WO2020245752A1

WO2020245752A1 - Method and system for assessing the island mode micronetwork stability

Info

Publication number: WO2020245752A1
Application number: PCT/IB2020/055258
Authority: WO
Inventors: Daniele MESTRINER; Alessandro Giuseppe LABELLA; Renato Procopio; Andrea BONFIGLIO; Massimo BRIGNONE
Original assignee: Universita' Degli Studi Di Genova
Priority date: 2019-06-05
Filing date: 2020-06-04
Publication date: 2020-12-10

Abstract

The invention relates to a method for evaluating the stability of a microgrid comprising a plurality (N) of sources (Si) controlled via a logic (mi, ni), the voltage / current relations of the network being expressed in matrix form (Y) as a function of the load on the network so as to be able to calculate the active power (Pi) and the reactive power (Qi) injected by the sources (Si) into the micronetwork for each load value, which method involves: a) measuring the amplitudes (Vi) and the phase shifts (δί) of the voltages output from each source (Si); b) receiving as input the values of the voltage / current relationship matrix (Y), such as, for example, the admittance matrix; c) receiving as input the parameters (mi, ni, Pni, Qni) of the micro-network control logic; d) calculating, for each pair of sources (Si, Sj), a set of parameters (Aij, Bij, Cij) as a function of the voltages Vi, the values of the matrix of the voltage / current relations (Y) and the parameters of the logic of network control (mi, ni); e) determining the stability of the micronetwork for a specific load condition by checking if the set of parameters (Aij, Bij, Cij) satisfies a condition for at least a subset (CS) of the possible combinations of pairs of sources (K). The invention also relates to a corresponding system.

Description

"Method and system for assessing the island mode micronetwork stability"

DESCRIPTION

The present invention relates to a method for evaluating the stability of micronetworks in island mode .

The evolution of the electrical system of our days has caused a strong development of the so-called micronetworks due to their flexibility, high reliability and ability to operate both connected to the traditional network and in island mode. Island mode micronetworks, i.e. not connected to the main network, represent the most interesting challenge of recent years, especially in those cases where there are no inertial sources (i.e., all generation sources are connected to the micronetwork thanks to power converters) . In this category, one of the most important problems concerns stability, which, due to the presence of power converters and the lack of inertia, cannot be assessed with the classic approaches designed for traditional power systems.

The importance of this topic is evidenced by a large number of studies carried out by many researchers. However, these studies were mainly carried out based on the theory of small signal stability; in particular, the models of inverters, synchronous generators, rectifiers and asynchronous motors have been studied in depth to be able to be applied to this theory. The main disadvantage of the approach to small signals is related to the fact that the validity of the results is limited to the neighborhood of the specific working point only, and this neighborhood is of unknown extent a priori, and therefore it is not possible to provide an estimate of the amplitude of the noise that can be tolerated without compromising the stability of the system. To overcome this problem, the system must be analyzed taking into account its non-linearities and this can be done in two ways: 1) by numerical simulations and 2) by applying Lyapunov Theory .

The first approach allows an accurate assessment of the stability of the system, but does not provide any analytical conditions. Therefore, every time you want to analyze a different configuration, a new system simulation must be done, with a consequent prohibitive increase in the computational cost.

On the other hand, approaches based on Lyapunov's theory have not been applied positively to the assessment of the stability of traditional electrical systems due to their complexity. In the case of micronetworks, which are much less extensive and complex, these approaches are much more applicable. In literature, some studies have been carried out to evaluate stability to large signals, but often rely on unrealistic hypotheses or on too simple infrastructures that make them inapplicable to low-voltage and medium- voltage micro-networks.

Given the current state of the art, the invention therefore aims to fill the void present in the evaluation of the stability of non-inertial micro- islands on the island, which can be generalized to any type of infrastructure.

The invention achieves the aim with a method for assessing the stability of a micronetwork comprising a plurality of sources controlled via a logic, the voltage / current relations of the network being expressed in matrix form as a function of the load on the network so as to be able to calculate the power active and the reactive power injected by the sources into the micronetwork as the load varies, which method involves :

a) measuring the amplitudes and phase shifts of the voltages output from each source;

b) receiving the values of the voltage / current relationship matrix as input, such as, for example, the admittance matrix;

c) receiving the parameters of the micro-network control logic at the input;

d) calculating, for each pair of sources, a set of parameters according to the voltages, the values of the voltage / current relationship matrix and the parameters of the network control logic;

e) determining the stability of the micronetwork for a specific load condition by checking if the parameter set satisfies a condition for at least a subset of the possible combinations of source pairs.

