A Comparative Analysis of Key Features for Tracing the Causes of Abnormal Electricity Prices in Market Environments

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A Comparative Analysis of Key Features for Tracing the Causes of Abnormal Electricity Prices in Market Environments

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Siting Dai,Wenmin Wu,Xiaolong Zhang,Weixin Hong,

Yan Li^*

Siting Dai,Wenmin Wu,Xiaolong Zhang,Weixin Hong,

Yan Li^*

This version is not peer-reviewed

Submitted:

29 July 2024

Posted:

30 July 2024

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Abstract

Electricity price signal is a direct reflection of the attributes of electric power commodities, but due to the influence of market supply and demand, power producers' offer, line capacity and other factors, electricity price signals often show a variety of abnormal forms. The identification and traceability of abnormal tariff signals is an important daily work of power trading centers at all levels. However, at present, it is common to rely on manual experience to analyze the causes of abnormal tariffs, which is inefficient and difficult to ensure the objective and scientific traceability of abnormal tariff causes. Since different types of abnormal tariff signals are caused by different dominant factors, if the key features of abnormal tariffs can be classified and matched, the identification efficiency can be effectively improved. For this reason, this paper proposes a traceability method of abnormal tariffs based on the comparison analysis of key features. First of all, based on the characteristics of historical tariff data, this paper completes the classification of tariff spike magnitude anomaly and tariff mean value anomaly, further establishes the key features of each type of abnormal tariff signal based on principal component analysis, and finally calculates the influence degree of each element within the key features one by one based on alternative algorithms, and then realizes the screening and traceability of the causes of abnormal tariffs based on the importance of the degree of influence ordering. The effectiveness of the proposed method has been verified in a large number of examples constructed based on the actual data of the electricity market, and the average correctness rate of the proposed method for the traceability of the causes of the average value and spike anomalies of the electricity price reaches more than 85%, which can reduce the labor cost in the process of the traceability of the causes of the anomalous electricity price.

Keywords:

Subject: Engineering - Electrical and Electronic Engineering

1. Introduction

The construction of the electricity market plays an important role in guiding the optimal allocation of electricity resources [1,2]. The power market construction plays an important role in guiding the optimal allocation of power resources. Major countries around the world have carried out power market reforms to better cope with the transformation of energy structure [3,4]. Electricity pricing is a key element of the electricity market that guides power resources. Electricity price, as a means to guide the distribution of electricity resources in the electricity market, is a direct reflection of the commodity attributes and market value of electricity. Since electricity price is subject to market supply [5], load demand [6], market structure [7], market forces [8], market policy[9] etc., changes in any one of the influencing factors may trigger changes in electricity prices, or even the formation of abnormal electricity prices, thus negatively affecting the credibility of the construction of the electricity market[10]. The prolonged and drastic fluctuations in electricity price signals will greatly increase the operational risks of the electricity market, thus hindering the enthusiasm of various market players to participate in the operation of the electricity market[11]. The market will be affected by a number of factors, including the following In particular, abnormal tariffs formed by abnormal offers of power producers and imperfect market policies can send wrong economic signals, thus reducing the allocation of market resources[12] and reduce the efficiency of market resource allocation[13]. In particular, abnormal tariffs formed by abnormal offers of power producers and perfect market policies can send wrong economic signals and reduce the efficiency of market resource allocation, or even jeopardize the safety of power system operation[14]. The market policy is perfect, and the formation of abnormal tariffs can send wrong economic signals. Therefore, it is necessary to systematically analyze the causes of abnormal tariffs in order to subsequently carry out targeted rectification of the power market clearing process, which is crucial to the improvement of the power market construction[15]. This is a crucial role in improving the construction of the power market.

At present, a large number of studies have been conducted at home and abroad on the analysis of the influencing factors of abnormal electricity prices. The existing methods can be roughly divided into two categories, one is to analyze for specific influencing factors, as discussed in [16,17] Focusing on the load as a single influencing factor, the correlation between different characteristic loads and electricity price is analyzed through the clustering method; researchers in [7] is to assess the impact of different generation resources on electricity market price through statistical methods, which includes correlation analysis methods and regression analysis methods, etc. The other category is to study the various types of influencing factors on electricity price as a whole. Scholars in [18] analyzes the relationship between electricity demand as well as electricity price and various influencing factors by means of graphical modeling method, which has the advantage of visualizing the direct or indirect influencing factors of electricity price; the study in [19] analyzes the determining influencing factors from the many influencing factors of the market price through the principal component analysis method. In order to realize the traceability of abnormal tariffs, the purpose of analyzing the key influencing factors of tariffs is to obtain the causal bias of different types of abnormal tariffs from multi-dimensional and multi-category influencing factors, so it is necessary to construct an overall analysis framework that can effectively consider all the influencing factors. At the same time, due to the fact that there must be differences between the sets of key influencing factors of different types of abnormal tariffs, the existing methods are mostly based on the characteristics of tariffs directly, ignoring the differences in the characteristics of different types of abnormal tariffs, which are very important in the subsequent analysis of the causes of abnormal tariffs, and therefore, the existing methods need to be improved.

