1 Introduction

A power system network topology provides information for managing and controlling physical and economical grid operations. The topological structure of an entire power network is a key input to solving core power system analysis problems such as state estimation, power flow, contingency analysis, and economic dispatch. Therefore, maintaining the correct topology information is of vital importance for reliable and efficient grid operations. The smart grid of the future will increasingly rely on data from fast sensing units such as phasor measurement units (PMUs) and smart meters in advanced metering infrastructures (AMIs). Therefore, unexpected data corruption will have severe impact on future grid operations. In particular, the corruption of data associated with power system network topology (e.g. the on/off status of circuit breakers (CBs)) could mislead system operators about the real-time topology conditions. As a result, the real-time market price, namely the locational marginal price (LMP), can be distorted because real-time LMP is calculated based on the network topology. This topology error due to data corruption can occur frequently in smart grid operations, which suggests the need for a more rigorous study on the impact of such errors on LMP.

The main objective of this paper is to analytically examine the impact of topology errors from data corruption on a change in LMP. In a control center, the power system network topology is built with two functions in an energy management system (EMS): topology processor and topology error processing. The topology processor converts a bus section/switch model into a bus/branch model using the on/off status data of CBs, which are collected by sensors in the supervisory control and data acquisition (SCADA) system. Topology error processing, as a subsequent process of state estimation, detects and identifies the erroneous circuit breakers. The corrected topology model is then fed into the state estimator and economic dispatch module to compute the optimal estimate of the system state and dispatch instruction, respectively. Real-time LMP, as the key variable for network congestion management [1], is obtained as the by-product of the security constrained economic dispatch (SCED) in the market management system (MMS). It should be noted that the formulation for SCED is primarily based on the estimated network topology by the state estimator in EMS. Obviously, the incorrect network topology estimate could result in a miscalculation of real-time LMP.

Figure 1 illustrates two types of impact flows corresponding to continuous (e.g., the power injection/flow and voltage magnitude) and discrete (e.g., the on/off status of CB) data corruption among SCADA, EMS, and MMS. This data corruption results from a natural error and/or human-made attack. Figure 1a, b represent the flow of data corruption-induced estimate of the system state and topology, respectively. In this paper, we assume the situation where the false status data of CBs successfully bypass topology error processing, thus generating an incorrect network topology. In this situation, we aim to investigate the economic impact (Fig. 1b) of undetected CB data corruption-induced topology errors on LMP calculation in SCED.

Fig. 1
figure 1

Illustration of the impact of continuous and discrete SCADA sensor data corruption on LMP calculation in SCED via state estimation

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A large body of research has been carried out to develop topology error processing for dealing with a natural error of CB status. The problem of detecting topological errors was first addressed in [2]. Methods for line topology error processing were then introduced in [3] and [4], based on a geometrically based test and a correlation index that explains the relationship between topology error and the suspected digital measurement set, respectively. Methods for checking the topology status at the substation level were proposed using linear programming [5], the largest normalized residual test [6], a modified two level least absolute value (LAV) state estimator [7], and computation-effective schemes based on a reduced model [8] and an implicit model [9]. A unified generalized state estimation that detects errors associated with both continuous/discrete data and device parameters was formulated in [10].

However, the authors in [6] pointed out the limitation of the existing topology error processing, incapable of detecting an irrelevant or critical line error in a certain measurement configuration. Moreover, the recently proposed topology data attack methods showed that an attacker could stealthily change the transmission line status by a malicious data injection into sensors [11, 12]. Using an algorithm based on the breadth-first search (BFS), an unobservable topology attack on AC state estimation was developed [13]. A novel defending mechanism was proposed to mitigate the risk of topology data attack by strategically shutting down some of the preselected transmission lines and therefore switching the network topologies [14]. The authors in [15] proposed a heuristic local topology attack and developed countermeasures to defend against such attack using a strategic placement of meters. A more realistic topology attack, namely a breaker-jammer attack, with limited resource requirements was proposed. This attack modifies the status of several circuit breakers, as well as jams (blocks in the communication) of flow measurements on transmission lines, which are technically easier to change than meter measurements [16]. More recently, it was shown that topology data attacks could have a severe impact on the economic operation of power systems, such as distorting an optimal generation cost in an optimal power flow (OPF) [17] and manipulating virtual bidding transactions for making a profit in a two-settlement electricity market (day-ahead and real-time) [18].

To the best of our knowledge, no previous work has been presented concerning the degree to which changes in topology data due to cyber attacks influence LMP calculation in real-time power markets. In particular, as the physical and economic risks from topology data attacks become increasingly threatening as mentioned in the above prior work, an analysis tool is required for system operators to accurately and quickly evaluate such risks from a cybersecurity perspective. We aim to develop a framework that is a computationally efficient closed-form application, with which system operators can rapidly predict and quantify the impact of topology data attack on LMP.