Advantageously, the method provides for determining the subset (CS) of the possible pairs of sources (Si) by measuring the angular differences between the voltages of all the possible pairs of sources (k) at an initial instant and selecting a predetermined number as a subset (CS) (N-l) of pairs having the largest angular differences. In this way it is possible to set the system of differential equations that governs the electrical quantities of the entire network to a Cauchy problem with analytically determinable solutions without the need to perform analysis on small signals around the equilibrium points as required by the state of the art. This is particularly advantageous because it allows a priori to determine whether a load condition on the network can give rise to instability even before it occurs, allowing the operator to intervene in advance to balance the network. The above without calculating the values of the phasors of the currents and voltages, but evaluating the existence of a simple mathematical relationship between parameters related to these quantities .

According to another aspect, the invention relates to a system for assessing the stability of a micro-network comprising a plurality of sources (Si) controlled via a logic, the voltage / current relations of the network being expressed in matrix form as a function of the load on the network so as to to be able to calculate the active power and the reactive power injected by the sources into the micronetwork as the load varies, which system includes:

a) a module for measuring the amplitudes of the voltages output from each source;

b) an interface configured to receive commands from a user ;

c) a memory suitable for containing program instructions ;

d) an input for reading and / or entering the values of the voltage / current relationship matrix, such as, for example, the admittance matrix; e) an input for reading and / or entering the parameters of the micro-network control logic;

f) a processor configured to execute the program instructions to implement the steps of the method according to the invention and send the results of the processing to an output, in particular an indication of the presence of stability or instability of the micro-network .

Further characteristics and improvements are the subject of the dependent claims.

The features of the invention and the advantages deriving from it will become clearer from the following detailed description of the attached figures, in which: Fig. 1 shows the general scheme of an inertia-free island micronetwork.

Fig. 2 shows an exemplified block diagram of the microgrid of the previous figure with stability control system according to an embodiment of the invention. Fig. 3 shows the detail of the processing group 1 of figure 2.

Fig. 4 shows the flow diagram of a method according to an embodiment of the invention.

Figure 1 shows a generic micronetwork, free of inertia, with N sources Si connected to the distribution network via a DC / AC converter Ii, a filter Fi and a transformer Ti .

The i-th source is managed by supplying the modulation index and the frequency of the modulating device to the corresponding DC / AC converter Ii by means of the logic Ci , indicated in the figure as distributed or separate for each source, but obviously also integrable in a single control module.

If we assume a first harmonic model for the converter and neglect the dynamics of the internal control loops, the i-th DC / AC converter Ii can be modeled as a controlled voltage source whose inputs are the amplitude of the voltage Vi is the angular frequency wi .

In case of droop-type control, these inputs are supplied to the converter through the following logic:

where the parameters P_{set , i} (active power reference) , Q_set,i (reactive power reference) , V_ni (voltage reference) and wn (nominal frequency of the microgrid) represent logic inputs that can be set by the user. Furthermore, Pi, Qi , Vi, wi are, respectively, the active and reactive power measurements, the voltage amplitude and the angular frequency while m_i and n_i are the droop coefficients (also called "statisms") of the active power and responsive. The angular frequency fi)i of the i-th converter is linked to the corresponding angle di by the following relationship:

where w_base is the base chosen for the angular frequency.

With the assumptions that:

• The part of the alternating current (AC) network is fully operational;

• Loads are modeled with constant impedance,

the AC network can be modeled with the extended admittance matrix and therefore the injections of active power Pi and reactive power Qi at the i-th bar can be calculated as follows:

where the operator * represents the conjugate of the complex number, , di is the voltage angle of the

i-th source and the matrices G_E and B_E represent, respectively, the conductance matrix and the susceptibility matrix that define the extended matrix of the admittances like YE = GE + jB_E .