Tracing the causes of abnormal electricity prices is essentially a process of finding the correspondence between the influencing factors and the electricity price signals, and realizing the process of inverting the corresponding influencing factors from the characteristics of the electricity price signals. Methods such as data-driven and machine learning are well suited to this goal, and relevant methods for analyzing price influencing factors have been proposed in the literature. For example, researchers in [20] proposes a machine learning-based framework for identifying the main influencing factors of price spikes in high percentage renewable electricity markets, on the other hand, proposes a data-driven based approach to analyze in detail the electricity price influencing factors in the PJM market [21]. However, both machine learning methods and data-driven methods rely on a large amount of historical data and require processing of historical data as well as analysis of influencing factors to provide sufficient samples, which cannot work satisfactorily in the absence of sufficient data or information for causal analysis. Another approach is to analyze the causes of abnormal electricity prices by modeling the economic behavior of the electricity market [22,23]. However, as we can see from the above, there are many influencing factors of electricity price, and it is difficult to analyze the multi-dimensional and multi-faceted influencing factors comprehensively and efficiently by relying only on the economic model or the market clearing model to trace the causes of abnormal electricity price.

In order to make up for the shortcomings of the existing methods in the analysis and traceability of the causes of abnormal tariffs, this paper proposes a traceability method for the causes of abnormal tariffs based on the comparative analysis of key influencing factors, which realizes the efficient traceability of the causes of abnormal tariff signals in the case of not being able to obtain sufficient data information. First, this paper establishes a market clearing model and simulates the electricity price signals under multiple scenarios according to the model parameters, and then organizes and classifies the simulated electricity price data according to the characteristics of the electricity price signals to form datasets with different types of abnormal electricity price signals. In each type of dataset, the corresponding set of causes of this type of abnormal tariff signal is obtained by principal component analysis. On this basis, the alternative algorithm calculates the influence degree of each element in the set of causes one by one and ranks them to realize the cause screening. Finally, the effectiveness of the cause screening method proposed in this paper is demonstrated through case analysis.

2. Electricity Spot Market Price Signal Generation and Assessment

At the present stage, the domestic power market generally adopts the mode of "power generation side quotes, user side quotes without quotes" to organize the day-ahead power energy market transactions. The former electric energy market adopts the mode of full energy declaration and centralized optimization and clearing, and the market units declare the offer information of the operation day in the former electric energy market, among which, the new energy trading units also need to declare the short-term power forecast information. Power sales companies and wholesale users declare the electricity demand curve on the day of operation in the former energy market, without declaring the price. The power dispatching agency takes into account the load forecast, the outgoing transmission and receiving curve, the non-market unit output curve, the generating unit maintenance plan, the generating unit operation constraints, the grid safety operation constraints and other factors, and takes the maximization of social welfare as the optimization objective, and clears the time-sharing generation output curve of the operating day and the time-sharing nodal tariff. The day-ahead electric energy market adopts the nodal tariff (LMP) pricing mechanism, and its calculation model is shown below.

2.1. Nodal Tariff Calculation Model

(1) Objective function

The node tariff calculation model makes the overall system operating cost minimum through optimization calculation, and the objective function is shown in Eq.(1):

\min \sum_{i = 1}^{N^{G}} \sum_{t = 1}^{T} (c_{i, t} P_{i, t}) + \sum_{l = 1}^{N^{L}} \sum_{t = 1}^{T} M^{pen} (L_{l, t}^{+} + L_{l, t}^{-})

(1)

Where N^G, N^G and

N^{L}

denote the total number of market units and the total number of system branches, respectively;

T

denotes the total number of time slots considered in the day-ahead electric energy market one-time out calculation;

c_{i, t}

denotes the declared price of market unit i in time slot t;

P_{i, t}

denotes the output of market unit i in time slot t;

M^{pen}

denotes the network tidal current constraint relaxation penalty factor used for the nodal tariff calculation;

L_{l, t}^{+}

and

L_{l, t}^{-}

denotes the forward and reverse tidal current slack variables for branch l in time period t.

(2) System load balance constraints

\sum_{i = 1}^{N^{G}} P_{i, t} = \sum_{k = 1}^{N^{B}} D_{k, t} \forall t \in T

(2)

Where

N^{B}

denotes the total number of nodes in the system;

D_{j, t}

denotes the load of node j at time period t.

(3) Upper and lower unit output constraints

P_{i, t}^{\min} u_{i, t} \leq P_{i, t} \leq P_{i, t}^{\max} u_{i, t} \forall i \in N^{G}, \forall t \in T

(3)

Where

P_{i, t}^{\min}

and

P_{i, t}^{\max}

are the upper and lower limits of the rated output of market unit i at time period t, respectively;

u_{i, t}

characterizes the start-stop status of market unit i at time period t, which is 0 if it is stopped and 1 otherwise.

(4) Unit Climbing Constraints

\{\begin{cases} P_{i, t} - P_{i, t - 1} \leq Δ P_{i}^{U} \forall i \in N^{G}, \forall t = 2, K, T \\ P_{i, 1} - P_{i, 0} \leq Δ P_{i}^{U} \forall i \in N^{G} \end{cases}

(4)

\{\begin{cases} P_{i, t} - P_{i, t - 1} \geq - Δ P_{i}^{D} \forall i \in N^{G}, \forall t = 2, K, T \\ P_{i, 1} - P_{i, 0} \geq - Δ P_{i}^{D} \forall i \in N^{G} \end{cases}

(5)

Where

Δ P_{i}^{U}

and

Δ P_{i}^{D}

denote the maximum uphill climb rate and maximum downhill climb rate of market unit i, respectively.

(5) Branch circuit tidal current constraints

\begin{array}{l} - P_{l}^{\max} \leq \sum_{i = 1}^{N^{G}} G_{l - i} P_{i, t} + \sum_{k = 1}^{N^{B}} G_{l - k} D_{k, t} - L_{l, t}^{+} + L_{l, t}^{-} \leq P_{l}^{\max} \\ \forall l \in N^{L} \end{array}

(6)

Where

P_{l}^{\max}

denotes the current transmission limit of branch l;

N^{L}

denotes the total number of branches in the system;

G_{l - i}

denotes the power transfer distribution factor of the node where market unit i is located to line l;

G_{l - k}

denotes the power transfer distribution factor of node k to line l.

