8.1.1 BNs at a Glance

Sequential data arise in many areas of science and engineering. The data may be a time series, generated by a dynamical system, or a sequence generated by a one-dimensional spatial process. In such problems, it is desirable to find the probability of future outcomes as a function of our inputs, and using the BNs shows a way do that. The BNs as a form of flexible and interpretable (graphical) models have been proffered as a promising framework for modeling complex systems such as power systems, as they can represent probabilistic dependence relationships among multiple interacting components and variables. The BN models illustrate the effects of system components upon each other in the form of an influence diagram. These models can be automatically derived from experimental data through a statistically founded computational procedure known as network inference. Although the relationships are statistical in nature, they can sometimes be interpreted as causal influence connections.

As mentioned, the BNs are a form of graphical modeling, in which dependencies among variables are described in a graph including nodes and arcs (edges). The nodes represent variables, and the arcs represent dependencies. Dependencies are statistical in nature. Figure 8.1 shows a simple BN with variables A, B, and C. Variables B and C are statistically dependent upon variable A. Therefore, an edge from A to B in Figure 8.1 indicates that knowing A can help to predict B. This may or may not indicate a causal relationship, i.e., one in which A directly or indirectly affects B.

The BNs are able to handle incomplete data sets. For example, consider a problem with two strongly anticorrelated input variables. Since they cannot encode the correlation between the input variables, when one of the inputs is not observable, producing an accurate prediction using most existing models is impossible. The BNs offer a natural way to encode such dependencies.

The BNs allow one to learn about the causal relationships between different variables. They allow making predictions in the presence of interventions. The BNs, in conjunction with Bayesian statistical techniques, facilitate the combination of domain knowledge and data. They have a causal semantics that makes the encoding of causal prior knowledge particularly straightforward. The Bayesian methods in conjunction with the BNs and other types of models offer an efficient and principled approach for avoiding the overfitting of data, while there is no need to hold out some of the available data for testing purposes. In other words, using the Bayesian approach, the study models can be smoothed in such a way that all available data can be used for training.

The BN approaches are not model based and can be easily scalable for large-scale systems, such as real-world power systems. They can also work well in a nonlinear environment with variable structures. A major advantage of the BNs over many other types of predictive and learning models, such as neural networks, is their possibility of representing the interrelationships among the data set attributes.

In general, a BN consists of (1) an acyclic graph S, (2) a set of random variables x = {x₁, …, x_n} (the graph nodes) and a set of arcs that determine the nodes’ (random variables’) dependencies, and (3) a conditional probability table (CPT) associated with each variable (p(x_i|pa_i)). Together, these components define the joint probability distribution for x. The nodes in S are in one-to-one correspondence with the variables x. In this structure, x_i denotes a variable and its corresponding node, and pa_i represents the parents of node x_i in S as well as the variables corresponding to those parents. The lack of possible arcs in S encodes conditional independencies. In particular, given structure S, the joint probability distribution for x is defined by

The probability encoded by a BN may be Bayesian or physical. In the case of building BNs from prior knowledge alone, the probabilities will be Bayesian, while learning these networks from data provides physical probabilities. The basic tasks related to the BNs are (1) the structure learning phase, i.e., finding the graphical model structure; (2) the parameter learning phase, i.e., finding nodes’ probability distribution; and (3) the BN inference.

The structure and parameter learning are based on the prior knowledge and prior data (training data) of the model. The basic inference task of a BN consists of computing the posterior probability distribution on a set of query variables q, given the observation of another set of variables e called the evidence (i.e., p(q|e)). Different classes of algorithms have been developed to compute the marginal posterior probability p(x|e) for each variable x, given the evidence e. No need to learn the inference data is an important characteristic in the BNs. Inference is a probabilistic action that obtains the probability of the query using prior probability distribution.

A graphical model to encode a set of conditional independence assumptions and a compact way of representing a joint probability distribution between random variables are the main features in a BN construction.