Several attempts have been made to analyze degree to which changes in data affect the operation of economic dispatch. The impact of system constraints (e.g. the limits of the line and generation capacity) in economic dispatch on LMP in an AC and DC optimal power flow model was studied in [19] and  [20]. The impact of unscheduled line outage on LMP is statistically investigated using a point estimation method [21]. In [22], we developed a framework to quantify the chain effect of continuous SCADA data corruption on real-time LMP via state estimation. In addition to this work, we derived a mathematical expression to calculate the change in congestion cost due to line topology errors and performed the impact analysis of LMP subject to line topology errors [23]. However, our prior work was limited to the analysis of a simple case that only corresponds to Case 1 in Sect. 3.1 of this paper (i.e. unchanging congestion line and marginal units after topology error).

Our current work involves a more realistic broad set of scenarios including Cases 2, 3, and 4 with the changing conditions for network congestion and marginal units (i.e. part-loaded generators) due to topology error. In each case, the sensitivity of LMP to changes in the costs of marginal units (i.e. marginal costs) and generation shift factors (GSFs) due to a line susceptance error on LMP is analyzed rigorously based on the developed LMP change equation, and the results for all four cases are compared. Furthermore, such impact is quantified at multiple dispatch intervals with the varying load conditions, along with the identification of the influential transmission line and the economically sensitive bus using the proposed system wide performance index.

The proposed framework can be primarily extended to develop offline analysis and planning tools to defense a smart grid against potential topology data attacks. Through re-executing SCED based on a large amount of historical operation data, system operators first classify the four different cases due to a targeted attack line error, which are illustrated in Sect. 3. For each case, they then evaluate the sensitivity of LMP with respect to changes in marginal costs and GSFs in an online manner.

The main contributions of this paper are summarized as follows.

  1. 1)

    We develop a computationally efficient closed-form analytical framework to assess LMP change in response to line status error from the line exclusion or inclusion when a single transmission line is congested. The derived LMP change equation based on an analytical congestion price formula is expressed in terms of local information such as marginal costs and GSFs, which are only related to the congested line without requiring information of an entire power system. Therefore, as the size of a power system increases, the relative computational saving compared with the extensive re-execution of SCED becomes more significant. In addition, the analytical congestion price formula can be extended to the LMP change equation that explains the relationship between LMP and the substation configuration error from bus splitting/merging.

  2. 2)

    The developed framework can be used as a historical data based-offline cybersecurity assessment tool to quantify the impact of a topology data attack on LMP. After classifying the topology error-induced network operating condition into four cases under different network congestion patterns and marginal units through re-execution of SCED, system operators could rapidly quantify the sensitivity of LMP at any bus to changes in the marginal cost and GSFs in an online manner. From a implementation perspective, the developed framework can be readily integrated as an offline tool into SCED without the need to change SCED formulation.

  3. 3)

    The simulation study under various physical and economical grid conditions validates the proposed framework and provides meaningful observations that help system operators to maintain a robust market operation against topology data attack: \(\textcircled {1}\) a fast sensitivity analysis of LMP to changes in marginal cost and GSF; \(\textcircled {2}\) identification of the most economically sensitive bus to any erroneous line; \(\textcircled {3}\) identification of the most influential line on LMP change. In view of the secure market operation, the simulation results from the proposed framework can be a key input to enhancing power system state estimation, topology error processing, and contingency analysis in EMS and prioritizing the upgrade of sensors that monitor the status of transmission lines. Furthermore, the sensitivity results associated with marginal costs and GSFs could provide practical guidelines for the development of a robust generation company’s bidding strategy and power system network planning against topology data attacks, respectively.

The remainder of this paper is structured as follows. In Sect. 2, two types of topology errors are introduced, the vulnerability of topology error processing is addressed, and a real-time power market model is formulated. In Sect. 3, LMP change expressions with respect to topology errors are formulated based on the analytical congestion price equation, and a performance index using these expressions is proposed. Section 4 verifies the formulated LMP change expressions and shows the simulation results through a rigorous case study in the IEEE-14 bus and 118-bus systems. Finally, concluding remarks and future work appear in Sect. 5.

2 Background

2.1 Two types of network topology errors

In general, power system network topology errors due to the incorrectly reported CB status can be categorized into the following two types [24]: line status error and substation configuration error. The former represent an incorrect exclusion/inclusion of transmission lines from the network model, whereas split/merging error of buses at the substation level belongs to the latter. Figure 2 summarizes these two types of topology errors in a two-bus system. In this paper, we perform the impact analysis of real-time LMP subject to the network topology change with the line status error.

Fig. 2
figure 2

Line status error and substation configuration error

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2.2 Vulnerability of topology error processing

For the line status error, topology error detection is performed using the normalized residual vector obtained from the linearized DC power flow measurement model [24]. Unfortunately, this topology error processing has a limited capability, in which it is unable to detect line status errors for irrelevant or critical lines [6]. The irrelevant line represents one with no measurement sensors. The critical line makes the network become unobservable if it is deleted from the network model. Recently, the feasibility of the human-made attack, namely the topology data attack, is proven mathematically, capable of completely bypassing topology error processing [11]. Figure 3 illustrates an example where the attacker changes the line status from on to off without being detected by system operators. This topology data attack must be undetectable through the manipulation of continuous sensor (power injection and flow) and discrete sensor (CB) data while maintaining KCL rule. Consequently, the incorrect network topology information might lead to the miscalculation of the optimal dispatch and LMP in SCED, which is formulated in the following subsection.

