
The year 2022 is characterized by a generalized energy crisis, which leads to steadily increasing electricity prices around the world, while the corresponding salaries remain stable. Therefore, examining trends in electricity prices relative to existing income levels can provide valuable insights into the overpricing/underpricing of energy consumption. In this article, we examine the tendencies of 35 European countries according to their national kWh prices and the average household incomes. We use a series of established clustering methods that leverage available information to reveal price and income patterns across Europe. We obtain important information on the balance between family earnings and electricity prices in each European country and are able to identify countries and regions that offer the most and least favorable economic conditions based on these two characteristics studied. Our analysis reveals the existence of four price and income patterns that reflect geographical differences across Europe. Countries such as Iceland, Norway, and Luxembourg exhibit the most favorable balance between prices and earnings. Conversely, electricity prices appear to be overpriced in many southern and eastern countries, with Portugal being the most prominent example of this phenomenon. In general, average household incomes become more satisfactory for European citizens as we move from east to west and south to north. In contrast, the respective national electricity prices do not follow this geographical pattern, leading to notable imbalances. After identifying significant cases of inflated prices, we investigate the respective causes of the observed situation with the aim of explaining this extreme behavior with exogenous factors. Finally, it becomes clear that the recent increase in energy prices should not be considered as a completely unexpected event, but rather as a phenomenon that has occurred and developed gradually over the years.
Citation: Dimitrios Saligkaras, Vasileios E. Papageorgiou. On the detection of patterns in electricity prices across European countries: An unsupervised machine learning approach[J]. AIMS Energy, 2022, 10(6): 1146-1164. doi: 10.3934/energy.2022054
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The year 2022 is characterized by a generalized energy crisis, which leads to steadily increasing electricity prices around the world, while the corresponding salaries remain stable. Therefore, examining trends in electricity prices relative to existing income levels can provide valuable insights into the overpricing/underpricing of energy consumption. In this article, we examine the tendencies of 35 European countries according to their national kWh prices and the average household incomes. We use a series of established clustering methods that leverage available information to reveal price and income patterns across Europe. We obtain important information on the balance between family earnings and electricity prices in each European country and are able to identify countries and regions that offer the most and least favorable economic conditions based on these two characteristics studied. Our analysis reveals the existence of four price and income patterns that reflect geographical differences across Europe. Countries such as Iceland, Norway, and Luxembourg exhibit the most favorable balance between prices and earnings. Conversely, electricity prices appear to be overpriced in many southern and eastern countries, with Portugal being the most prominent example of this phenomenon. In general, average household incomes become more satisfactory for European citizens as we move from east to west and south to north. In contrast, the respective national electricity prices do not follow this geographical pattern, leading to notable imbalances. After identifying significant cases of inflated prices, we investigate the respective causes of the observed situation with the aim of explaining this extreme behavior with exogenous factors. Finally, it becomes clear that the recent increase in energy prices should not be considered as a completely unexpected event, but rather as a phenomenon that has occurred and developed gradually over the years.
This work is motivated by the networked control system (NCS) or remote control system where the media that connects the controller and actuator causes random packet delays or dropouts. This type of control systems are anticipated to have vast applications. The readers are referred to the review paper [1]. The common features of NCS are random packet delays or dropouts, competition of multiple nodes in the network, data quantization, etc., and this makes the modeling and stability analysis very challenging. Markovian type regime switching models have been proved to be very successful in modeling NCS [2,3]. However, the stability of such systems still remains a challenge. Most stability criteria are given as complicated LMI conditions [4,5] via Lyapunov function approach, or dwell time [6], or delay Riccati equations [7]. A stability condition using hybrid system analysis is found in [8], which is again a sufficient condition and is not easy to verify. A neat sufficient and necessary condition is in great need.
In order to overcome the constraints of NCS, such as random packet delays or dropouts, it is a natural idea to estimate system state and compensate for packet disorders [9]. One believes that in this way better system performance and better system stability can be achieved. Predictive control has been effectively applied to NCS [10,11,12,13]. This control method generates a sequence of future control variables, which can be used to compensate for packet disorder or dropouts [14], and to estimate unknown system state as well [15,16]. However, as mentioned earlier, the stability conditions are usually hard to verify. To analyze the stability issue, one must have a thorough understanding of the dynamics of NCS with random time delays and state estimate.