8.1.2 Graphical Models and Representation

In a BN, probabilistic relationships are usually represented by a simple directed acyclic graph (g). In the graph, the nodes represent variables and the edges represent dependencies. Therefore, the lake of edge between two variables indicates a conditional independency.¹³ In a directed acyclic graph (DAG), the edges are single-ended arrows, originating from one node (parent node, e.g., A in Figure 8.1) and ending in another (child node, e.g., B and C in Figure 8.1). Furthermore, a DAG does not include directed cycles, e.g., feedback from a node to itself. The type and value of dependency for each variable to its parent(s) is quantitatively described via a conditioned probability distribution (CPD). The CPD, which is consistent with the conditional independencies implied by g, can be described by a vector of its parameters. The variables in a BN may be continuous or discrete. However, in this chapter, for the AGC design issue, discrete variables are used. A BN represents the joint probability distribution for a finite set of random variables x, where x_i ∈ x may take on a value from a specific domain.

Graphical models are generated by probability and graph theories to introduce a natural tool for dealing with two problems that occur throughout applied mathematics and engineering. In particular, they play a significant role in the synthesis and analysis of machine learning algorithms. Building a complex system by combining simpler parts is the fundamental idea of a graphical model. The probability theory provides the glue whereby the parts are combined, ensuring that the system as a whole is consistent, and providing ways to interface models to data. The graph theoretic side of graphical models provides both an intuitively appealing interface by which humans can model highly interacting sets of variables, and a data structure that lends itself naturally to the design of efficient general purpose algorithms.²¹

The graphical model formulation provides a natural framework for the design of new systems. Many of the classical multivariate probabilistic systems studied in the fields of engineering and science are special cases of the general graphical model formulation. The graphical model framework provides a way to view real-world systems by transferring specialized techniques that have been developed in one field to other areas.

As mentioned, in the probabilistic graphical models, the nodes represent random variables and the arcs represent conditional independence assumptions between variables. The arcs’ pattern presents the graph structure. Hence, it provides a compact representation of joint probability distributions. For example, for N binary random variables, an atomic representation of the joint p(x₁, …, x_n) needs O(2ⁿ) parameters, whereas a graphical model may need exponentially fewer, depending on which conditional assumptions are considered.²¹

Usually, to construct a graphical model, the following two steps should be considered: (1) the model definition or learning phase for recognizing random variables (as graphical model nodes), the nodes’ probability distribution estimation, and the nodes’ dependencies identification (as model arcs), and (2) model inference for computing the posterior probability distribution on a set of query variables, given the observation of another set of variables (i.e., p(q|e)). Graphical models can be classified as undirected and directed models. The undirected graphical models (or Markov networks) are more suitable in the physical applications, while the directed graphical models (or BNs) are more popular with the AI and machine learning processes.

8.1.3 A Graphical Model Example

For the sake of illustration, consider an example on the blackout phenomenon following a significant disturbance/fault in a power system. In the event of a significant disturbance, generators and prime mover controls become important, as well as system control loops and special protections. In the case of improper coordination, it is possible for the system to become unstable, and generating units or loads may ultimately be tripped, possibly leading to a system blackout. As a power system fails, because of dynamic complexity and its high-order multivariable structure, more than one source of instability (frequency, angle, and voltage) may ultimately emerge. In a power system, the dynamic performance is influenced by a wide array of devices with different responses and characteristics. Hence, instability in a power system may occur in many different ways, depending on the system topology, operating mode, and form of the disturbance.

The impacts of a disturbance may also involve much of the system, depending on the power system topology, form of the disturbance, and operating mode. For example, a disturbance on a critical part, followed by its isolation by protective relays, will cause fluctuation in tie-line power flows, rotor angle speeds, and bus voltages. The machine speed variations will actuate prime mover governors and system frequency, the voltage deviation will affect both generator and voltage regulators, and the frequency and voltage deviation will affect the system loads to varying degrees, depending on their responses and different characteristics.²² Furthermore, the various protective relays and devices may respond to these variations, and thereby affect the power system performance, possibly even leading to a system blackout.

Based on the above description, the graph shown in Figure 8.2 can be considered an example to explain the consequence of events leading to a blackout. One may represent the dependencies among more important variables in a BN graph as shown in Figure 8.2. This BN represents the joint probability distribution between a serious disturbance (D), frequency deviation (F), rotor angle variation (A), system voltage deviation (V), protective devices and load response (PL), and system blackout (B).

In this graph, A and V are the children of D, F is the child of A and D, PL is the child of F, A, and V, and B is the child of PL. D has no parents and is called a root node. Assume each variable can take on the value 1 (in the case of existing) or 0 (in the case of absenting).