Fig. 3
figure 3

Simple topology data attack

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2.3 Real-time power market model

A real-time power market consists of the two main pricing models: Ex-ante (e.g. in NY ISO) and Ex-post (e.g. in ISO New England, PJM, and Midwest ISO) [25]. Since Ex-ante and Ex-post models rely on the network topology and the cost functions of generators, our results in Sect. 3 are applicable to both models. In this paper, we consider a real-time Ex-ante market model where LMPs are computed before the actual deployment of dispatch orders.

For the system operator, the Ex-ante dispatch is formulated as follows.

$$\begin{aligned} \min _{p_{i}} \sum _{i\in G} C_ip_i \end{aligned}$$
(1)

s.t.

$$\begin{aligned}&\lambda :\;\sum _{n=1}^{N_b} P_{g_n} = \sum _{n=1}^{N_b} L_{d_n} \end{aligned}$$
(2)
$$\begin{aligned}&\varvec{\tau }:\;p_i^{\text {min}} \le p_i \le p_i^{\text {max}} \quad~~~~~~~~~~~~~~~~~~~~~~~~~~\forall i\in G \end{aligned}$$
(3)
$$\begin{aligned}&\varvec{\mu }:F_{l}^{\text {min}} \le \sum _{n=1}^{N_b} H_{l,n}(P_{g_n}-L_{d_n}) \le F_{l}^{\text {max}}\nonumber \\&\quad \forall l=1,2,\ldots ,N_l \end{aligned}$$
(4)

In this formulation, the objective function is to minimize the total generation costs with energy cost \(C_i\) for generator i in (1). Equation (2) is the system-wide energy balance equation for total real power output \(P_{g_n}\) and fixed demand \(L_{d_n}\) at bus n for all buses \(N_b\). Equation (3) is the physical capacity constraints of each ith generator’ real power \(p_i\). Equation (4) is the transmission line constraints with min/max flow limits \(F_{l}^{\text {max}}\) and \(F_{l}^{\text {min}}\) for transmission line l. \(\lambda\), \(\varvec{\tau }\), and \(\varvec{\mu }\) are the dual variables associated with the aforementioned equality and inequality constraints. \(\varvec{\tau }\) and \(\varvec{\mu }\) are expressed as \(\varvec{\tau }=[\varvec{\tau }_{\text {max}}^\mathrm {T},\varvec{\tau }_{\text {min}}^\mathrm {T}]^\mathrm {T}\) and \(\varvec{\mu }=[\varvec{\mu }_{\text {max}}^\mathrm {T},\varvec{\mu }_{\text {min}}^\mathrm {T}]^\mathrm {T}\), respectively, where subscript max(min) represents the max(min) inequality constraint. \(H_{l,n}\) is the element at the lth row and nth column of the \(N_{l}\times N_b\) generation shift factor matrix \(\varvec{H}\). This matrix explains the sensitivity of branch flows to nodal injection powers. The real-time LMP vector \(\varvec{\pi }\) is computed using the following [26]:

$$\begin{aligned} \varvec{\pi } = \lambda \cdot \mathbf {1}_{N_b} - \varvec{H}^\mathrm {T}\left( \varvec{\mu }_{\text {max}} - \varvec{\mu }_{\text {min}}\right) \end{aligned}$$
(5)

In (5), a scalar \(\lambda\) is called as an energy component, which is LMP at a slack bus. The other vector term, \(\varvec{H}^\mathrm {T}\left( \varvec{\mu }_{\text {max}} - \varvec{\mu }_{\text {min}}\right)\), is called a congestion component, which represents the congestion prices at all buses. If a marginal unit is connected to a slack bus, \(\lambda\) is equal to the marginal cost). Otherwise, \(\lambda\) has some implicit value involving the congestion condition.

3 Analysis of LMP change to topology error

In this section, we develop an analytical framework to quantify the change in LMP to the line status error. We consider the situation where an adversary can successfully result in the line status error without being detected by topology error processing when a single transmission line is congested with and without topology error. Recently, some work has proposed a profitable attack strategy by manipulating power flow solution though the injection of malicious data into analog sensors. With this research trend, it is expected that a profitable topology attack strategy through the corruption of discrete sensor data can be developed in the near future. The development of such profitable topology attack strategy would be initiated in a simple scenario with a single line congestion because multiple congestion scenario requires more complex attack algorithm and analysis. For analytical purposes this paper presents a first step toward developing a cybersecurity tool to quantify the impact of a simple profitable topology attack on LMP. In addition, most dispatch time periods in a real-time market have only a few line congestions or no congestion. Thus, the assumption of a single line congestion after the attack is quite reasonable and valid for analytical purposes. A more general framework to involve multiple line congestions would be referred to a future work.

3.1 Formulation of the change in LMP to topology error

First, we introduce our prior result [23], the following Proposition 1, in which the shadow price associated with the congested transmission line is written as a function of the energy costs of marginal units and their corresponding generation shift factors related to the congested line.