(6) Nodal tariff calculation

After solving the above day-ahead electricity energy market clearing model, the pairwise multipliers of the system load balance constraints and branch flow constraints for each time period can be obtained, and then the tariff of node i at time period t can be given by Eq.(7) :

λ_{k, t} = μ_{t} - \sum_{l = 1}^{N^{L}} (σ_{l, t}^{\max} - σ_{l, t}^{\min}) G_{l - k}

(7)

Where

λ_{k, t}

denotes the tariff of node k at time period t;

μ_{t}

denotes the dyadic multiplier of the system load balance constraint at time period t;

σ_{l, t}^{\max}

and

σ_{l, t}^{\min}

denote the dyadic multipliers of the maximum forward and reverse tidal current constraints of branch l, respectively. According to Eq.(6), it can be shown that when the branch tidal current constraint is over the limit, i.e., the branch tidal current slack variables

L_{l, t}^{+}

and

L_{l, t}^{-}

are not zero, the

σ_{l, t}^{\max}

and

σ_{l, t}^{\min}

are the network tidal current constraint relaxation penalty factor, which is set by the power dispatch organization.

2.2. Current Electricity Market Price Assessment Process

In the current domestic successfully operated power market, the anomaly analysis and evaluation of spot electricity price is still mainly based on manual experience, and the overall process is shown in Figure 1. First of all, the power market operation organization will get the market clearing results including nodal tariffs and unit outputs through the market clearing model according to the market clearing rules and market operation boundary conditions. According to the results of market clearing, market traders will observe and classify abnormal electricity prices according to the characteristics of electricity price signals, and match the causes of abnormal electricity price signals according to manual experience and analyze them accordingly, and ultimately, the market regulator will issue a daily report on the electricity spot market and publish it to the society in order to better serve the construction of the electricity spot market.

However, by analyzing the nodal tariffs of the Guangdong electricity spot market in 2022, it can be seen that the frequency of various types of abnormal tariffs is relatively frequent, accounting for nearly 20% of the total number of trading days in the market. At the same time, due to the many factors affecting electricity prices and the fact that the analysis of the causes of abnormal electricity prices mostly relies on manual experience, the results of the traceability analysis of the causes are often unable to comprehensively and accurately cover all the causes, and the analysis process is time-consuming.

2.3. Project Research Ideas

In order to solve the above problems, this paper firstly realizes the cause analysis of abnormal tariff signal through the classification and extraction of key features of abnormal tariff signal, and then through the two steps of cause matching and traceability. The specific idea diagram is shown in Figure 2.

First, based on the historical dataset, the dataset of different categories of abnormal tariff signals is constructed, and the key feature sets of different abnormal tariff signals are extracted by principal component analysis, so as to avoid analyzing the low correlation features in the process of cause screening. Second, according to the characteristics of the abnormal tariff signal to be traced back to the cause of the abnormal tariff signal classification and matching the corresponding set of key features, and then sequentially carry out the impact function calculation and analyze the contribution of each feature to the abnormal signal, and finally based on the size of the contribution of the individual features, to achieve the cause of the screening. The details of the two steps are described in Chapters 2 and 3, respectively.

3. Classification and Extraction of Key Features of Abnormal Tariff Signals

Different types of abnormal tariffs are often caused by different dominant factors, and categorizing abnormal tariffs to extract key features can initially sift out the key directions that need to be traced in terms of causes among a large number of tariff features, and discard the influence of unimportant factors on the efficiency of traceability. This section will introduce the classification of abnormal tariffs and the key feature extraction method.

4.1. Classification of Abnormal Tariffs

There are various types of tariff anomalies, among which tariff spike anomalies, i.e., "price pegs", and tariff average anomalies are the most common ones, which represent sudden changes of tariff signals between time periods and abnormalities of the overall level of tariffs, respectively. Taking these two types of anomalies as examples, we classify the types of tariff anomalies and analyze their causes.

4.1.1. Classification of Tariff Spike Anomalies

First, based on Eq.(7) to obtain the expression of the tariff for each node for each time period, calculate the weighted tariff magnitude of a single time period

λ_{t}^{Value}

as shown in Eq.(8):

λ_{t}^{Value} = (\sum_{k = 1}^{N^{B}} λ_{k, t}) / N^{B}

(8)

And then through the Eq.(9) and Eq.(10) respectively, to calculate the current tariff magnitude and the rate of change of tariffs in the previous and subsequent time periods

η_{t}^{pre}

and

η_{t}^{next}

η_{t}^{pre} = \frac{λ_{t}^{Value} - λ_{t - 1}^{Value}}{λ_{t - 1}^{Value}}, t = 2, 3, K, T

(9)

η_{t}^{next} = \frac{λ_{t}^{Value} - λ_{t + 1}^{Value}}{λ_{t + 1}^{Value}}, t = 1, 2, K, T - 1

(10)

Only

η_{t}^{pre}

and

η_{t}^{next}

both exceed the set threshold before the tariff at time period t can be judged as a spike anomaly.