While the graph appears to represent variable dependencies, its primary purpose is actually to encode conditional independencies, critical for their ability to confer a compact representation to a joint probability distribution.²³ In the above example, B is dependent upon PL, PL is dependent upon F, A, and V, A and V are dependent upon D, and F is dependent upon D and A.

However, when the system is stabilized following a fault, B becomes independent of F, A, and V, as well as D. If we already know that a frequency deviation (voltage or angle variation) is occurring at F = 1 (V = 1, A = 1), then knowing something about the presence of D will not help us to determine the value of B as well as PL. Therefore, B and PL are conditionally independent of D given F, A, and V. Formally, x is conditionally independent of y given z if

The graph (g) encodes the Markov assumptions. As a consequence of the Markov assumption, the joint probability distribution over the variables represented by the BN can be factored into a product over variables, where each term is local conditional probability distribution of that variable, conditional on its parent variables (Equation 8.1).

The above chain rule states that the joint probability of independent entities is the product of their individual probabilities. A main advantage of the BN is its compact representation of the joint probability distribution. With no independence assumption, the joint probability distribution over the variables D, V, A, F, PL, and B is

For binary variables, this is ${\mathrm{\Sigma }}_{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathrm{6}}{\mathrm{2}}^{\mathit{i}\mathrm{-}\mathrm{1}}\mathrm{=}\mathrm{1}\mathrm{+}\mathrm{2}\mathrm{+}\mathrm{4}\mathrm{+}\mathrm{8}\mathrm{+}\mathrm{1}\mathrm{6}\mathrm{+}\mathrm{3}\mathrm{2}\mathrm{=}\mathrm{6}\mathrm{3}$ parameters. Employing the conditional independencies, the joint probability distribution for the graphical model shown in Figure 8.2 becomes

where, for binary variables of the example at hand, it is equal to 1 + 2 + 2 + 4 + 8 + 2 = 19 parameters.

In the case of the multinomial distributions, each CPD can be presented in a CPT. For binary variables, in order to specify all the parameters of the CPD for variable x_i with parent(s) pa_i, each CPT must have 2^k entries, where k is the number of parents. Returning to the blackout example, it is assumed that the variables are binary. The parameters of the CPTs are shown in Figure 8.3.

The previous CPTs include nineteen parameters that are specified for the present BN. While the graph reveals the conditional independencies, the CPTs demonstrate the strength of dependencies. The previous tables indicate various probabilities; for instance, the probability of a disturbance (D) occurring is 0.5, the probability of having a frequency deviation (F) when a disturbance (D) and angle deviation (A) are present is 0.99, and the probability of a blackout after affecting the system load and the response of protective devices (PL) is 0.15. The tables show that although both D and A can cause F, D alone has a stronger effect than A alone:

In the above calculations, the third variable is fixed at zero. Ignoring this assumption, the results will be computed as follows, which shows a different result:

8.1.4 Inference

Inference and reasoning from evidence and factual knowledge is the most common task in the BN applications, even for incomplete information. In an inference task, usually it is desirable to know the value of a particular node, without accessing that information. Usually, for this purpose, the evidence is in use. For example, in the previous example it is desirable to know if the disturbance (D) happened when the system frequency deviation (F) appeared (P(D|F)). Here, the F variable is called evidence. When evidence is available, one may reason about a cause of the instantiated variable. Bayes’ rule is used to compute such a probability:

Here, D is the unknown (hidden) node and F is the observed evidence. Without evidence, it is necessary to sum the joint probability distribution over all possible values of the other variables. For the given example, Equation 8.5 is calculated as follows:

and for P(D = 0|F = 1),

As another example, it is desirable to know if the frequency deviation (F) happened when the system load and protective devices were affected (P(F|PL)). This probability can be computed as follows:

Therefore,

Similarly, the probabilities for P(F = 1|PL = 0) and P(F = 0|PL = 0) can be calculated:

It is noteworthy that even for the binary case, the joint probability distribution has size O(2ⁿ), where n is the number of nodes. The full summation/ integration over the joint probability distribution using discrete/continuous variables is called exact inference, and for large networks takes a long time. In response to this challenge, approximate inference methods have been proposed in the literature, such as the Monte Carlo sampling method (which provides gradually improving estimates as sampling proceeds²⁴) and Markov chain Monte Carlo (MCMC) methods (including the Gibbs sampling and Metropolis– Hastings algorithm²⁵).