3.1.1 Proposition 1

Let i and j be two marginal units with \(C_j > C_i\), belonging to different buses. Then, the shadow price for the congested transmission line l is expressed as:

$$\begin{aligned} \mu _{l} = \frac{\mathrm {\Delta } C(j,i)}{\mathrm {\Delta } H_l(i,j)} \end{aligned}$$
(6)

where \(\mathrm {\Delta } C(j,i) = C_j-C_i\) ; \(\mathrm {\Delta } H_l(i,j) = H_{l,i} - H_{l,j}\).

The result in Proposition 1 is used to express LMP change formulation in the following corollary, the result of which accounts for the effect of the line status error on the change in LMP.

3.1.2 Corollary 2 (LMP change index to line status error)

Denote the lines l and \(l'\) by the congested lines before and after topology error, respectively. Let (i, j) and (p, q) be two pairs of marginal units with cost functions \(C_j > C_i\) and \(C_q > C_p\), corresponding to before and after the topology error, respectively. The vector of LMP changes (i.e., the differences between LMPs without and with topology error at all buses) is written as:

$$\begin{aligned} \begin{aligned} \mathrm {\Delta } \varvec{\pi }_l^k =&\underbrace{(\lambda - \tilde{\lambda }^k)\mathbf {1}_{N_b}}_{\text {(I)}} + \underbrace{\mathrm {\Delta } C(q,p) \left[ \frac{(\varvec{H}_{l'}^{k})^\mathrm {T}}{\mathrm {\Delta } \widetilde{H}_{l'}^k(p,q)}\right] }_{\text {(II)}} \\&- \underbrace{\mathrm {\Delta } C(j,i) \left[ \frac{\varvec{H}_l^\mathrm {T}}{\mathrm {\Delta } H_l(i,j)}\right] }_{\text {(III)}} \end{aligned} \end{aligned}$$
(7)

The indices \(\mathrm {\Delta } \varvec{\pi }_l^k\) for the change of LMP with respect to the exclusion of a line k are categorized into four different cases as follows:

  1. 1)

    Case 1: both unchanging congestion line and marginal units.

$$\begin{aligned} \mathrm {\Delta } \varvec{\pi }_l^k = \varvec{W}_{l,l}^{(1)} \left[ \begin{array}{c} \mathrm {\Delta } C(j,i) \\ -\mathrm {\Delta } C(j,i) \end{array}\right] \end{aligned}$$
(8)
  1. 2)

    Case 2: unchanging congestion line, changing marginal units.

$$\begin{aligned} \mathrm {\Delta } \varvec{\pi }_l^k = \mathrm {\Delta } \varvec{\varLambda } + \varvec{W}_{l,l}^{(2)} \left[ \begin{array}{c} \mathrm {\Delta } C(q,p) \\ -\mathrm {\Delta } C(j,i) \end{array}\right] \end{aligned}$$
(9)
  1. 3)

    Case 3: changing congestion line, unchanging marginal units.

$$\begin{aligned} \mathrm {\Delta } \varvec{\pi }_l^k = \varvec{W}_{l,l'}^{(3)} \left[ \begin{array}{c} \mathrm {\Delta } C(j,i) \\ -\mathrm {\Delta } C(j,i) \end{array}\right] \end{aligned}$$
(10)
  1. 4)

    Case 4: both changing congestion line and marginal units.

$$\begin{aligned} \mathrm {\Delta } \varvec{\pi }_l^k = \mathrm {\Delta } \varvec{\varLambda } + \varvec{W}_{l,l'}^{(4)} \left[ \begin{array}{c} \mathrm {\Delta } C(q,p) \\ -\mathrm {\Delta } C(j,i) \end{array}\right] \end{aligned}$$
(11)

For \(n=1,2,\ldots , N_b\),

$$\begin{aligned} \varvec{W}_{l,l}^{(1)}= \left[ \begin{array}{l} \left.\frac{\widetilde{H}_{l,n}^{k}}{\mathrm {\Delta } \widetilde{H}_{l}^{k}(i,j)}\,\right|\frac{H_{l,n}}{\mathrm {\Delta } H_{l}(i,j)} \end{array}\right] _{(N_b\times 2)} \end{aligned}$$
(12)
$$\begin{aligned} \varvec{W}_{l,l}^{(2)}= \left[ \begin{array}{l} \left.\frac{\widetilde{H}_{l,n}^{k}}{\mathrm {\Delta } \widetilde{H}_{l}^{k}(p,q)}\,\right|\frac{H_{l,n}}{\mathrm {\Delta } H_{l}(i,j)} \end{array}\right] _{(N_b\times 2)} \end{aligned}$$
(13)
$$\begin{aligned} \varvec{W}_{l,l'}^{(3)}= \left[ \begin{array}{l} \left.\frac{\widetilde{H}_{l',n}^{k}}{\mathrm {\Delta } \widetilde{H}_{l'}^{k}(i,j)}\,\right|\frac{H_{l,n}}{\mathrm {\Delta } H_{l}(i,j)} \end{array}\right] _{(N_b\times 2)} \end{aligned}$$
(14)
$$\begin{aligned} \varvec{W}_{l,l'}^{(4)}= \left[ \begin{array}{l}\left. \frac{\widetilde{H}_{l',n}^{k}}{\mathrm {\Delta } \widetilde{H}_{l'}^{k}(p,q)}\,\right|\frac{H_{l,n}}{\mathrm {\Delta } H_{l}(i,j)} \end{array}\right] _{(N_b\times 2)} \end{aligned}$$
(15)
$$\begin{aligned} \mathrm {\Delta } \varvec{\varLambda }= & (\lambda - \tilde{\lambda }^k)\mathbf {1}_{N_b} \end{aligned}$$
(16)