By inserting equation (4) in (3) and in (1) , it can be obtained that, for each i = 1 ... N:

The proposed system of equations is an algebraic- differential system of equations, whose general form is :

where

while all the other parameters (droop coefficients, elements of the admittance matrix of the network, etc.) are grouped in a multi-dimensional variable k. Obviously, f and g are multi-dimensional functions that reproduce the relationships expressed in (5) .

Suppose that, for the micro-network considered, there is an equilibrium point such that:

Where each variation in the network (e.g. a load variation, the opening of a line, etc.) can be coded in a change of variable k from ko to ki. Consequently, the dynamics of the micronetwork is described by system (6), setting k = k₁ and the initial values y(0) = y₀.

With this consideration, the stability of the micronetwork corresponds to the existence of a new equilibrium point and the possibility of reaching it. The approach commonly adopted to address this problem is based on Lyapunov's linearization method: if the system linearized around the new equilibrium point yi is stable (i.e. all the Jacobian eigenvalues of the system are on the left side of the complex plane) , then the equilibrium point is asymptotically stable locally. However, as shown below, this result does not guarantee that the new equilibrium point can be reached starting from the starting point yo . In other words, in order to apply this theory, it would be necessary to verify that the initial condition belongs to the domain of attraction of the final equilibrium point.

The inventors have therefore designed an alternative method for evaluating the stability of the signals of a micronetwork.

To better understand the teachings underlying the invention, a simplified case will now be described which provides only two regulated sources with only the droop control on the active power enabled. Disabling the control on reactive power means that n± in the second system equation (1) is zero and, therefore, the voltages Vi are constant.

The results obtained will then be generalized for each inertia-free micronetwork with both activated droop controls (both active and reactive power) and for a number N of sources.

Micronetwork with two sources

Rewriting the first equation of the system (5) with N = 2 and adding the two differential equations, we obtain the following one-dimensional Cauchy problem:

and having defined:

Since f is a continuous and differentiable function, (9) is a Cauchy problem with existence and uniqueness of the solution. So if f (yo) = 0 then the solution is y(t) = _Yo. Otherwise if f (yo) ¹ 0, the solution can be easily obtained as:

where

whose invertibility is a consequence of the continuity of f and therefore of the existence of a neighborhood of yo where f has a constant sign. Rewriting f in the following way:

it is possible to demonstrate that the following condition is sufficient to make sure that y is asymptotic and limited:

Proof: if (16) were not verified, equation (14) would always be different from zero, therefore H would be unlimited and defined everywhere; this implies that y diverges. Otherwise, if the condition expressed in (16) is verified, the zeros of the function f belong to the following set:

The domain of H can be obtained by evaluating the order of the zeros of f. For this purpose, it is necessary to know the value of the first and second derivative of f for each:

From equations (18) and (19) it is clear that if y is a zero of the first order then A²-B²-C²< 0, otherwise if A²-B²-C²= 0, y is a zero of the second order. In both cases, the order is greater than or equal to one and therefore the integrand function that defines H diverges into . Hence, if (16) is verified, the domain of H is , where and is the

largest zero of f such that and is the smallest

zero of f such that

. Furthermore, if f is positive (negative) in I, then and

and therefore:

This proves that condition (16) is sufficient to obtain a solution of y that becomes constant for t large enough, i.e. for the existence of a final working point such that yo belongs to its domain of attraction.

Extension of the proposed approach to a generic micronetwork with N sources

In order to extend the results proposed previously to a more realistic configuration, the following must be taken into consideration:

· The need to consider a micronetwork with more than two sources controlled with droop logic;

• The inclusion of the droop control on reactive power (considering the second system equation (5) )

This is done by applying some simplifying hypotheses on system (5) that allow to describe the dynamics of the micronetwork in an approximate way through N-l decoupled differential equations of the type described in (9) - (10) and therefore to evaluate the stability of the micronetwork through N-1 conditions such as that described in (16) . With this objective, it should be noted that the second equation of (5) can be seen as a system of N algebraic equations having as unknown the amplitude of the voltages Vi (with i = 1..N) . The system solution provides the correlation between the amplitude of the voltages and the phases. However, since this system is rarely analytically solvable, a simplified approach according to an embodiment of the invention is proposed below.