8.1.5 Learning

In practical applications, the BN is usually unknown, and it is necessary to learn it from performed training data, expert knowledge, and other prior information. Learning in a BN can refer to the BN structure (graph topology) or the BN parameters of the related joint probability distribution. Over the years, several learning methods/algorithms have been developed, such as MCMC, expectation maximization (EM), maximum likelihood estimation (MLE), and local search (LS) through model space techniques.¹⁷ Based on the BN structure (known/unknown) and rate of observability (full/partial), the existing BN learning methods can be classified into four classes, as shown in Table 8.1.

8.2.1 Frequency Control and Wind Turbines

The dynamic behavior of a power system in the presence of wind power units might be different from that in conventional power plants.⁷ The power outputs of such sources are dependent on weather conditions, seasons, and geographic location. When wind power is a part of a power system, additional imbalance is created when the actual wind power deviates from its forecast due to wind velocity variations (see Chapter 6). So, scheduling conventional generator units to follow the load (based on the forecasts) may also be affected by wind power output.

Furthermore, the effect of wind farms on the dynamic behavior of a power system may cause a different system frequency response to a disturbance event (such as load disturbance). Since the system inertia determines the sensitivity of the overall system frequency, it plays an important role in this consideration. Lower system inertia leads to faster changes in the system frequency following a load-generation variation. The addition of synchronous generation to a power system intrinsically increases the system inertial response.

In practice, an AGC system must be designed to maintain the system frequency and tie-line power deviations within the limits of specified frequency operating standards. In the existing standards, the acceptable range of frequency deviation for near-normal operation (AGC operating area) is small. This range is mainly determined by the available amount of operating reserved power in the system. It is noteworthy that the total AGC power reserve in a real power system is usually about 15%, and the AGC loop can usually track the load variation in the range of 1 to 10% of the overall load demand.

The AGC system is designed to operate during a relatively small and slow change in real power load and frequency. For large imbalances in real power associated with rapid frequency changes that occur during a fault condition, the AGC system is unable to restore the frequency. There is a risk that these large frequency deviation events might be followed by additional generation events, load/network events, separation events, or multiple contingency events. For such large frequency deviations and in a more complex condition, the emergency control and protection schemes must be used to restore the system frequency.¹

This intrinsic increase does not necessarily occur with the addition of wind turbine generators (WTGs) due to their differing electromechanical characteristics.⁴ So, the impact of wind farms on power system inertia is a key factor in investigating the power system frequency behavior in the presence of a high penetration of wind power generation.

As mentioned in Chapter 6, to analyze the additional variation caused by WTGs, the total effect is important, and every change in wind power output does not need to be matched one for one by a change in another generating unit moving in the opposite direction. The slow WTG power fluctuation dynamics and total average power variation negatively contribute to the power imbalance and frequency deviation, which should be taken into account in the well-known AGC control scheme. This power fluctuation must be included in the conventional area control error (ACE) formulation.

8.2.2 Generalized ACE Signal

The conventional AGC model is discussed in Chapter 2, Bevrani¹ and Kundur.²⁶ Following a load disturbance within the control area, the frequency of the area experiences a transient change and the feedback mechanism generates an appropriate rise or lower signal in the participating generator units according to their participation factors to make the generation follow the load. In the steady state, the generation is matched with the load, driving the tie-line power and frequency deviations to zero. As there are many conventional generators in each area, the control signal has to be distributed among them in proportion to their participation.²⁷

The frequency performance of a control area is represented approximately by a lumped load-generation model using equivalent frequency, inertia, and damping factors.²⁸ Because of the range of use and specific dynamic characteristics, such as a considerable amount of kinetic energy, the wind units are more important than the other renewable energy resources. The equivalent system inertia in a power system with wind units can be defined as

where H_sys is equivalent to the inertia constant. H_C and H_W are the total inertia constants due to conventional and wind turbine generators, respectively. The inertia constant for wind power is time dependent. The typical inertia constant for the wind turbines is about 2 to 6 s.¹

Similar to the given analysis and problem formulation in Section 6.1, the updated ACE signal should represent the impacts of wind power on the scheduled flow over the tie-line. The ACE signal is traditionally defined as a linear combination of frequency and tie-line power changes as follows:²⁶

where ∆f is frequency deviation, β is frequency bias, and ΔP_tie is the difference between the actual (act) and scheduled (sched) power flows over the tie-lines.