Equation (7) represents the LMP change equation for Case 4. This equation can be modified to illustrate the other three cases with varying conditions for network congestion and marginal units due to topology error. It should be noted that in (7) three components influence the topology error-induced LMP change. First, the components (II) and (III) are the contribution of congestion price with and without topology error to the change in LMP, respectively. Note that each component is written as the multiplication form of two independent functions relying on the energy costs of marginal units and GSFs. The former derives from an economical layer associated with market operations, whereas the latter derives from a cyber-physical layer associated with physical and communication network. On the other hand, the component (I) is the contribution of the energy price difference between without and with topology error to LMP change. This component belongs to both economical and cyber-physical layers because the calculation of the energy price implicitly relies on the values of the marginal costs and GSFs. Evaluation of the change in component (I) to the marginal costs and GSFs can be conducted in an online manner by judiciously reselecting the slack bus as described in Remark 1.

3.1.3 Remark 1

There are two remarks:

  1. 1)

    LMP at any bus with a marginal unit is equal to the marginal cost of the marginal unit.

  2. 2)

    The varying location of a slack bus has no influence on the LMP change under a lossless SCED model. Therefore, in the case where a slack bus has no marginal unit, searching and reselecting for a slack bus with a marginal unit assures no change in component (I) when marginal costs (except at a reselected slack bus) and GSFs vary, thus leading to the desired on-line analysis.

Using three components from both economical and cyber-physical layers, the LMP change to the topology error is quantified in the control/computation layer. Figure 4 illustrates Case 1 in the aforementioned three-layer framework where each piece of information associated with physical and economical layers is fed into an upper control/computation layer for the impact analysis of LMP subject to topology error.

Fig. 4
figure 4

Three-layer framework illustrating LMP change to topology error

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On the other hand, LMP change index to line status error can be extended to substation configuration error as shown in Remark 2.

3.1.4 Remark 2 (LMP change index to substation configuration error)

Suppose that a power system with \(N_b\) buses has a congested line l. Then, for Case 1 the LMP change equations with respect to the merging error at bus n and splitting error at bus m are expressed as:

$$\begin{aligned} \mathrm {\Delta } \varvec{\pi }_l^n&= \mathrm {\Delta } C(j,i) \left[ \frac{(\varvec{H}_{l(N_b\times 1)}^{n})^\mathrm {T}}{\mathrm {\Delta } \widetilde{H}_l^n(i,j)} - \frac{\varvec{H}_{l(N_b\times 1)}^\mathrm {T}}{\mathrm {\Delta } H_l(i,j)}\right] \end{aligned}$$
(17)
$$\begin{aligned} \mathrm {\Delta } \varvec{\pi }_l^m&= \mathrm {\Delta } C(j,i) \left[ \frac{(\varvec{H}_{l[(N_b+1)\times 1]}^{m})^\mathrm {T}}{\mathrm {\Delta } \widetilde{H}_l^m(i,j)} - \frac{\varvec{H}_{l[(N_b+1)\times 1]}^\mathrm {T}}{\mathrm {\Delta } H_l(i,j)}\right] \end{aligned}$$
(18)

where

$$\begin{aligned} {\left\{ \begin{array}{ll} (\varvec{H}_{l(N_b\times 1)}^{n})^\mathrm {T}=\left[ \varvec{H}_{l[1\times (N_b-1)]}^{n}~0\right] ^\mathrm {T}\\ \varvec{H}_{l[(N_b+1)\times 1]}^\mathrm {T}=\left[ \varvec{H}_{l}~0\right] ^{\mathrm {T}} \end{array}\right. } \end{aligned}$$
(19)

The number 0 corresponds to the added or deleted bus after topology change. Similarly, the above equations can be extended to other cases.

3.2 Performance metrics

For the four cases in Corollary 1, the LMP changes at each dispatch interval t can be expressed in a matrix form:

$$\begin{aligned} \varvec{\varPsi }^{t}= \left[ \begin{array}{cccc} \mathrm {\Delta } \varvec{\pi }_{l,1}^{1,t} &\mathrm {\Delta } \varvec{\pi }_{l,2}^{1,t} &\cdots &\mathrm {\Delta } \varvec{\pi }_{l,N_b}^{1,t} \\ \mathrm {\Delta } \varvec{\pi }_{l,1}^{2,t} &\mathrm {\Delta } \varvec{\pi }_{l,2}^{2,t} &\cdots &\mathrm {\Delta } \varvec{\pi }_{l,N_b}^{2,t} \\ \vdots &\vdots &\ddots &\vdots \\ \mathrm {\Delta } \varvec{\pi }_{l,1}^{N_k,t} &\mathrm {\Delta } \varvec{\pi }_{l,2}^{N_k,t} &\cdots &\mathrm {\Delta } \varvec{\pi }_{l,N_b}^{N_k,t} \end{array}\right] \end{aligned}$$
(20)