Consider the second equation of (5) ; assuming that the voltage of the i-th converter is weakly influenced by changes in the angles and voltages of the other converters, it can be assumed that:

Where d₀ , d_k0 and d_k0 are the initial values of angles and stresses. Thanks to this hypothesis, the second equation of the system (5) becomes a set of algebraic and decoupled second degree equations that can be solved to find the values of Vi .

Hypothesis (22) has two main consequences: 1) allows to solve the non-linear algebraic system in an approximate but analytical way and 2) provides a value for the voltages Vi to be used in the first equation of (5) .

This implies that the right part of the differential equations in (5) is again a linear combination of trigonometric functions. This property is fundamental in order to extend the results obtained with two sources to the current case.

The last step needed to extend the results of the simplified section with two machines is to consider the angular differences instead of the dynamics of the single angle.

Rewriting the first equation of (5) as a function of angular differences, it follows that:

for every

In a micronetwork with N sources there are K = (N² -N)/2 possible angular differences. Among them, only N-l are linearly independent. Following the criterion followed for the stability of traditional electrical systems, suppose you divide the set of angular differences into two groups: the first, henceforth called Critical Sources (CS) , contains N-1 differences that satisfy the following property:

• They are linearly independent

• They are those which, at the initial moment, assume the largest N-1 values among all the possible differences

The remaining K-N + 1 differences are referred to as Non-Critical Sources (NCS) .

Considering the right part of equation (23) , in the hypothesis that, the following approximation can be made :

The meaning of (24) is that the angular difference belonging to the NCS set can be kept constant at its initial value, while the angular difference belonging to the CS set needs to maintain explicit dependence on the term. The idea that supports the hypothesis presented in (24) is that the sources, whose angles are close to the beginning of the transient, remain close even during the entire transient since both the starting point and the end point are determined by the same control law.

Thanks to equation (24) and using the properties of the trigonometric functions of sine and cosine functions, we obtain that the system of equations (23) is reduced to the following set of N-l independent differential equations:

which have the same shape as the equation (10) . The dynamics of the micronetwork converge therefore in an equilibrium point if the N-l conditions of the type (16) are satisfied, that is:

With ij belonging to the angular differences of the CS set

And in which applying the equation (11) to the generic case instead of to the specific case with i = 1 and j = 2 described above, the following definitions are obtained:

Figure 3 shows the block diagram of a system capable of exploiting the teachings of the present invention. In figure 2 said system, indicated by the dotted box, is shown integrated in the micronetwork of figure 1. A single system is sufficient to evaluate the stability of an entire micronetwork, but it is obviously possible to also provide for the use of multiple systems of the same type able to act on parts of the same in a distributed way.

The system includes:

a) a measurement module 2 of the amplitudes (Vi) and of the phase shifts (di) of the voltages output from each source Si;

b) an interface 101 configured to receive commands from a user;

c) a memory 201 adapted to contain program instructions ;

d) an input 301 for reading and / or entering the values of the voltage / current Y relationship matrix, such as, for example, the admittance matrix;

e) an input 401 for reading and / or inserting the parameters of the control logic Ci of the micronetwork, such as the references ( Pni , Qni) and the droop coefficients (mi, ni) of the active and reactive power; f) a 501 processor configured to execute the program instructions to implement the steps of the method according to the invention. With reference to the example shown in figure 4, these steps provide for: reading the values of the measurements of the amplitudes of the voltages Vi coming from the measuring module 2 ;

calculating, for each pair of sources (Si, Sj), a set of parameters (Aij , Bij , Cij) as a function of the voltages Vi, the values of the matrix of the voltage / current relations Y and the parameters of the network control logic such as analytically described by equations (11) suitably rewritten with reference to the indices i, j and not to the specific case with i = 1 and j = 2 explained above thus obtaining the following definition for the said parameters

; determining the stability of the micronetwork for a specific load condition by checking whether the set of parameters (Aij , Bij , Cij) satisfies a condition for at least a subset (CS) of the possible combinations of pairs of sources (K) ;

The system also includes:

g) an output 601 to provide the results of the processing, in particular an indication of the presence of stability or instability of the micro-network.