For a considerable amount of wind (W) power, its impact must also be considered with conventional (C) power flow in the overall area tie-line power. Therefore, the updated ΔP_tie can be expressed as follows:

The total wind power flow change is usually smooth compared to variation impacts from the individual wind turbine units. Using Equations 8.7 and 8.9, an updated ACE signal can be completed as

where P_tie-C,act, P_tie-C,sched, P_tie-W,act, and P_tie-W,estim are actual conventional tie-line power, scheduled conventional tie-line power, actual wind tie-line power, and estimated wind tie-line power, respectively.

In typical AGC implementations, the system frequency gradient and ACE signal must be filtered to remove noise effects before use. The ACE signal then is often applied to the controller block. Control dead-band and ramping rate are different for various systems.¹ The controller can send higher or lower pulses to generating plants if its ACE signal exceeds a standard limit.

8.3.1 Control Framework

The overall view of the proposed control framework for a typical area i is conceptually shown in Figure 8.4. The control scheme in each area presents two agents: an estimator agent for the estimation amount of load change, and an intelligent BN-based controller agent to provide an appropriate supplementary control action signal. The objective of the proposed design is to regulate the frequency and tie-line power in the power system concerning the integration of wind power units with various load disturbances and achieve a desirable control performance.

The two-agent schema entailed the minimum number of measurement/ monitoring and control activities in a control area to track the AGC tasks. The controller agent uses ΔP_tie (Equation 8.9) and load demand change (ΔP_L) signals to provide the control action signal (∆P_C). The estimator agent is responsible for estimating the amount of load change.

8.3.2 BN Structure

To find a clear view of the BN structure, it is better to start by determining the necessary variables for modeling. This initial task is not always straightforward. As part of this task, one should (1) correctly identify the modeling objective, (2) investigate important relevant observations, (3) determine what subset of those observations is worthwhile to model, and finally, (4) organize the observations into variables having mutually exclusive and collectively exhaustive states.

In the process of a BN construction for the AGC issue, the aim is to achieve the AGC objective and keep the ACE signal within a small band around zero using the supplementary control action signal. Then, the query variable in the posterior probability distribution is a ∆P_C signal and the posterior probabilities according to possible observations relevant to the problem (as shown in Figure 8.5) are as follows:

According to Equation 8.11, there are many observations that are related to the AGC problem; however, the best ones, which have the least dependency on the model parameters (e.g., frequency bias factor, etc.) and cause the maximum impact on the frequency deviation, and consequently ACE signal changes, are load disturbance and tie-line power deviation signals. Then the appropriate posterior probability that should be found is p(∆P_C|∆P_tie,∆P_L).

The ∆P_tie can be practically obtained using measurement instruments. However, ∆P_L is one of the input parameters that is not measurable directly, but it can be easily estimated using a numerical/analytical method. A simple method to estimate the amount of load change following a load disturbance is discussed in the next section. This estimation method is initially based on the measured frequency gradient and the specified system characteristics. Considering the AGC duty cycle (timescale), the total consumed time needed for the estimation process is not significant.

Another approach for constructing the graphical model of a BN can be described based on the following observations:²⁹ From the chain rule of probability, it is known that

For every x_i, there will be some subset π_i ⊆ {x₁, …, x_i–1} such that x_i and {x₁, …, x_i–1}/π_i are conditionally independent given π_i. That is, for any x,

Combining Equations 8.12 and 8.13,

Comparing Equations 8.8 and 8.10 shows that the variables sets (π₁, …, π_n) correspond to the BN parents (pa₁, …, pa_n), which in turn fully specify the arcs in the network structure S. Consequently, to determine the structure of a BN, one must (1) order the variables somehow and (2) determine the variables sets that satisfy Equation 8.13 for i = 1, …, n.

8.3.3 Estimation of Amount of Load Change

As mentioned, the estimator agent estimates the total power imbalance (amount of load change sensed in control area ΔP_L) by an assigned algorithm based on the following explained analytical method.