where the row and column of the \(N_k \times N_b\) matrix \(\varvec{\varPsi }^{t}\) represent the locations of the line k exclusion (\(k=1,2,\ldots , N_k\)) and the bus n (\(n=1,2,\ldots , N_b\)), respectively. \(\varvec{\varPsi }^{k,t}(l,n)\) is denoted by the element at the lth row and nth column of \(\varvec{\varPsi }^{t}\) and represents \(\mathrm {\Delta } \varvec{\pi }_{l,n}^{k,t}\), which explains the LMP change at bus n with respect to the line k error when the line l is initially congested at dispatch interval t.

The matrix \(\varvec{\varPsi }^{t}\) can be used to provide the system-wide metrics that help system operators to find the following grid assets given the congested line l.

  1. 1)

    The \(k_1\)th most and \(k_2\)th least influential transmission line on LMP, on average, with respect to a total of \(N_b\) buses and T dispatch intervals:

    $$\begin{aligned} k_1&= \arg \max _{k_1}\left( \sum _{n=1}^{N_b}\sum _{t=1}^T\left| \varvec{\varPsi }^{k_1,t}(l,n)\right| /N_b T\right) \end{aligned}$$
    (21)
    $$\begin{aligned} k_2&= \arg \min _{k_2}\left( \sum _{n=1}^{N_b}\sum _{t=1}^T\left| \varvec{\varPsi }^{k_2,t}(l,n)\right| /N_b T\right) \end{aligned}$$
    (22)
  2. 2)

    The \(n_1\)th most and \(n_2\)th least sensitive bus in LMP, on average, with respect to a total of \(N_k\) transmission line exclusions and T dispatch intervals:

    $$\begin{aligned} n_1&= \arg \max _{n_1}\left( \sum _{k=1}^{N_k}\sum _{t=1}^T\left| \varvec{\varPsi }^{k,t}(l,n_1)\right| /N_k T\right) \end{aligned}$$
    (23)
    $$\begin{aligned} n_2&= \arg \min _{n_2}\left( \sum _{k=1}^{N_k}\sum _{t=1}^T\left| \varvec{\varPsi }^{k,t}(l,n_2)\right| /N_k T\right) \end{aligned}$$
    (24)
Fig. 5
figure 5

IEEE 14-bus system including bus-breaker model

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4 Numerical study

In this section, we evaluate the impact of network topology errors on real-time LMP using the proposed LMP change index. The numerical study is conducted in IEEE-14 bus and IEEE-118 bus systems with the real-time ex-ante power market model where the LMP change at any bus with respect to any line exclusion error is evaluated. Figure 5 shows the detailed bus-breaker model in the IEEE-14 bus system [24]. This figure illustrates the scenario where the status error of one circuit breaker at bus 11 leads to the undetectable lines 6–11 exclusion error, consequently modifying the congestion pattern from the the lines 5, 6 congestion to the lines 2–4 congestion. The specifications of five generators in the IEEE 14-bus system are provided in Table 1.

Table 1 Generator parameters of IEEE 14-bus test system
Full size table

4.1 Impact of energy costs of marginal units on LMP

In this subsection, we investigate the effect of the marginal costs (in economical layer illustrated in Fig. 4) on an LMP change given fixed GSFs (in cyber-physical layer illustrated in Fig. 4). Figure 6 show the impact of varying \(\mathrm {\Delta } C(q,p)\) on the LMP change. The results in this figure correspond to Case 4 where a pair of the marginal units (i, j) = (1, 8) appear in the lines 5, 6 congested network before the topology error. After the lines 3, 4 exclusion, the marginal units are changed to (p, q) = (1, 2), and lines 2–4 becomes congested. Indeed, in this situation the generator at slack bus 1 is still the marginal unit with the topology error. Therefore, the contribution of energy component \(\mathrm {\Delta } \varvec{\varLambda }\) to the LMP change is not considered in these figures.

Fig. 6
figure 6

Impact of a varying gap between the marginal costs with and without topology error on LMP change in Case 4: a pair of marginal units (p, q) with topology error

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We can observe from Fig. 6 that as the gap between the energy costs of the marginal units increases, the value of LMP change increases at only buses 2 and 3 with positive \(W_{l,l'}^{(4)}\). This observation results from  (11), which explains that the increase of \(\mathrm {\Delta } C(1,2)\) leads to the increase of \(\mathrm {\Delta } \varvec{\pi }_l^k\) at buses associated with only \(W_{l,l'}^{(4)}>0\). Let us denote the element at the ith row and jth column of \(W_{l,l'}^{(4)}\) by \(W_{l,l'}^{(4)}(i,j)\). Then, in Fig. 6 \(W_{l,l'}^{(4)}(2,1)\) and \(W_{l,l'}^{(4)}(3,1)\) have a positive value of 1 whereas other elements \(W_{l,l'}^{(4)}(i,1)~(i\ne 2,3)\) have negative values.