In this paper we propose a predictive control method of NCS with random packet delays and state estimate/compensation, and clearly show the structure of this system. This control method has much less computational burden compared to classical predictive control method. Then the system is modeled as a regime switching system, and an upper bound on the number of regimes is provided. By the word ``scale'' we mean the number of distinct regimes (denoted by
Throughout this paper we use
Unlike most NCSs where the objective systems on the actuator side are time invariant, here we assume that the objective system is a regime switching system with Markovian jumps, and we call it a Markov Jump System (MJS). The application of switching systems in engineering can be found, for example, in [17,18,19]. We let
We begin with simple and practically reasonable assumptions on this system.
● The actuator receives signals with random delays.
The actuator receives control signals from the controller and performs the required actions. Due to the random packet delays, the actuator might receive no signal, or receive a packet that is randomly delayed, or receive multiple packets. If more than one packet arrive at the same time, the actuator executes the most recent one.
● Obsolete signals are discarded.
If, at a certain time instant, a control variable has been applied, then any control signal older than it but arrives at later times will be discarded. For example, if
● The delay times are bounded.
The backward time delay
● The controller sends out signals at every discrete instant.
The controller receives state feedback from the actuator via randomly delayed packets, and the controller estimates the future system state and sends out a new control signal. If no state feedback is received at a certain time instant, the controller simply sends a zero control signal.
● Past control signals are recorded.
The controller keeps a record of past control signals that have been sent. This information is used to calculate a new control signal whenever a feedback packet arrives.
● Packet size is limited.
The network bandwidth is very limited so that each time the controller and actuator just send out one small packet to the other party. That means, sending out a large packet containing a long sequence of controls or system states is not applicable. (Otherwise this system reduces to classical control system without time delay.)
The controlled system on the actuator side becomes
xk+1=a(ik)xk+b(j)(ik)uk−τ2, | (1.1) |
where
The NCS is a special dynamical system that might be easy to describe in words, but is hard to understand because its dynamics looks chaotic. One must get down to the bottom and have a careful dissemination to obtain a clear understanding. In what follows we shall do this work.
We use the triplet
If no control signal arrives at time
The expression
We use a similar mechanism to represent the situation at the controller side. The pair
Notice that the received information depends on the sequence of the past states
Proposition 2.1. An upper bound on the number of different regimes of this problem (denoted by
m2+T1⋅(1+T2)2+T1⋅(2+T2)/2⋅(1+T1)⋅(2+T1)/2 | (2.1) |
Proof. On the controller side we consider the pair
Similarly, on the actuator side, any pair
Due to the feedback delay, a sequence with length
The controller then calculates and sends out a new control variable
However, recall that not all the transitions
m⋅(1+T2)⋅(2+T2)/2⋅[m⋅(1+T2)]1+T1=m2+T1⋅(1+T2)2+T1⋅(2+T2)/2. |
Now the proof is complete.
It is clear that to represent the complete system scenario, we need
In each scenario in
zk=[xk(ik),xk−1(ik−1),...,xk−T1(ik−T1),uk−1,uk−2,...,uk−T1−T2]T, | (3.1) |
where the superscript