Consider the ith generator swing equation for a control area with N generators (i = 1, …, N):

where ΔP_mi is the mechanical turbine power, ΔP_Li is the load demand (electrical power), H_i is the inertia constant, D_i is load damping, and ΔP_di is the load-generation imbalance. By adding N generators within the control area, one obtains the following expression for the total load-generation imbalance:

Equation 8.16 shows the multimachine dynamic behavior by an equivalent single machine. Using the concept of an equivalent single machine, a control area can be represented by a lumped load-generation model using an equivalent frequency Δf, system inertia H, and system load damping D.¹

The magnitude of the total load-generation imbalance ΔP_D, after a while, can be obtained from Equation 8.16.

where

and ∆P_tie is defined in Equation 8.9. The total mechanical power change indicates the total power generation change due to governor action, which is in proportion to the system frequency deviation:²⁶

Equations 8.19, 8.20, and 8.23 give

Thus, the total load change in a control area is proportional to the system frequency deviation. Neglecting the network power losses, ∆P_D(t) shows the load-generation imbalance proportional to the total load change. Using Equation 8.16, the magnitude of total load-generation imbalance immediately after the occurrence of disturbance at t = 0⁺ s can be expressed as follows:

Equations 8.20 and 8.25 give

Here, ∆f is the frequency of the equivalent system. To express the result in a suitable form for sampled data, Equation 8.26 can be represented in the following difference equation:

where T_S is the sampling period and ∆f₁ and ∆f₀ are the system equivalent frequencies at t₀ and t₁ (the boundary samples within the assumed interval).

8.4.1 BN Construction

After determining the most worthwhile subset of the observations (∆P_tie, ∆P_L), in the next phase of the BN construction, a directed acyclic graph that encodes assertion of conditional independence is built. It includes the problem random variables, conditional probability distribution nodes, and dependency nodes.

The basic structure of the needed graphical model for the AGC issue, which is shown in Figure 8.5, is built based on prior knowledge of the problem. According to Equation 8.7, the ACE signal is dependent on the frequency and tie-line power deviations; then they will be the parent nodes of the ACE signal (control input) in the BN graphical model, and since frequency deviation is dependent on the load disturbance and tie-line power deviation, they will be parent nodes of ∆f. Since ∆P_C as the controller output is considered to be dependent on the ACE signal only, the ACE node will be the parent node for the control action signal.

Using the ordering (∆P_tie, ∆P_L, ∆f, ACE, ∆P_C) and according to Equation 8.13, the conditional dependencies are described as follows:

In the graphical model of a BN, each node presents a system variable. The edges between nodes present dependency between the system variables. In a BN, the aim is to find the probability distribution of the graphical model nodes from training data (parameter learning), and then do an inference task according to the observation. In the graphical model each node has a probability table and nodes with parents have conditional probability tables (because they are dependent on their parents).

The graphical model of the AGC problem (Figure 8.5) is based on the right side of the described relationships in Equation 8.28. In the next step of the BN construction (parameter learning), the local conditional probability distributions p(x_i|pa_i) must be computed from the training data. Probability and conditional probability distributions related to the AGC issue, according to Figure 8.5, are p(∆P_L), p(∆P_tie), p(∆f|∆P_L,∆P_tie), p(ACE|∆P_tie,∆f), and p(∆P_C|ACE). To find these probabilities, the training data matrix should be provided.

Here, the Bayesian networks toolbox (BNT)¹⁷ is used for probabilistic inference of the model. The BNT uses the training data matrix and finds the conditional probabilities related to the graphical model variables (as the parameter learning phase).

Once a BN is constructed (from prior knowledge, data, or a combination), various probabilities of interest from the model can be determined. For the problem at hand, it is desired to compute the posterior probability distribution on a set of query variables, given the observation of another set of variables called the evidence. The posterior probability that should be found is p(∆P_C|∆P_tie,∆P_L). This probability is not stored directly in the model, and hence needs to be computed. In general, the computation of a probability of interest given a model is known as probabilistic inference.

8.4.2 Parameter Learning

As mentioned and shown in the graphical model of a control area (Figure 8.5), the essential parameters used for the learning phase among each control area of a power system can be considered as ∆P_tie, ∆P_L, ∆f, ACE, and ∆P_C.

In order to find a related set of training data (∆P_tie, ∆P_L, ∆f, ACE, ∆P_C) for the sake of the parameter learning phase, one can provide a long-term simulation for the considered power system case study in the presence of various disturbance scenarios. This large learning set is partly complete and can be used for the parameter learning issue in the power system with a wide range of disturbances. Since the BNs are based on inference and new cases (which may not be included in the training set) can be inferred from the training table data, it is not necessary to repeat the learning phase of the system for different amounts of disturbances occurring in the system.