It should be noted that having local information of \(W_{l,l'}^{(4)}\) enables system operators to quickly categorize all buses into two bus groups with an increasing or decreasing LMP with respect to the change in marginal costs. Furthermore, this result can provide a practical guideline for the generation companys bidding strategy in case of topology error.

4.2 Impact of GSFs on LMP

In this subsection, we examine the impact of the line exclusion-induced GSFs on LMP change given fixed marginal costs in all four cases. From a large number of simulations, we can obtain the following observations:

  1. O1:

    The buses with the most positive and negative LMP changes correspond to those at the ends of the union of the congested line and the erroneous line.

  2. O2:

    In general, each pair of buses at the ends of the congested line and the erroneous line shows opposite directions of LMP change.

  3. O3:

    However, if a bus is connected to the same end of both the congested line and the erroneous line, (O2) does not always hold true.

  4. O4:

    For Cases 1 and 3, buses with positive and negative LMP change can be quickly identified by checking the following vector indices:

    $$\begin{aligned} \varvec{V}^{(1)}_{l,l (N_b\times 2)}&=\left[ W_{l,l}^{(1)}(n,1)\right] -\left[ W_{l,l}^{(1)}(n,2)\right] \end{aligned}$$
    (25)
    $$\begin{aligned} \varvec{V}^{(3)}_{l,l' (N_b\times 2)}&=\left[ W_{l,l'}^{(3)}(n,1)\right] -\left[ W_{l,l'}^{(3)}(n,2)\right] \end{aligned}$$
    (26)

    which are the difference between the first and second columns in  (12) and (14) matrices, respectively. Using these two indices, all buses in the entire power system are quickly grouped into two groups with positive and negative LMP changes. Afterwards, system operators can readily predict a market participant’ profit or loss. For Cases 2 and 4, the component (I) in (16) is added to (25) and (26), which is recalculated in the same way as that Remark 1.

Fig. 7
figure 7

LMP changes in Case 1 to the exclusion of the lines 2–4 and the lines 4, 5 when the lines 5, 6 is congested

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Fig. 8
figure 8

LMP sensitivity to changes in the susceptance of the lines 6–13 and the lines 4, 5 when the lines 5, 6 is congested

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Figure 7 show the LMP changes in Case 1 where lines 5, 6 is congested with the exclusions of lines 2–4 and 4, 5, respectively. The result in Fig. 7a is consistent with (O1) and (O2). Buses 6 and 4 have the most positive and negative LMP change since they correspond to one end of the congested lines 5, 6 and the erroneous lines 2–4, respectively. In addition, each pair of buses associated with the congested line and the erroneous line has an LMP change with the opposite direction. On the other hand, we can observe from Fig. 7b that the lines 4, 5 exclusion makes LMPs at the pair of buses 5 and 6 change in the same direction. This is because bus 5 is connected to the same end of the congested line and the erroneous line. This phenomenon verifies (O3). Lastly, (O4) is verified since \(V^{(1)}_{l,l}\) is negative at only buses 3, 4, 5 in Fig. 7a and buses 2, 3, 4 in Fig. 7b so that these buses have negative LMP changes.

Figure 8 shows the impact of the varying line susceptance for two different lines on LMP at six different buses when the lines 5, 6 is congested. The original susceptance of the line is divided into forty equal steps, and the fortieth step in the x-axis represents the corresponding line exclusion. Using the results in these figures, we can fairly compare the sensitivity of LMP at any bus to susceptance change in some targeted line. For example, we observe from Fig. 8a that bus 13 has the highest sensitivity between the first and thirty fourth steps and bus 6 between the thirty fifth and fortieth steps. The former and latter buses correspond to the end of the excluded line and the congested line, respectively. These sensitivity results can be valuable input to enhance parameter estimation process in EMS in view of secure market operations.

Fig. 9
figure 9

LMP changes in the four cases under the lines 9, 10 congestion

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Figure 9 shows the LMP changes for four cases with different line exclusions under the identical line congestion (lines 9, 10 congestion). In this figure, four plots represent the highest LMP change in each case, which is calculated rapidly by the proposed LMP change index. In Case 2, a pair of marginal units 1 and 6 become changed to a pair of units 3 and 6. In Case 3, the congested line varies from lines 9, 10 to lines 1, 2. Case 4 includes the same changing marginal units in Case 2, along with the different congested lines 5, 6. From this figure, we make the following observations:

  1. O5:

    Case 1 shows the smallest effect on LMP change among four cases. In addition, Cases 3 and 4 have a more significant effect on LMP than Cases 1 and 2. We can conjecture from this that the impact of the line exclusion with varying marginal units or the congested line on LMP is larger than without them. Furthermore, the change in congestion pattern appears to be more influential on LMP than in the marginal unit.

  2. O6:

    The line exclusion closer to the congested line does not always have a larger impact on LMP at buses at both ends of the congested line. For instance, lines 13, 14 is closer to the congested lines 9, 10 than lines 2–4. However, the lines 13, 14 exclusion has less effect on LMP at buses 9 and 10 than that of the lines 2–4 exclusion. This fact allows potential topology data attackers to manipulate LMPs significantly in a more distributed and unexpected way irrespective of the distance between the congested and attacked lines.