zk+1=A(C(k)2,τ(k)2,ik)zk+B(C(k)2,τ(k)2,ik)uk, | (3.2) |
which is an extension of (1.1). Since each entry in (3.1) is necessary to describe the system dynamics in a certain regime, we are clear that
If
A(C(k)2,τ(k)2,ik)=(a(ik) 0 0 0 0⋯1 0 0 0 0⋯0 1 0 0 0⋯⋯⋯⋯⋯⋯⋯0(∗)⋯ 0 0 0⋯0⋯ 0 1 0⋯⋯⋯⋯⋯⋯⋯),B(C(k)2,τ(k)2,ik)=[0 0 0⋯ 1(∗) 0 0⋯]T, |
where
If
If
A(0,τ(k)2,ik)=(a(ik) 0⋯b(ik) 0 0⋯1 0 0 0 0 0⋯0 1 0 0 0 0⋯⋯⋯⋯⋯⋯⋯⋯0(∗)⋯ 0 0 0 0⋯0⋯ 0 1 0 0⋯⋯⋯⋯⋯⋯⋯⋯),B(0,τ(k)2,ik)=[0 0 0⋯ 1(∗) 0 0⋯]T, |
where the position of
On the controller side we have
uk=−F(C(k)1,τ(k)1,C(k−τ(k)1−1)2,τ(k−τ(k)1−1)2,ik−τ(k)1−1)zk. | (3.3) |
If
uk−τ(k)1−τ(k−τ(k)1−1)2,uk−τ(k)1−τ(k−τ(k)1−1)2+1,...,uk−1, | (3.4) |
the controller is able to estimate the system state
Recall that if the controller does not receive any feedback packet at present time
We need to point out that there is much flexibility in calculating the new control variable, which might be the most resourceful research in NCS. In this paper we present our control method as follows. Firstly, based on the most recent information
ˆxk−τ(k)1+1=a(ˆik−τ(k)1)xk−τ(k)1+b(ˆik−τ(k)1)uk−τ(k)1−τ(k−τ(k)1−1)2, | (3.5) |
and
ˆxk−τ(k)1+2=a(ˆik−τ(k)1+1)ˆxk−τ(k)1+1+b(ˆik−τ(k)1+1)uk−τ(k)1−τ(k−τ(k)1−1)2+1, | (3.6) |
and so forth until
One may notice that the system mode information
Based on the probability transition matrix
To calculate a new control
ˆik−τ(k)1,ˆik−τ(k)1+1,⋯,ˆik+τ(k−τ(k)1−1)2−1,ˆik+τ(k−τ(k)1−1)2,..,ˆik−τ(k)1+L−1, | (3.7) |
where
J(ˆxk+τ2,ˆik+τ(k−τ(k)1−1)2−1,τ(k)1,τ(k−τ(k)1−1)2)=L−τ(k)1∑j=τ(k−τ(k)1−1)2+1ˆxTk+jQ(k+j)ˆxk+j+L−τ(k)1−τ(k−τ(k)1−1)2−1∑j=0ˆuTk+jR(k+j)ˆuk+j, | (3.8) |
where
Unlike the usual cost function which is the expected value along all possible future paths, this cost function (3.8) is calculated along the deterministic path (3.7), and because the optimization is performed on one path, a huge amount of calculation is saved. To see this in more details, let us consider a control method which chooses future controls
J(ˆxk+τ2,ˆik+τ(k−τ(k)1−1)2−1,τ(k)1,τ(k−τ(k)1−1)2)=L−τ(k)1∑j=τ(k−τ(k)1−1)2+1E[ˆxTk+jQ(k+j)ˆxk+j|xk−τ(k)1]+L−τ(k)1−τ(k−τ(k)1−1)2−1∑j=0ˆuTk+jR(k+j)ˆuk+j, | (3.9) |
where
But for our proposed simplified predictive control method, a natural question remains: what if the path we predicted does not actually occur? Remember that so far we have not used rolling optimization which is the true power of RHC. The calculation of future control sequences is repeated at each sample instant, and if a prediction error occurs, it will be corrected by re-planning at the next sample instant.
Actually we can make this calculation faster by noticing that the control format in RHC with quadratic criterion function (3.8) is given by linear feedback form, see, e.g., [20]. We borrow the iterative algorithm in [20] for the calculation of state feedback control in time varying case, then we obtain
uk=−F(ˆik+τ(k−τ(k)1−1)2−1,τ(k)1,τ(k−τ(k)1−1)2)ˆxk+τ(k−τ(k)1−1)2. | (3.10) |
Recall that
Dk−τ(k)1+1=(0 ⋯0 a(ˆik−τ(k)1) 0 ⋯0 b(ˆik−τ(k)1)⋯0 ⋯⋯ ⋯ ⋯ 0 1 0⋯0 ⋯⋯ ⋯⋯ 1 0 0⋯⋯⋯⋯ ⋯⋯⋯⋯ ⋯⋯), |
where
Dk−τ(k)1+1zk=(ˆxk−τ(k)1+1 uk−τ(k)1−τ(k−τ(k)1−1)2+1⋯⋯uk−1)T |
by (3.5).
Now we define
Dk−τ(k)1+2=(a(ˆik−τ(k)1+1) b(ˆik−τ(k)1+1)0 0⋯0 01 0⋯0 00 1⋯⋯ ⋯⋯ ⋯⋯), |
and get
Dk−τ(k)1+2Dk−τ(k)1+1zk=(ˆxk−τ(k)1+2 uk−τ(k)1−τ(k−τ(k)1−1)2+2 ⋯uk−1)T |
by (3.6). Continue this process until we reach
Dk+τ(k−τ(k)1−1)2⋯Dk−τ(k)1+1zk=ˆxk+τ(k−τ(k)1−1)2. | (3.11) |
Substituting (3.11) in (3.10) yields
uk=−F(ˆik+τ(k−τ(k)1−1)2−1,τ(k)1,τ(k−τ(k)1−1)2)⋅Dk+τ(k−τ(k)1−1)2⋯Dk−τ(k)1+1zk. |
Since
F(C(k)1,τ(k)1,C(k−τ(k)1−1)2,τ(k−τ(k)1−1)2,ik−τ(k)1−1)=F(ˆik+τ(k−τ(k)1−1)2−1,τ(k)1,τ(k−τ(k)1−1)2)⋅Dk+τ(k−τ(k)1−1)2⋯Dk−τ(k)1+1. | (3.12) |
In the next section we shall investigate the stability of this control algorithm.