After providing the training set, the training data related to control areas are given to the BNT separately. The BNT uses the input data and does the parameter learning phase for each control area’s parameters. It finds prior and conditional probability distributions related to that area’s parameters, which, according to Figure 8.5, are p(∆P_L), p(∆P_tie), p(∆f|∆P_L,∆P_tie), p(ACE|∆P_tie,∆f), and p(∆P_C|ACE). Following completion of the learning phase, the power system simulation will be ready to run and the proposed model uses the inference phase to find an appropriate control action signal (∆P_C) for each control area.

During the simulation stage, the inference phase is done as follows: At each simulation time step, the corresponding controller agent of each area gets the input parameters (∆P_tie,∆P_L) of the model and digitizes them for the BNT (the BNT does not work with continuous values). The BNT finds the posterior probability distribution values p(∆P_C|∆P_tie,∆P_L) related to each area. Then, the controller agent finds the maximum posterior probability distribution from the return set, and gives the most probable evidence ∆P_C in the control area.

8.5.1 Thirty-Nine-Bus Test System

The thirty-nine-bus test system is widely used as a standard system for testing of new power system analysis and control synthesis methodologies, as well as in Chapters 6, 7, and 11 of this book. A single line diagram of the system is given in Figure 8.6. This system has ten generators, nineteen loads, thirty-four transmission lines, and twelve transformers. The well-known test system is updated by adding two wind farms in buses 5 and 21, as shown in Figure 8.6. The system is divided into three areas.

The total system installed capacity is 841.2 MW of conventional generation and 45.34 MW of wind power generation. There are 198.96 MW of conventional generation, 22.67 MW of wind power generation, and 265.25 MW load in area 1. In area 2, there are 232.83 MW of conventional generation and 232.83 MW of load. In area 3, there are 160.05 MW of conventional generation, 22.67 MW of wind power generation, and 124.78 MW of load.

The simulation parameters for the generators, loads, lines, and transformers of the test system are given in Bevrani et al.³¹ All power plants in the power system are equipped with a speed governor and power system stabilizer (PSS). However, only one generator in each area is responsible for the AGC task: G1 in area 1, G9 in area 2, and G4 in area 3.

In the present work, similar to the real-world power systems, it is assumed that conventional generation units are responsible for providing spinning reserve for the purpose of load tracking and the AGC task. For the sake of simulation, random variations of wind velocity have been considered. The dynamics of WTGs, including the pitch angle control of the blades, are also considered. The start-up and rated wind velocities for the wind farms are specified as about 8.16 and 14 m/s, respectively. Furthermore, the pitch angle controls for the wind blades are activated only beyond the rated wind velocity. The pitch angles are fixed to 0° at the lower wind velocity below the rated one.

To cope with real-world power systems, in the performed application the important inherent requirement and basic constraints, such as governor dead-band and generation rate constraint imposed by physical system dynamics, are considered. For the sake of simulation, three step load disturbances are simultaneously applied to the three areas: 3.8% of the total area load at bus 8 in area 1, 4.3% of the total area load at bus 3 in area 2, and 6.4% of the total area load at bus 16 in area 3. Using the simulation, the training table rows can be built in the format shown in Table 8.2. The applied step load disturbances ∆P_Li (pu), the output power of wind farms P_WT (MW), and the wind velocity V_W (m/s) are considered as shown in Figure 7.5.

A simple presentation of probability tables using the proposed graphical model (Figure 8.5), according to the training data, after the parameter learning phase for the test system, is shown in Table 8.3. Some samples of returned posterior probability distribution values p(∆P_C|∆P_tie,∆P_L) from BNT environment are also shown in Table 8.4.

The frequency deviation (∆f) and area control error (ACE) signals of the closed-loop system are shown in Figures 8.7 and 8.8. As a sample, the produced mechanical power by the AGC participant unit in area 2, corresponding electrical power, and overall tie-line power for the same area are shown in Figure 8.9.

In the proposed simulations, the wind power impacts on the overall system frequency behavior can be clearly seen. The fast movements in wind power output are combined with movements in load and other resources. The power system response is affected by the wind power fluctuation. When wind power is a part of the power system, an additional imbalance is created when the actual wind output deviates from its forecast. Before the load disturbance occurred in 10 s, there was also a little oscillation during the simulation that was initially caused from random variations of wind velocity, and it was magnified due to the resulting power fluctuation.