Specifically, the processor can be configured to determine the subset (CS) of the possible pairs of sources (Si) by measuring the angular differences between the voltages of all the possible pairs of sources (k) at an initial instant and selecting as a subset (CS) a predetermined number (N-1) of pairs having the largest angular differences.

In particular, the processor provides an indication of the stability of the micronetwork if, starting from an initial value of the amplitudes (Vio) and of the phase shifts (di0) of the voltages for an initial configuration of the network (ko) , a new load configuration (k1) of the network implies that the vector (y) of the phase shifts (di) is asymptotic and unlimited, said stability situation being evaluated in analytical form by the processor, checking if a relationship between network parameters is satisfied.

According to an embodiment, the processor provides an evaluation of the stability of the micro-network using a relationship between the elements (di) of the vector (y) of the voltage phase shifts of the type:

where i, j belonging to the set of pairs of sources (CS) with angular difference between the non-constant phase shifts, in which Aij , Bij , Cij are parameters calculated on the basis of the voltage / current relationships imposed by the micro-network control .

In particular, the processor determines the parameters Aij , Bij , Cij and outputs an indication of stability if the following relationship is satisfied:

where i, j belonging to the set of pairs of sources (CS) with angular difference between the non-constant phase shifts.

Claims

1. Method for assessing the stability of a micronetwork comprising a plurality (N) of sources (Si) controlled through a logic (mi, ni) , the network voltage/current relationships can be expressed in the matrix form (Y) as a function of the network load so to be able to calculate the active power (Pi) and the reactive power (Qi) injected by the sources (Si) into the micronetwork as a function of the load, which method provides for:

a) measuring the amplitudes (Vi) and the phase shifts ( di) of the output voltages of each source (Si) ; b) receiving the input of the values of the matrix of the voltage/current relationships (Y) , such as, for example, the admittance matrix;

c) receiving the input of the parameters (mi , ni , P_ni , Q_ni) of the micronetwork control logic;

d) calculating, for each pair of sources (Si, Sj) , a set of parameters (Aij , Bij , Cij) as a function of the voltages Vi, of the values of the matrix of voltage/current relationships (Y) and of the parameters of the network control logic (mi, ni) ;

e) determining the micronetwork stability for a specific load condition by checking if the set of parameters (Aij , Bij , Cij) meets a condition for at least one subset (CS) of the possible combinations of pairs of sources (K) .

2. Method according to claim 1, wherein the step of determining the subset (CS) of the possible pairs of sources (Si) is provided by measuring the angular differences between the voltages of all the possible pairs of sources (k) at an initial time and by selecting as subset (CS) a predetermined number (N-1) of pairs having the largest angular differences.

3. Method according to claim 1 or 2 , wherein the micronetwork stability is assessed by considering a relationship between the vector (y) of the phase shifts (di) and that (x) of the amplitudes (Vi) of the source voltages of the type:

where f and g are multidimensional functions and k is a vector containing the network parameters for an established load condition.

4. Method according to one or more of the preceding claims, wherein the micronetwork is considered stable if, starting from initial value of the voltage amplitudes (Vio) and phase shifts (di0) for an initial configuration of the network (ko) , a new load configuration (k1) of the network results in the vector (y) of the phase shifts (di) being asymptotic and unlimited, said stability situation being evaluated analytically by checking if a relationship among the network parameters is satisfied.

5. Method according to one or more of the preceding claims, wherein the control logic is of the droop type wherein:

where wi and w_n are, respectively, the angular frequency of the output voltage of the i-th source and the angular reference frequency of the micronetwork, mi the droop coefficient of the active power Pi injected into the network by the i-th source having reference Pni, ni the droop coefficient of the reactive power Qi injected into the network by the i source having reference Q_ni and Vi is the voltage amplitude at the i- th source having reference V_ni.

6. Method according to one or more of the preceding claims , wherein the relationship between the vector (y) of the phase shifts (di) and that (x) of the amplitudes (Vi) of the source voltages is of the type:

where w_base is the base selected for the angular frequency, Vi the amplitude of the output voltage of the source i, the matrices G_E and B_E represent, respectively, the conductance matrix and the susceptance matrix that define the admittance matrix as Y = GE + jB_E.