4.1.2 Average Tariff Anomalies Classification

According to Eq.(11) Calculate the average value of electricity price over a complete dispatch cycle

\bar{λ}

\bar{λ} = \frac{\sum_{t = 1}^{T} λ_{t}^{Value}}{T} = \frac{\sum_{t = 1}^{T} \sum_{k = 1}^{N^{B}} λ_{k, t}}{T \cdot N^{B}}

(11)

Comparing the average value of electricity price

\bar{λ}

compared with the threshold, it is possible to determine whether the electricity price in the corresponding dispatch cycle is an anomaly of the mean value.

4.2. Establishment of Key Feature Set of Abnormal Tariff Based on Principal Component Analysis Method

Assuming that there are n samples for each category of anomalous tariffs, and each sample has p features, the

x_{i i}

denotes the eigenvalue of the jth feature of the ith sample, so that the feature matrix

X

of the sample can be obtained by Eq.(12):

X = [\begin{matrix} x_{11} & x_{12} & L & x_{1 p} \\ x_{21} & x_{22} & L & x_{2 p} \\ M & M & O & M \\ x_{n 1} & x_{n 1} & L & x_{n p} \end{matrix}] = [x_{1}, x_{2}, L, x_{p}]

(12)

The purpose of principal component analysis is to find a transformation matrix

L

, thus transforming the original p features into m features with smaller dimensions, and realizing feature dimensionality reduction while guaranteeing the characteristics of the data. The obtained feature matrix of the data after dimensionality reduction is

Z

, the process is shown in Eq.(13):

\begin{array}{l} Z = X L \\ \Rightarrow {[\begin{matrix} z_{1} \\ z_{2} \\ M \\ z_{m} \end{matrix}]}^{⊤} = {[\begin{matrix} x_{1} \\ x_{2} \\ M \\ x_{p} \end{matrix}]}^{⊤} [\begin{matrix} l_{11} & l_{12} & L & l_{1 m} \\ l_{21} & l_{22} & L & l_{2 m} \\ M & M & O & M \\ x_{p 1} & x_{p 2} & L & x_{p m} \end{matrix}] \\ \Rightarrow \{\begin{cases} z_{1} = l_{11} x_{1} + l_{21} x_{2} + L + l_{p 1} x_{p} \\ z_{2} = l_{12} x_{1} + l_{22} x_{2} + L + l_{p 2} x_{p} \\ M \\ z_{m} = l_{1 m} x_{1} + l_{2 m} x_{2} + L + l_{p m} x_{p} \end{cases} \end{array}

(13)

The process of principal component analysis method is shown below:

① Data feature standardization:

Due to the existence of different measures between different dimensions of data features, in the process of subsequent dimensionality reduction processing, in order to ensure that different data dimensions can be fairly handled between them, the data feature matrix

X

needs to be standardized processing. According to Eq.(14), the average value of each feature

\bar{x_{j}}

is calculated. According to Eq.(15) the standard deviation of each feature

S_{j}

is calculated.

\bar{x_{j}} = \frac{1}{n} \sum_{i = 1}^{n} x_{i j}

(14)

S_{j} = \sqrt{\frac{\sum_{i = 1}^{n} {(x_{i j} - \bar{x_{j}})}^{2}}{n - 1}}

(15)

According to the calculation to get the mean and standard deviation, each feature of each sample is standardized by Eq.(16) to obtain the standardized feature matrix

X^{stand}

as shown in Eq.(17).

X_{i j} = \frac{x_{i j} - \bar{x_{j}}}{S_{j}}

(16)

X^{stand} = [\begin{matrix} X_{11} & X_{12} & L & X_{1 p} \\ X_{21} & X_{22} & L & X_{2 p} \\ M & M & O & M \\ X_{n 1} & X_{n 1} & L & X_{n p} \end{matrix}] = [X_{1}, X_{2}, L, X_{p}]

(17)

② Calculate the covariance matrix for the standardized sample:

The covariance between any two data features is calculated through Eq.(18) Then form the covariance matrix shown in Eq.(19).

r_{i j} = \frac{1}{n - 1} \sum_{k = 1}^{n} (X_{k i} - \bar{X_{i}}) (X_{k j} - \bar{X_{j}})

(18)

R = [\begin{matrix} r_{11} & r_{12} & L & r_{1 p} \\ r_{21} & r_{22} & L & r_{2 p} \\ M & M & O & M \\ r_{p 1} & r_{p 1} & L & r_{p p} \end{matrix}]

(19)

③ Calculate the covariance matrix

R

eigenvalues and eigenvectors of the covariance matrix:

Calculate the covariance matrix

R

The eigenvalues of the covariance matrix are sorted in the order of the absolute magnitude of the eigenvalues as shown in Eq.(20) shown. In addition, calculate the eigenvectors corresponding to each eigenvalue and normalize each eigenvector to obtain the eigenvector corresponding to the jth eigenvalue ordered by the size of the absolute value of the eigenvalue, as shown in Eq.(21):

|λ_{1}| \geq |λ_{2}| \geq L \geq |λ_{p}|

(20)

L_{j} = {[L_{1 j}, L_{2 j}, L, L_{p j}]}^{⊤}, j = 1, 2, L, p

(21)

④ Give the principal components:

Based on obtaining the eigenvectors in Eq.(21) and Eq.(13) , the principal component expression after arbitrary dimensionality reduction can be obtained as Eq.(22) shown:

z_{i} = l_{1 i} x_{1} + l_{2 i} x_{2} + L + l_{p i} X_{p}, i = 1, 2, L, p

(22)

⑤ Calculate the contribution of each principal component and the cumulative contribution:

The contribution of the ith principal component

F_{i}

is shown in Eq.(23):

f_{i} = \frac{|λ_{i}|}{\sum_{k = 1}^{p} |λ_{k}|}, i = 1, 2, L, p

(23)

The cumulative contribution of the first i principal components is shown in Eq(24):

F_{i} = \frac{\sum_{j = 1}^{i} |λ_{j}|}{\sum_{k = 1}^{p} |λ_{k}|}, i = 1, 2, L, p

(24)

Usually, we select a few principal components such that the cumulative contribution reaches a set value

η_{PCA}

, which is used to form the key feature set.