Putting (3.3) in (3.2) yields
zk+1=A(C(k)2,τ(k)2,ik)zk−B(C(k)2,τ(k)2,ik)⋅F(C(k)1,τ(k)1,C(k−τ(k)1−1)2,τ(k−τ(k)1−1)2,ik−τ(k)1−1)zk. | (4.1) |
Denote
H=H(C(k)2,τ(k)2,ik,C(k)1,τ(k)1,C(k−τ(k)1−1)2,τ(k−τ(k)1−1)2,ik−τ(k)1−1)=A(C(k)2,τ(k)2,ik)−B(C(k)2,τ(k)2,ik)⋅F(C(k)1,τ(k)1,C(k−τ(k)1−1)2,τ(k−τ(k)1−1)2,ik−τ(k)1−1), |
then we obtain the following dynamics
zk+1=H⋅zk, | (4.2) |
where
The usual stability criteria on MJSs are mean convergence (MC) and mean square convergence (MSC). We say that a system is uniformly MC if
F=diag(H)⋅(P′⊗Id),A=diag(H⊗H)⋅(P′⊗Id2), | (4.3) |
where
Theorem 4.1. The MJS (4.1) is uniformly MC if and only if
Proof. The proof can be found in [22]. Here we just point out that the matrices
In the numerical example we shall test both stabilities.
Remark 4.1. We are able to provide this sufficient and necessary condition on stability, and this is largely due to the fact that the structure of this NCS has been clearly revealed.
For the complete representation
(C(k)2,τ(k)2,ik,C(k)1,τ(k)1)→(C(k+1)2,τ(k+1)2,ik+1,C(k+1)1,τ(k+1)1). | (5.1) |
As we mentioned earlier, the regimes do not have full reachability, but the transition (5.1) can be decomposed into three parts that are treated independently. The first part
We consider an example where the objective system, which is a MJS, has two modes (
[x(1)k+1x(2)k+1]=[1.20.2 10 ][x(1)kx(2)k]+[0.05 0.1]uk−τ2, |
and
[x(1)k+1x(2)k+1]=[0.70.2 10 ][x(1)kx(2)k]+[0.10.3]uk−τ2, |
respectively. This system is controlled by a controller through a network with random time delays. It is assumed that
The matrix
Pm=(0.8 0.20.2 0.8). |
To construct
(0,0)(0,1) (0,2)(1,0) (1,1)(2,0)(0,0) 0.70 00.3 00 (0,1) 0.40.4 00 0.20 (0,2) 0.50.2 0.30 00 (1,0) 0.30.4 00 00.3 (1,1) 0.50.2 0.30 00 (2,0) 0.40.2 0.40 00 | (6.1) |
Now that
P1=(0.6 0 0.40.5 0.5 00.7 0.3 0). |
By the derivation in this paper, we need to keep
(0,0)→(0,0)→(0,0)(0,0)→(0,0)→(1,0)(0,0)→(1,0)→(0,0)(0,0)→(1,0)→(0,1)(0,0)→(1,0)→(2,0)⋯⋯⋯, |
and the total number of all possible 3-pair paths is 45. The estimate from (2.1) on this part is given by
The vector
Without any control we have
We choose the prediction horizon
The MJS stability certainly depends on the transition probabilities among the regimes. For instance, if we use
Pm=(0.85 0.150.25 0.75),P1=(0.7 0 0.30.6 0.4 00.8 0.2 0), |
and
P2=(0.8 0 0 0.2 0 00.5 0.3 0 00.2 00.6 0.3 0.1 00 00.4 0.30 00 0.30.5 0.20.3 00 00.7 0.20.1 00 0), |
then again, without control we have
Then we apply the proposed predictive control method and have
To overcome the constraints of NCS such as random packet delays/disorders, it is a natural idea to estimate the system states and send out a control signal that compensates the time delay, hence the expectation of a better control performance. In this paper we modeled the NCS as a regime switching system and proposed a simplified predictive control method. The regime estimate (2.1), which is one of the main contributions of this paper, illustrates that this seemingly small system actually has a very large scale. The structure of this system has been clearly revealed, and a concise sufficient and necessary condition on stability is obtained. Numerical examples clearly show the features of this dynamical system.
This work is supported by Barrios Technology Faculty Fellowship from University of Houston - Clear Lake, 2017-2018.
The author declares no conflict of interest in this paper.
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