta}_{d}=0$
for developing the controller for Eqn. (8). See [21, 22] for more detailed
analysis where the authors use singular perturbation and time-scale separation
to show that it is indeed the case for mechatronic systems of this form. Next,
consider the filtered tracking error

$\displaystyle z\triangleq\dot{\varepsilon}+2\zeta_{0}\omega_{n0}\varepsilon.$
(19)

Taking first derivative of Eqn. (19) yields

$\displaystyle\dot{z}=$
$\displaystyle\ddot{\varepsilon}+2\zeta_{0}\omega_{n0}\dot{\varepsilon}$
$\displaystyle=$
$\displaystyle\ddot{\theta}+2\zeta_{0}\omega_{n0}\dot{\theta},$

then inserting the pitch actuator dynamics in Eqn. (8), and the fault model in
Eqn. (9) yields

$\displaystyle\dot{z}$
$\displaystyle=\delta\omega_{n0}^{2}(\theta_{r}-\theta)-\delta\omega_{n0}^{2}\left(\frac{2\left(\rho\zeta_{0}\omega_{n0}-\zeta_{0}\omega_{n0}\right)}{\delta\omega_{n0}^{2}}\right)\dot{\theta}$
$\displaystyle=\delta\omega_{n0}^{2}\left(\theta_{r}-\theta-\eta\dot{\theta}\right),$
(20)

where

$\displaystyle\eta\triangleq\left(\frac{2\left(\rho\zeta_{0}\omega_{n0}-\zeta_{0}\omega_{n0}\right)}{\delta\omega_{n0}^{2}}\right)$

is an uncertain parameter due to actuator faults. Consequently, the low-level
control law is designed as

$\displaystyle\theta_{r}=\theta-k_{\theta}z+\widehat{\eta}\dot{\theta},$ (21)

where $k_{\theta}>0$ is the control gain, and $\widehat{\eta}$ is the
parameter estimation. Next, inserting the control law in Eqn. (21) into the
open-loop dynamics in Eqn. (20) yields

$\displaystyle\dot{z}=-\delta\omega_{n0}^{2}k_{\theta}z+\delta\omega_{n0}^{2}\widetilde{\eta}\dot{\theta},$
(22)

where $\widetilde{\eta}=\widehat{\eta}-\eta$ is the parameter estimation
error. Consequently, the adaptive law is designed as

$\displaystyle\dot{\widehat{\eta}}=-\alpha z\dot{\theta},$ (23)

where $\alpha>0$ is an adaptation gain.

######  Theorem 4.3.

Consider the low-level control law in Eqn. (21), together with the adaptive
law in Eqn. (23), if the following conditions hold

$\displaystyle k_{\theta}>0,\hskip 5.69054pt\alpha>0,$ (24)

then the closed-loop error system in Eqn. (22) is globally asymptotically
stable.

######  Proof 4.4.

Denote the positive definite radially unbounded candidate Lyapunov function

$\displaystyle
V=\frac{1}{2}z^{2}+\frac{\delta\omega_{n0}^{2}}{2\alpha}\widetilde{\eta}^{2},$

taking first derivative yields

$\displaystyle\dot{V}=z\dot{z}+\frac{\delta\omega_{n0}^{2}}{\alpha}\widetilde{\eta}\dot{\widehat{\eta}},$

Inserting the closed-loop dynamics in Eqn. (22) results in

$\displaystyle\dot{V}=-\delta\omega_{n0}^{2}k_{\theta}z^{2}+\delta\omega_{n0}^{2}z\widetilde{\eta}\dot{\theta}+\frac{\delta\omega_{n0}^{2}}{\alpha}\widetilde{\eta}\dot{\widehat{\eta}}.$
(25)

Next, inserting the adaptation law in Eqn. (23) yields

$\displaystyle\dot{V}$ $\displaystyle=-\delta\omega_{n0}^{2}k_{\theta}z^{2}.$
(26)

This indicates that $\dot{V}$ is negative semidefinite implying that
$V\in\mathcal{L}_{\infty}$; which implies that $z\in\mathcal{L}_{\infty}$, and
$\widetilde{\eta}\in\mathcal{L}_{\infty}$. Thus, it implies that
$\varepsilon$, and $\dot{\varepsilon}$ ($\dot{\theta}$) are also bounded. Now,
consider Eqn. (22), as $z$, $\widetilde{\eta}$, and $\dot{\theta}$, are
bounded, then $\dot{z}$ is also bounded implying that $z$ is uniformly
continuous. Next, taking the integral of both sides in Eqn. (26) yields

$\displaystyle\int_{0}^{\infty}\dot{V}d\tau$
$\displaystyle=\int_{0}^{\infty}-\delta\omega_{n0}^{2}k_{\theta}z^{2}d\tau,$
$\displaystyle V(\infty)-V(0)$
$\displaystyle=-\int_{0}^{\infty}\delta\omega_{n0}^{2}k_{\theta}z^{2}d\tau,$
(27)

Since $V$ is bounded, then $z\in\mathcal{L}_{2}$. Since
$z\in\mathcal{L}_{2}\cap\mathcal{L}_{\infty}$, and also it is uniformly
continuous, according to the Barbalat’s lemma[23], $z$ converges to zero
asymptotically. Thus, $\varepsilon\rightarrow 0$, and $\dot{\theta}\rightarrow
0$ as $t\rightarrow\infty$.