After completing the establishment of the key feature set of each abnormal tariff signal type, the corresponding feature set can be matched according to the type of abnormal tariff signal to be analyzed in the subsequent abnormal tariff cause tracing process, so as to filter the features with smaller correlation relationship and determine the features that need to be focused on to avoid low-return inputs.

5. Sensitivity-Based Analysis of the Causes of Abnormal Tariffs

5.1. Impact Function of Key Features on Electricity Prices

After obtaining the key feature set of various types of abnormal tariffs, it is still necessary to focus on the analysis of each feature element within the set. In this process, the sensitivity of the tariffs to the feature elements can be a good reflection of the degree of influence of the feature elements on the tariffs. The prerequisite for obtaining the sensitivity information is to have a comprehensive grasp of the market clearing information, which by nature is not fully disclosed to the society, but as the regulator of the electricity market, it can be requested to be provided by the power dispatch department in the future market operation process. After obtaining the complete market clearing information, the influence function of each characteristic element on the electricity price can be obtained through the method of multi-parameter planning[24] and then analyze the contribution of each element to the abnormal electricity price by sensitivity calculation.

The process of obtaining the influence function of the characteristic elements on the electricity price through the multi-parameter planning method is shown below.

First, for ease of derivation, the optimization problem(1) -(6) can be rewritten in a compact form as Eq.(25) -(26) shown:

\min A w

(25)

s . t . E w \leq b : δ

(26)

Where

w

is the vector of variables;

A

is the matrix of coefficients in the objective function;

E

is the matrix of coefficients in the constraints;

b

is the vector of constraints' constants;

δ

is the vector of dyadic multipliers corresponding to the constraint.

According to the Eq.(25)-(26), the nodal tariff shown in Eq.(7) can be rewritten as Eq.(27) shown:

λ_{k, t} = M_{k, t} δ

(27)

Where

M_{k, t}

denotes the matrix of coefficients corresponding to Eq.(7) the matrix of coefficients corresponding to those in Eq.

The KKT condition of optimization Eq.(25)-(26) is given in Eq.(28) -(30):

A - E^{⊤} δ = 0

(28)

(E w - b) ⊙ δ = 0

(29)

δ \geq 0

(30)

Since the eigenelements exist only in the coefficient matrix

A

and

b

in the coefficients matrix, the following will be discussed in two cases.

①The generator's offer or the branch crossing penalty factor is the feature to be analyzed (i.e., what is changed is the matrix

A

the relevant elements in the matrix)

At this point, according to the Eq.(26) ,(29) The constraints can be categorized into acting constraints and non-functioning constraints, respectively, as Eqs.(31) and(32) shown:

E^{AC} w = b^{AC} : δ^{AC}

(31)

E^{DC} w < b^{DC} : δ^{DC} = 0

(32)

Where

[(E^{AC})^{⊤} (E^{DC})^{⊤}] = E^{⊤}

[(b^{AC})^{⊤} (b^{DC})^{⊤}] = b^{⊤}

[(δ^{AC})^{⊤} (δ^{DC})^{⊤}] = δ^{⊤}

Bring

δ^{DC} = 0

into Eq.(28), eliminating the dyadic multipliers corresponding to the non-functioning constraints, we end up with Eq.(33) :

A = {(E^{AC})}^{⊤} δ^{AC}

(33)

By the optimality theory[24] that

(E^{AC})^{⊤}

is a full rank matrix, Eq.(33) can be obtained by matrix variation as Eq.(34) :

δ^{AC} = {({(E^{AC})}^{⊤})}^{- 1} A

(34)

The features to be analyzed in the matrix are labeled as

A^{*}

. From Eq.(34), it can be seen that the dyadic multipliers that act as constraints are linear functions with respect to the feature

A^{*}

of the matrix. Since the dyadic multiplier of the non-functioning constraint

δ^{DC} = 0

can also be regarded as a linear function of the feature

A^{*}

of a linear function, combining the two leads to Eq.(35) :

δ = k A^{*} + g

(35)

Where

k

and

g

are the vectors obtained by collapsing the transformations.

Substituting Eq.(35) into Eq.(27) can obtain the node tariff expression as shown in Eq.(36):

λ_{k, t} = M_{k, t} (k A^{*} + g)

(36)

From Eq.(36) and the above derivation process, it can be seen that when the optimization problem(25) -(26) of the starting constraint set does not change, the node tariff is a linear function about the feature to be analyzed, and this linear relationship does not change in this scenario, we remember that the feasible domain of such a feature with the starting constraint set not changing is

R_{j}

. Once the starting constraint set changes, a new linear functional relationship exists. The final node tariff can be characterized as a segmented linear function of the feature

A^{k}

as Eq.(37) shown:

λ_{k, t} = \{\begin{cases} f_{1} (A^{*}), \forall A^{*} \in ℜ_{1} \\ L \\ f_{j} (A^{*}), \forall A^{*} \in ℜ_{j} \\ L \\ f_{N} (A^{*}), \forall A^{*} \in ℜ_{N} \end{cases}

(37)

Where

f_{j} (A^{*})

denotes the expression of the jth segmented linear function.