TABLE 8.4
Some Samples of Probabilities According to the Graphical Model

It is shown that using the proposed method, the area control error and frequency deviations in all areas are properly driven close to zero in the presence of wind turbines and load disturbance. Furthermore, the areas’ frequency deviation is less than the frequency deviation in the system with multiagent RL-based controllers.

8.5.2 A Real-Time Laboratory Experiment

Since the AGC as a supplementary control is known as a long-term control problem (few seconds to several minutes), it is expected that the proposed AGC methodology will be successfully applicable in real-world power systems. To illustrate the capability of the proposed control strategy in real-time AGC applications, an experimental study has been performed on the large-scale Analog Power System Simulator (APSS)¹ at the Research Laboratory of the Kyushu Electric Power Company (Japan). For the purpose of this study, a longitudinal four-machine infinite bus system is considered as the test system. A single line diagram of the study system is shown in Figure 8.10a. All generator units are of the thermal type, with separately conventional excitation control systems. The set of four generators represents a control area (area I), and the infinite bus is considered as the other connected systems (area II).

The detailed information of the system, the parameters of generator units, and turbine systems (including the high-pressure, intermediate-pressure, and low-pressure parts) are given in Bevrani and Hiyama.²⁸ The wind farm (WF) consists of 200 units of 2 MW rated variable speed wind turbines (VSWTs). The VSWT parameters are indicated in Table 8.5. Although in the given model the number of generators is reduced to four, it closely represents the dynamic behavior of the West Japan Power System. As described in Bevrani and Hiyama,²⁸ the most important global and local oscillation modes of the actual system are included.

The whole power system (shown in Figure 8.10a) has been implemented using the APSS. Figure 8.10b shows an overview of the applied laboratory experimental devices, including the generator panels, monitoring displays, and control desk. The proposed control scheme, including estimator and controller agents, was built in a personal computer and connected to the power system using a digital signal processing (DSP) board equipped with analog-to-digital (A/D) and digital-to-analog (D/A) converters. The converters act as the physical interfaces between the personal computer and the analog power system hardware.

The performance of the closed-loop system is tested in the presence of load disturbances. The nominal area load demands are also fixed at the same values given in Bevrani and Hiyama.²⁸ More than 10% of the total demand power is supplied by the installed wind farm in bus 9.

For the first scenario, the power system is tested following a step loss of 0.06 pu conventional generation. The participation factors for Gen 1, Gen 2, Gen 3, and Gen 4 are fixed at 0.4, 0.25, 0.20, and 0.15, respectively. The applied step disturbance and the closed-loop system response, including frequency deviation (Δf) and tie-line power change (ΔP_tie), are shown in Figure 8.11. This figure shows that the frequency deviation and tie-line power change are properly maintained within a narrow band.

TABLE 8.5
The VSWT Parameters

As a severe test scenario, the power system is examined in the presence of a sequence of step load changes. The load change pattern and the system response are shown in Figure 8.12. The obtained results show that the designed controllers can ensure good performance despite load disturbances. It is shown that the proposed intelligent AGC system acts to maintain area frequency and total exchange power closed to the scheduled values by sending a corrective smooth signal to the generators in proportion to their participation in the AGC task.

The experiment results illustrate that the system performance using the proposed BN-based multiagent controller is quite better than the multiagent RL technique. It has been shown^30,31 that the multiagent RL design presents greater performance than the conventional (well-tuned) proportional-integral (PI)-based AGC systems. Therefore, in comparison with an existing conventional AGC system, the closed-loop performance of the present BN-based multiagent scheme is significantly improved. In summary, flexibility, a higher degree of intelligence, model independency, and handling of incomplete measured data (uncertainty consideration) can be considered some important advantages of the proposed methodology.

In the real-time simulations, the detailed dynamic nonlinear models of wind turbines are used without applying an aggregation model for the turbine units. That is why in the simulation results, in addition to the long-term fluctuations, the faster dynamics on a timescale of seconds are also observable.

Although the nonlinear simulation results for the test systems illustrate the capability of the proposed intelligent-based AGC scheme, certainly the observed dynamic response is still far from a real-world large-scale power system, with tens of conventional power plants and numerous distributed wind generators and other power sources. In a real large-scale power system with a high penetration of real power, the power system does not need to respond to the variability of each individual wind turbine.⁸ Rapidly varying components of system signals are almost unobservable due to various filters involved in the process.