For the IEEE 118-bus system, with 54 generation buses and 186 transmission lines as shown in Fig. 10, we assume that the lines 15–17 is congested. Figure 11 show Case 1 and Case 3 that correspond to without and with change of the line congestion (from the lines 15–17 to the lines 1, 2) due to three different line exclusions, respectively. The observations from the IEEE 14-bus test cases are also verified in the larger IEEE 118-bus system. For example, in the case of the lines 12–16 exclusion in Fig. 11b, the economically sensitive buses to this line exclusion are buses 2 and 16 that are associated with the ends of the congested line and the excluded line, respectively. Through the comparison between Fig. 11a, b, we observe more impact of the change in congestion pattern on LMP than without it. In addition, we also verify that the relative LMP change compared with the IEEE 14-bus test cases is smaller in the IEEE 118-bus system. This observation indicates that the larger power systems become the less topology data attack has an impact on LMP. However, as the power system size increases, the number of potential attack points also increases. Therefore, more coordinated and distributed topology attacks against many attack points could have a more detrimental impact on LMP in larger power systems.

Fig. 10
figure 10

IEEE 118-bus system

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Fig. 11
figure 11

LMP changes to different line exclusions in the IEEE 118-bus system

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Lastly, we conduct a simulation study of this LMP change at multiple dispatch intervals with varying load conditions. For this study, we set the limits of lines 9, 10 and lines 3, 4 to 70 and 80 MW, respectively. In this setting, both Case 3 and Case 4 have the lines 9, 10 congestion with topology error. Other lines are assumed to have sufficient flow capacity limits. Figure 12 show LMP results under varying seasonal load profiles in the congested network. Figure 12a shows four different Ercot 15-min load data, each of which implies a representative load profile for each season. Figure 12b shows the frequency of four cases subject to any line exclusion, based on the 07/15 load data in Fig. 12a. In Fig. 12b, the line exclusion with zero frequency involves an infeasible dispatch or no congestion condition after the corresponding line exclusion.

Fig. 12
figure 12

LMP results under varying load conditions

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We can identify from Fig. 12c the most influential lines 2–5, 3, 4, and 4, 5 (the line index \(k_1=5,6,7\)) under 01/15, 07/15, and 10/15 load conditions, respectively. In addition, economically sensitive bus 3 (the bus index \(n_1=3\)) to all line exclusions is verified in four load conditions from Fig. 12d. This fact justifies (O1) since bus 3 is connected to one end of the congested lines 3, 4. (O6) is also verified in Fig. 12c. For example, for 07/15 plotted in Fig. 12c, the exclusion of the lines 4–7 (\(k=8\)) much closer to the congested lines has less impact on LMP than the lines 1, 2 (\(k=1\)) exclusion.

Finally, the novelty and application of the proposed LMP change index as well as the observations from simulation studies can be summarized as follows:

  1. 1)

    Derivation of analytical LMP change index to topology errors. For various operation conditions (Cases 1 to 4), the analytical equations are proposed to quantify the impact of a single line topology error on LMP change.

  2. 2)

    Fast assessment of LMP change in response to varying marginal cost and GSF. Without exhaustively executing SCED, in each case the impact of marginal cost and GSF on LMP is quantified using the proposed LMP index. As a result, all the buses can be quickly categorized into two bus groups with a positive or negative LMP change direction to the change in marginal cost and GSF. This could provide information for the generation company to make its bidding strategy as well as with the system operator to predict a market participants profit or loss.

  3. 3)

    Identification of economically vulnerable buses and influential lines on LMP. Simulation studies show that the buses with the most sensitivity to topology errors are those at the ends of the union of the congested line and the erroneous line. This also implies that the erroneous line and/or the congested line are the most influential ones on LMP.

  4. 4)

    Cybersecurity evaluation. The results from (3) above can be used as a practical guideline for the development of cybersecurity index to quantify the performance of a cyber data attack against smart grid operation. For example, continuous and discrete sensors monitoring the erroneous and congested lines must be protected with high priority to mitigate the financial risk of topology data attacks. Furthermore, the proposed LMP change index can be potentially embedded in EMS/MMS applications such as power system state estimation, topology error processing, and contingency analysis to enhance secure market operations.

5 Conclusion

In this paper, we develop a closed-form analytical framework to assess LMP change in response to line status error when a single transmission line is congested. The developed framework provides system operators with a computationally efficient analysis tool because it requires only local information related to the congested line, such as marginal cost and GSF. Using this framework, system operators quickly evaluate the sensitivity of LMP to changes in marginal cost and GSF in an online manner. Simulation results provide insights into the identification of the most (or least) influential transmission line on LMP change and the most (or least) economically sensitive bus to any line status error in an entire power system. The proposed framework can be primarily used as an offline tool using historical data to quantify and predict the impact of topology data attacks on LMP, as well as provide guidelines for robust generation company’s bidding strategy/network planning and the prioritization of sensor upgrade against topology data attacks.

Future work includes the development of a framework to conduct an in-depth analysis of LMP changes to a more general topology error including substation configuration error under multi-line congestion. The developed framework should be verified and tested in larger test systems under various topology attack scenarios.