##  5 Numerical Results

The developed controller is validated on a 5MW variable pitch WT model via the
Fatigue Aerodynamics Structures and Turbulence (FAST) simulator developed by
the US National Renewable Energy  Lab (NREL) [17].

Figure 2: Block diagram of the developed controller Figure 3: Step wind signal
Figure 4: Rotor speed response, and control authority subjected to the step
wind signal Figure 5: Stochastic wind signal Figure 6: Rotor speed response,
and control authority subjected to the stochastic wind signal

The desired value for the rotor speed is $\omega_{r0}=1.267rad/s$, and the
operating points for the pitch angle, and the wind speed are
$\theta_{0}=19.94deg$, $\nu_{0}=22m/s$. The nominal actuator parameters are
$\zeta_{0}=0.6$, $\omega_{n0}=11.11rad/s$. The parameters $\kappa=7622.7$,
$J=43784700kg.m^{2}$. Regarding the parameters $p_{1}$, $p_{2}$, and $p_{3}$
in Eqn. (6), and Eqn. (7), their values depend on the operating point of the
WT. In [18], their values are obtained experimentally. Here, their best values
at the operating point are obtained via solving an optimization problem, which
minimizes the deviation from the experimental values subject to (5):

$\displaystyle\textsf{Minimize:
}\left\|\mathbf{p}-\bar{\mathbf{p}}\right\|^{2}$ Subject to:
$\displaystyle\left(\frac{\kappa\nu_{0}^{3}}{J\omega_{r0}}\left(\frac{\nu_{0}}{\omega_{r0}}-p_{1}\right)-\frac{\kappa\nu_{0}^{3}}{3J\omega_{r0}}p_{3}\left\|\bm{\theta}_{0}\right\|^{2}\right)\check{p}_{2}-\frac{P_{0}}{J\omega_{r}}=0,$

where $\mathbf{p}=\begin{bmatrix}p_{1}&\check{p}_{2}&p_{3}\end{bmatrix}$, with
$\check{p}_{2}=\textsf{e}^{-p_{2}\frac{\nu_{0}}{\omega_{r0}}}$, are the
optimization variables, and the vector $\bar{\mathbf{p}}$ contains the
corresponding reported experimental values. The solution gives $p_{1}=5.4148$,
$p_{2}=0.0682$, and $p_{3}=0.029$. Since the wind speed is bounded in the
full-load region as $11.4m/s\leq v\leq 25m/s$, and the pitch angle is also
bounded as $\theta\in[0^{\circ},90^{\circ}]$, the upper bound parameters are
calculated as $\bar{\rho}_{\nu}=1$, $\bar{\rho}_{\omega}=1.5$. Moreover,
considering that
$\bm{\rho}_{\theta}=\frac{-2\kappa\eta_{\nu}^{3}p_{3}}{3J\eta_{\omega}}\textsf{e}^{\left(-p_{2}\frac{\eta_{\nu}}{\eta_{\omega}}\right)}\bm{\eta}_{\theta}$,
and choosing $\bm{\rho}_{0}=[-1\hskip 5.69054pt-1\hskip 5.69054pt-1\hskip
5.69054pt]^{\top}$, then $\bm{\rho}_{\theta}^{\top}\bm{\rho}_{0}\geq 0.15$ is
obtained. Consequently, for a given $\gamma=0.25$, and a fixed $\psi=0.5$, and
then applying the sufficient condition in Eqn. (17) gives the condition for
the control gain as $k\geq 54.17$ in the high-level control loop, which the
choice of $k=55$ is chosen.  Regarding (17), decreasing $\gamma$ increases the
gain $k$, and since
$\left\|\widetilde{\omega}_{r}\right\|\leq\left\|\sigma\right\|\leq\gamma\left\|\widetilde{v}\right\|$,
decreasing $\gamma$ attenuates the impact of the wind disturbance on the rotor
speed. Hence, increasing $k$ decreases the rotor speed fluctuations against
the wind disturbance. The design parameters for the low-level controller are
chosen such that the sufficient condition in Eqn. (24) is satisfied, then the
design parameters are chosen as $\alpha=0.3$, $k_{\theta}=2.5$. Note that
$k_{\theta}$ adjusts the speed of the error convergence while $\alpha$ tunes
the speed of the parameter estimation. The control structure of the developed
controller is illustrated in Fig. 2. The proposed controller is compared with
the well-tuned PI-gain-scheduling controller (baseline controller) developed
by NREL [17], and an adaptive controller proposed in [11], where in the later
an adaptive directional forgetting scheme is utilized to identify the model.
Two cases, with deterministic and a stochastic wind signals, are used to show
the performance of the proposed control system. The stochastic wind signal has
a mean value of $22m/s$, and the turbulence intensity of $20\%$ produced by
the TurbSim software, [24]. The fault is ramped up in the time interval
$150s-180s$, and then completely activated within $180s-220s$, and ramped down
within $220s-250s$. Figure 3 shows the deterministic wind signal applied to
the WT. Figure 4 shows the rotor speed response subjected to the deterministic
wind signal.