② The system boundary is the feature to be analyzed (i.e., the change is the vector

b

associated elements in the vector)

According to the same derivation process one can obtain Eq.(34). Since the matrix

A

is constant, so the pairwise multipliers corresponding to the starting constraints are also constant, and combined with the pairwise multipliers of the non-functioning constraints

δ^{DC} = 0

, it can be inferred that when the set of starting constraints does not change, the nodal tariff is similarly a constant.

Marks the features to be analyzed in the vector

b

The feature to be analyzed in

b^{*}

when

b^{*}

occurs, the optimization problem (25) -(26) of the starting constraint set changes accordingly, and the corresponding node tariffs change, and the final node tariffs can be characterized as a segmented constant function as shown in Eq.(38):

λ_{k, t} = \{\begin{cases} q_{1}, \forall b^{*} \in ℜ_{1} \\ L \\ q_{j}, \forall b^{*} \in ℜ_{j} \\ L \\ q_{N}, \forall b^{*} \in ℜ_{N} \end{cases}

(38)

Where

q_{j}

denotes the constant value of the node tariff in the jth feasible domain.

Essentially Eq.(38) is a special segmented linear function which, together with Eq.(37) can be combined and characterized by the same formula, as Eq.(39) shown:

λ_{k, t} = \{\begin{cases} f_{1} (x_{i}), \forall x_{i} \in ℜ_{1} \\ L \\ f_{j} (x_{i}), \forall x_{i} \in ℜ_{j} \\ L \\ f_{N} (x_{i}), \forall x_{i} \in ℜ_{N} \end{cases}

(39)

Where

x_{i}

denotes the ith feature in the set of corresponding key features;

f_{j} (x_{i})

denotes the expression of the jth segmented linear function.

With the above analysis, it can be seen that the electricity price at each node is a segmented linear function about the key features, so the weighted average of the electricity price at each time period is also a segmented linear function about the key features. Since the process of obtaining the above functional relationship through the multi-parameter planning method is slow, the sampling method can be used to approximate the functional relationship as shown in Eq.(39).

5.2. Methodological Process of Analyzing the Causes of Abnormal Electricity Prices

After completing the establishment of the key feature sets of different categories of abnormal tariffs, the abnormal tariff signals waiting for causal traceability analysis are matched with the corresponding key feature sets according to the process shown in Figure 3 and the degree of influence is calculated one by one to complete the causal screening.

The specific steps are as follows:

① Data Input: Import the abnormal tariff signal data to be analyzed for causal traceability, as well as the corresponding clearing boundary conditions and clearing model parameter information into the analysis program, and execute step ②;

② Abnormal categorization: categorize the abnormal tariff signals according to the abnormal tariff signal recognition results, and execute steps ③ and ④ synchronously;

③ Normal signal characteristics derivation: Based on the normal tariff signal threshold and the abnormal tariff signal characteristics to be traced back to the cause, deduce the range of normal tariff signal value characteristics, and jump to step ⑥;

④ Key feature set matching: according to the classification results of abnormal tariffs, match the corresponding key feature sets and select features one by one according to the order of importance of the elements in the set;

⑤ Influence function calculation: through the multi-parameter planning method, the nodal tariff can be characterized as a segmented linear function form of the selected parameters and combined with the characteristics of the normal tariff signal data obtained in step ③ for step ⑥;

⑥ Calculation of the degree of impact: Based on the formula(39) the influence function, inverse the value of the corresponding characteristics under the normal tariff data characteristics, calculate the difference between the data before and after, and normalization, perform step ⑦;

⑦Feature traversal judgment: determine whether all elements in the corresponding key feature set have performed the degree of influence calculation, if not, then skip to step ④, otherwise continue to execute step ⑧;

⑧ Sort all the influence degree coefficients, and according to the sorting results, screen the abnormal tariff signal causes.

6. Case Study

The data sources used in this paper are simulated and generated by the IEEE-30 node system through the fluctuation of the clearing boundary parameters, and a total of 15,194 clearing results for 24 time periods are generated. In this section, the key feature sets of the two categories of abnormal tariff signals, namely, tariff spike anomaly and tariff mean anomaly, are shown based on the generated data; the traceability process of the proposed method on the causes of abnormal tariffs is shown for a typical anomaly case; and finally, the accuracy rate of the proposed method is analyzed, which proves the effectiveness of the proposed method of traceability of abnormal tariffs based on the comparative analysis of the key features. Finally, the accuracy rate of the proposed method is analyzed to prove the effectiveness of the method based on key features comparison analysis.

6.1. Explanation of the Construction of the Set of Key Features of Anomalous Tariffs

The 15,000 sets of tariff clearing data obtained from the simulation are classified by the anomaly tariff classification method described in Section 2.1. Since the different manifestations of tariffs under the same anomaly tariff category may be caused by different influential features, we further subdivide the anomalous tariff types by classifying the spike anomalies into the upper and lower spikes. According to the classification results, 12,969 upper-peak tariff periods and 2,0148 lower-peak tariff periods were obtained to form the corresponding dataset, and the key influential features of these two types of anomalies were obtained through principal component analysis, as shown in Table 1.

Similarly, for the tariff mean value anomaly was also subdivided into two categories, mean value too high and mean value too low, respectively, 2270 sets of mean value too high data and 2822 sets of mean value too low data were obtained, and the key impact feature sets of these two categories of anomalies were obtained through principal component analysis, as shown in Table 2. For the category of tariff mean value anomalies, the difference between the key imaging features of different tariff anomaly manifestations is more obvious.