7. Method according to one or more of the preceding claims , wherein the relationship between the vector (y) of the phase shifts (di) and that (x) of the amplitudes (Vi) of the source voltages is simplified by assuming that

where are the initial angle and

voltage values.

8. Method according to one or more of the preceding claims , wherein the relationship between the vector (y) of the phase shifts (di) and that (x) of the amplitudes (Vi) of the source voltages, used to assess the micronetwork stability, is of the type:

with , where N is the number

of sources, w_base is the base selected for the angular frequency, V_± is the amplitude of the output voltage of the i-th source, the matrices G_{E, ij} and B_E,ij represent, respectively, the elements of the conductance matrix and susceptance matrix, and m_i the droop coefficient of the active power of the i-th source .

9. Method according to one or more of the preceding claims , wherein the relationship between the vector (y) of the phase shifts (d_±) and that (x) of the amplitudes (Vi) of the source voltages is simplified by assuming that the angular difference between voltages of different sources can be considered constant at its initial value for a subset (CS) of the possible source pairs.

10. Method according to one or more of the preceding claims, wherein, in order to assess the micronetwork stability, a relationship among the elements (di) of the vector (y) of the phase shifts is used, of the type:

with i, j belonging to the set of the pairs of sources (CS) with not constant angular difference between the phase shifts, wherein Aij , Bij , Cij are parameters calculated based on the voltage/current relationships imposed by the micronetwork control logic .

11. Method according to claim 10, wherein the micronetwork is considered stable if the following relationship is satisfied:

with i, j belonging to the set of pairs of sources (CS) with not constant angular difference between the phase shifts.

12. Method according to one or more of the preceding claims, wherein , for a micronetwork (i = 1, 2) with two single sources with droop control logic of the type w i - w 0 = mi ( P_ni - Pi) , where w i e coo are, respectively, the angular frequency of the output voltage of the source i and the angular reference frequency of the micronetwork, m_i the droop coefficient of the active power Pi injected into the network by the source i having reference P_ni , the parameter set comprises three values (A, B, C) defined as

where w_base is the base selected for the angular frequency, Vi the amplitude of the output voltage of the source i, the 2x2 matrices GE and B_E represent, respectively, the conductance matrix and the susceptance matrix that define the admittance matrix Y = GE + jBE .

13. Method according to claim 12, wherein the micronetwork is considered stable if the following relationship is satisfied:

14. System for assessing the stability of a micronetwork comprising a plurality (N) of sources (Si) controlled through a logic (Ci) , the network voltage/current relationships can be expressed in the matrix form (Y) as a function of the network load so to calculate the active power (Pi) and the reactive power (Qi) injected by the sources (Si) into the micronetwork as a function of the load, which system comprises :

a) a measuring module (2) to measure the amplitudes (Vi) and phase shifts (di) of the output voltages of each source (Si) ;

b) an interface (101) configured for receiving controls by a user;

c) a memory (201) adapted to contain program instructions ;

d) an input (301) for reading and/or inputting the values of the matrix of the voltage/current relationships (Y) , such as, for example, the admittance matrix;

e) an input (401) for reading and/or inputting the parameters (mi, ni , P_ni , Q_ni) of the micronetwork control logic (Ci) ;

f) a processor (501) configured for executing the program instructions for:

calculating, for each pair of sources (Si, Sj), a set of parameters (Aij , Bij , Cij) as a function of the voltages Vi, of the values of the matrix (Y) of voltage/current relationships and of the parameters of the network control logic (mi, ni) ;

determining the micronetwork stability for a specific load condition by checking if the set of parameters (Aij , Bij , Cij) meets a condition for at least one subset (CS) of the possible combinations of pairs of sources (K) ;

g) an output (601) for outputting the processing outcomes, in particular an indication about the presence of micronetwork stability or instability.

15. System according to claim 14, wherein the processor is configured for determining the subset (CS) of the possible pairs of sources (Si) by measuring the angular differences between the voltages of all the possible pairs of sources (k) at an initial time and by selecting as subset (CS) a predetermined number (N- 1) of pairs having the largest angular differences.

16. Island mode micronetwork comprising a plurality (N) of sources (Si) controlled through a logic (mi, ni) , characterized by comprising a system for determining the stability as a function of the load according to claim 14 or 15.