Figure 7: Faulty pitch actuator and the estimated parameter subjected to the
step wind signal Figure 8: Faulty pitch actuator and the estimated parameter
subjected to the stochastic wind signal

It clearly illustrates that the proposed controller outperforms others in
rejecting the actuator fault, and has significantly less fluctuations in the
rotor speed, when the wind speed is changing. Figure 5 shows the stochastic
wind signal applied to the WT. Figure. 6 shows the rotor speed response, and
the control authority to the stochastic wind signal. It shows that other
methods have huge fluctuations between 300s-450s due to a sudden drop in the
wind speed. Note that the rotor speed responses are noisy.

Figure. 7 (deterministic), and Fig. 8 (stochastic) illustrate the results of
the low-level closed-loop system. The results show that when the fault
happens, the adaptive parameter estimation starts to increase, and when the
fault is linearly vanishing, at the same time, the adaptive parameter
converges to its faultless value. Hence, the adaptive mechanism rejects this
time-varying-incipient fault. Also, Fig. 8 shows that the proposed controller
has more pitch activities to deal with the stochastic wind signal. Table 1
shows the root mean square (rms) of the rotor speed error with respect to the
PI-gain-scheduling controller. It shows that the developed controller
significantly reduces the rotor speed tracking error. Note that the numbers
are relative to that of the NREL’s baseline controller and does not represent
the absolute error for each controller. This was done because the absolute
numbers for the proposed controller are really small. Table 2 compares the rms
of the error of the rotor speed with respect to the fault-free baseline
controller during the time that the fault can happen, i.e, between
$150sec-250sec$. Table 2 shows that the rms of the error for the proposed
controller is not only the smallest, but also has no increment when fault
occurs compared to other controllers. Figure 9 compares the rms of the rotor
speed error subjected to stochastic wind signals with different mean values,
the TI of $20\%$, and $24\times 24$ girds. It shows that the proposed
controller attenuates the wind disturbance and has very small increment in the
rms of the error when the wind speed increases.

Table 1: Relative Rms of the rotor speed error compared with NREL’s baseline controller Wind type | Deterministic | Stochastic
---|---|---
Baseline Controller [17] | 100% | 100%
Adaptive Controller [11] | 142.68% | 175.59%
Proposed Controller | 29.85% | 28.97%
Table 2: Relative Rms of the rotor speed error compared with the fault-free NREL’s baseline controller subjected to stochastic wind signal Fault condition | Fault-free | Faulty
---|---|---
Baseline Controller [17] | 100% | 102.65%
Adaptive Controller [11] | 136.34% | 141.91%
Proposed Controller | 34.22% | 34.22%
Figure 9: Root means square of the rotor speed tracking error subjected to
stochastic wind signals at different operating points

##  6 Conclusion

This paper has proposed a nonlinear robust-adaptive controller for a WT with
faulty pitch actuators. It was shown that the proposed $\mathcal{L}_{2}$
controller provides a finite-gain $\mathcal{L}_{2}$ stable mapping from the
wind disturbance to the rotor speed tracking error and the proposed adaptation
mechanism ensures a globally asymptotically stable pitch angle error in the
low-level layer with time-varying uncertainty in the control input
effectiveness. Simulation results show a considerable reduction in
fluctuations of the output in spite of exposure to an unmeasurable largely
fluctuating wind disturbance and time-varying actuator faults.

##  7 Future Work

While this paper designed an effective robust-adaptive controller, it did not
address the problem for individual actuators. Thus, to improve the efficiency
of the controller, faulty actuators should be indentified, and then a splitter
should be designed to distribute the control authority from the high-level
control to the low-level layer based on the degree of the faults. In addition,
more aggressive faults, and actuator failure, will be considered.

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