The set of key impact features for each type of abnormal tariff type can be used for subsequent causal traceability analysis, and the specific usage will be shown in the next section.

6.2. Validation of the Effectiveness of the Process of Tracing the Causes of Abnormal Electricity Prices

Taking the data related to a certain abnormal mean value as an example, this paper shows the process of the method proposed in this paper for the cause traceability analysis. The average price of this set of data is 991.06 yuan /MWh. However, according to the statistics of the generated data, the average price of electricity is higher than 300 yuan /MWh in more than 5% of the cases, and this value is recorded as the upper limit of the normal range of the average price.

First of all, according to the key characteristics corresponding to the abnormal type of high mean value and their ranking, the functional characterization of the mean price of each key feature is obtained in turn, as shown in Figure 4

In Figure 4, the average value of the tariff in the "too high average" case is shown as a function of the average net load change, load peak and valley value change, unit capacity change, unit output lower limit change, feeder capacity upper and lower limits change, creepage coefficient change, and feeder tidal current constraint overrun penalty factor change, respectively. The case where the tariff is zero indicates that there is no solution for the clearing procedure in this scenario. Based on this, the sensitivity calculation can be analyzed to calculate the degree of influence of each feature on the mean value of the tariff. In this case, the degree of influence of each feature on the mean value of electricity price is shown in Table 3.

According to the results of the contribution calculation, it can be found that in this "too high average value" anomaly case, the average value of the tariff is the most sensitive to the change of the average net load, followed by the size of the upper and lower limits of the tributary current. It should be noted that, as shown in Figure 4-(e), since the change in the number of times the tributary current constraints are exceeded is not very obvious and it is difficult to analyze the degree of change quantitatively, the upper and lower tributary current constraints are used as a substitute for the analysis. Except for the two influencing characteristics mentioned above, the average value of electricity price in this case is not very sensitive to changes in other characteristics. This is also consistent with the results obtained from manual empirical analysis.

6.3. Verification of the Accuracy of the Traceability of the Causes of Abnormal Tariffs

20 groups of spike anomalies and 20 groups of mean value anomalies are selected from the generated anomalous tariff cases respectively to validate the effectiveness of the proposed method, and the results obtained by manual experience are used as the standard to judge whether the results are accurate or not, and the results of the traceability of the causes of anomalous tariffs are shown in Table 4. and Table 5. respectively.

From Table 4, it can be found that for electricity price spike anomalies, the formation causes are more concentrated, the electricity price signal is more sensitive to the elements in the key influence set, and the proposed method can identify the main influencing factors more sensitively, while the manual empirical method can not do a precise analysis of the data, and therefore the traceability results usually have a wider range. Overall, for electricity price spike anomalies, the traceability results of the proposed method in this paper are consistent with the manual method in about 90% of the cases.

From Table 5, it can be found that for the tariff mean anomaly, the formation causes will be more complicated, and when the tariff mean anomaly is traced through the method proposed in this paper, about 85% of the cases get the same traceability results as the manual empirical method.

In summary, the comprehensive traceability accuracy of the method proposed in this paper reaches more than 85%, and at the same time, relative to the manual empirical method, it can give a scientific and objective ordering of the contribution of the influencing factors, so that the traceability results of the causes of abnormal electricity prices can be based on the evidence and traces.

8. Conclusion

The traceability method of the causes of abnormal tariffs based on the comparative analysis of key features proposed in this paper makes full use of the difference relationship between the causes of tariff types in different anomalies, and establishes the key influencing feature sets of different abnormal tariff types through the method of principal component analysis, which avoids the unnecessary repetitive analysis of a large number of influencing factors. On this basis, based on the sensitivity information, we compare and analyze the degree of influence of the elements in the key influence feature set on the tariff signal one by one and calculate the contribution ranking, so as to provide objective and scientific results of the traceability analysis of the causes of abnormal tariffs without the need to analyze the influencing factors and other related preprocessing operations of the historical data, which makes up for the shortcomings of the existing methods of traceability of the causes of abnormal tariffs in the scenarios of insufficient data, and it is of great significance for the stable operation of the electric power market. The stable operation of the electricity market is of great significance.

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Figure 1. Schematic diagram of the current electricity market price assessment process.

Figure 2. Schematic diagram of the proposed methodology for tracing the causes of abnormal electricity prices based on the comparative analysis of key features.

Figure 3. Flow chart of feature importance ranking method based on substitution algorithm.

Figure 4. The characterization of the functional features representing the relationship between the average electricity price within the scheduling period and various key influencing factor.

Table 1. A collection of key features of abnormal electricity price spikes.

Order	Electricity prices on the spike	Electricity prices under the spike
1	Branches' tidal current reach rate	Branches' tidal current reach rate
2	net load	net load
3	Total unit capacity	Total unit capacity
4	Change in load from previous time period	Change in load from previous time period
5	Change in load compared to a later time period	Change in load compared to a later time period
6	Quoted Maximum Value	Quoted Maximum Value
7	average value of quoted prices	average value of quoted prices

Table 2. A collection of key features of abnormal electricity price averages.

Order	Too high average value	Too low mean value
1	Average net load	Average net load
2	peak-to-valley variation in load	peak-to-valley variation in load
3	Total unit capacity	Total unit capacity
4	Total minimum output of the unit	Maximum load uphill climb
5	Number of times branch circuit tidal current constraints are exceeded	Climbing under maximum load
6	Climbing capacity of the unit	Maximum offer for the unit
7	Trend-bound overstepping penalty factor	Average unit price quoted

Table 3. The degree to which each feature influences the average electricity price.

Order	Key features	Absolute value of the percent change in feature	Contributionarrange in order
1	Average net load	Approximately equal to 10%	1
2	Peak-to-valley variation in load	Greater than 44%	3
3	Total unit capacity	Greater than 50%	4
4	Total minimum output of the unit	Not very relevant	7
5	Upper and lower limits of branch tidal currents	Approximately equal to 14%	2
6	Climbing capacity of the unit	Much greater than 50 per cent	6
7	Trend-bound overstepping penalty factor	Much greater than 50 per cent	5

Table 4. Traceability results of abnormal electricity price spikes.

Order	Traceability results of the proposed method	Artificial empirical traceability results	Order	Traceability results of the proposed method	Artificial empirical traceability results
1	Load level	Load level	11	Changes in loads with later time periods	Changes in loads with later time periods
2	Load changes between pre and post time periods	Load level, change in load between pre and post time periods	12	Change in load from previous time period	Load level, change in load between pre and post time periods
3	Branches' tidal current reach rate	Upper and lower branch circuit current limits, load	13	Change in load from previous time period	Change in load from previous time period
4	Load changes between pre and post time periods	Load changes between pre and post time periods	14	Load change from the previous time period, branch tidal current attainment rate	Load level, change in load between pre and post time periods
5	Branch Circuit Trend Boundary Rate, Load Change Between Pre- and Post-Time Periods	Load changes between pre and post time periods	15	Load changes between pre and post time periods	Load changes between pre and post time periods
6	Load changes between pre and post time periods	Load level, load change between pre and post time period branch current upper and lower limits	16	Change in load from previous time period	Change in load from previous time period
7	Change in load from previous time period	Load changes between pre and post time periods	17	Load level	Upper and lower branch circuit trend limits, load levels, and load changes from the previous time period
8	Load level	Change in load from previous time period	18	Load level	Upper and lower branch circuit trend limits, load levels, and load changes from later time periods
9	Change in load from previous time period	Load levels, load changes between pre and post time periods, upper and lower limits of branch currents	19	Change in load from previous time period	Upper and lower limits of branch circuit currents, load levels
10	Change in load from previous time period	Change in load from previous time period	20	Changes in loads with later time periods	Changes in loads with later time periods

Table 5. Traceability results of abnormal electricity price averages.

Order	Traceability results of the proposed method	Artificial empirical traceability results	Order	Traceability results of the proposed method	Artificial empirical traceability results
1	Branch circuit tidal current upper and lower limits, average net load, load peak-to-valley difference, tidal constraint overrun penalty factor	Upper and lower branch circuit trend limits, lower supply-demand ratio, load peak-to-valley difference, and trend constraint overrun penalty factor	11	Average net load, total unit capacity	High supply-demand ratio
2	Average net load, upper and lower branch current limits	Upper and lower limits of branch tidal currents, lower supply-demand ratios	12	Average net load, total unit capacity	High supply-demand ratio
3	Average net load, load peak-to-valley differential, unit creep capacity, tidal restraint overrun penalty factor	Upper and lower feeder tidal limits, lower supply-demand ratio, and tidal constraint overrun penalty factors	13	Average net load, quoted average	High supply-demand ratio, average value of offers
4	Average net load, upper and lower branch current limits	Average net load, load peak-to-valley difference, upper and lower limits of branch tidal currents	14	Average net load, quoted average	High supply-demand ratio, average value of offers
5	Average net load, upper and lower branch circuit tidal current limits, unit climbing capacity	Average net load, upper and lower branch tidal limits, tidal constraint overrun penalty factor	15	Average net load, unit vertical capacity, quoted average value	High supply-demand ratio, average value of offers
6	Upper and lower limits of branch tidal currents	Low supply-demand ratio	16	Average net load, unit vertical capacity, quoted average value	High supply-demand ratio, average value of offers
7	Average net load, upper and lower branch current limits, total unit capacity	Lower supply-demand ratio, upper and lower branch tidal current limits	17	Average net load	High supply-demand ratio
8	Upper and lower limits of branch tidal currents	Upper and lower limits of branch tidal currents	18	Average net load, total unit capacity	High supply-demand ratio
9	Upper and lower limits of branch tidal currents	Upper and lower limits of branch tidal currents	19	Average net load, total unit capacity	High supply-demand ratio
10	Average net load	Upper and lower limits of branch tidal currents	20	Total unit capacity, average net load	High supply-demand ratio, average value of offers

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A Comparative Analysis of Key Features for Tracing the Causes of Abnormal Electricity Prices in Market Environments

Abstract

1. Introduction

2. Electricity Spot Market Price Signal Generation and Assessment

2.1. Nodal Tariff Calculation Model

2.2. Current Electricity Market Price Assessment Process

2.3. Project Research Ideas

3. Classification and Extraction of Key Features of Abnormal Tariff Signals

4.1. Classification of Abnormal Tariffs

4.1.1. Classification of Tariff Spike Anomalies

4.2. Establishment of Key Feature Set of Abnormal Tariff Based on Principal Component Analysis Method

5. Sensitivity-Based Analysis of the Causes of Abnormal Tariffs

5.1. Impact Function of Key Features on Electricity Prices

5.2. Methodological Process of Analyzing the Causes of Abnormal Electricity Prices

6. Case Study

6.1. Explanation of the Construction of the Set of Key Features of Anomalous Tariffs

6.2. Validation of the Effectiveness of the Process of Tracing the Causes of Abnormal Electricity Prices

6.3. Verification of the Accuracy of the Traceability of the Causes of Abnormal Tariffs

8. Conclusion

References

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