Optimized Hierarchical Control for an AC Microgrid Under Attack

Distributed frequency control

An inverter based AC microgrid is controlled following a hierarchical frame. In the first level, the primary control avoids the appearing of high circulating currents when inverters are connected in parallel. Primary control drops the frequency according to the active power demanded by loads ^[3], this linear relation is represented by (1)

where ^_fref is the frequency of reference in Hz, ^_mi is the droop coefficient, ^_Pi is the medium power at inverter i, and ^_Pi * is the active power reference for inverter i.

The power reference value ^_Pi * can be modified based on the requirements of each inverter or some programmed event. However, reference values cannot be achieved only by primary control, because in an isolated microgrid not always the power generated equals the power demanded ^[7]. In fact, those values are better generated by a tertiary control capable of optimize them according to power availability criteria.

Primary control also assures that each inverter supplies power according to its maximum active power value, a condition known as power-sharing presented in (2)

Then, the equation that represents the frequency for inverter i in an a.c microgrid with primary droop control, and secondary frequency control ^[4] is given by

where ^{_{δ fi}} is the secondary control term.

The secondary control shifts the droop control function returning the frequency to its reference value. Secondary frequency control can include a leader node (not necessarily unique) that knows the reference value. So, consensus problem changes into a synchronization problem. Secondary control is given by (4)

where ^_Nij is the set of neighbors of agent i, ^_aij represents the ij-th elements of the adjacency matrix, ^_fi y ^_fj are the frequencies measured in the inverters i and j, respectively. ^_gi represent the elements of the pinning vector, and ^_fref represents the frequency reference value.

Cyber-attacks over an AC microgrid

A distributed control gathers data from neighbors through a local communication system, becau- se of this is more susceptible to cyber-attacks. According to Figure 1 attacks can be classified as attacks to local controllers, and attacks to the communications links between nodes ^[8].

Figure 1: Types of attacks in a microgrid

If the attack occurs over the controller of the i-th inverter, the frequency at inverter i is given by (5)

where ^{_{fi c}} is the corrupted frequency at inverter i after the attack, ^_ηi indicates if there is an attack or not, and f _i ^a is a perturbation injected by the controller.

Similarly, if the attack occurs in the link between two nodes this is model as it is shown in (6)

where ^{_{fi j}} is the frequency measured between inverters i and j, and ^_ηi indicates the presence of the attack ^[8].

The error in inverter i is given by (7)

where the values fˆ _i and fˆ _j correspond to the observed values in the nodes i and j, respectively.

So, it is defined as the norm of the error (7) as

and the sum of the norms of the errors as

Then, the following factor is defined

where ∆ _i represents a threshold value that considers the effect of other factors rather than attacks. Having into account this factor, the confidence factor is defined as

The confidence factor only can take values between zero and one (0 ? ^_Ci ? 1). The control of frequency is modified as it is shown in (12)

The trust factor is another coefficient that determines the reliability of the measures that a node received from its neighbors. The measurements that exceed the threshold value are rejected. First, it is calculated the factor defined in (16)

where Θ _i corresponds to a threshold value, and ? ^_Ni ? is the number of neighbors of the inverter. The trust factor is defined as

The control of frequency is modified as it is shown in (16)

Assuming an AC microgrid with predominantly inductive transmission lines, the instantaneous active power exchange is given by the following expression

where ? ^_Yij ? is the magnitude of the line admittance between inverter i and j, ^_Vi and ^_Vj are the bus voltages at inverters i and j, respectively.

Taking the first derivative of p along time, and considering the relationexpression (17) becomes

Taking the first derivative of (3) along time expression is written as

Then replacing (18), and (16) in (19) the following expression is obtained

Expression (20) is written in matrix form as follows

where M is a diagonal matrix with the droop coefficients, ^_Lc is a Laplacian matrix for the physical connection among inverters whose values are given by ωij = ^_EiEj ? ^_Yij ?, F is the vector that contains the frequency measured at each inverter, C and T are diagonal matrices whose values are the coefficient and trust parameters, respectively. L is the Laplacian matrix who represents the connections among agents. G is the diagonal matrix that determines if an agent is a leader or not.

Expression (21), can be written as ∆˙F = ( ^{_{M Lc}} +CT (L+G))F , the solution of this differential equation is of the form F(t) = Ae ^{− t ( _MLc} +CT (L+tt)). This expression is asymptotically stable if term ^{_{M Lc}} + CT (L + G) remains positive. Notice that, Laplacian matrices are positive defined, as matrices M ,C, T , and G are diagonal matrices whose values are real and positive, the product of those matrices is positive defined too, then the expression holds.

Economic Dispatch Problem

Tertiary control is considered the last layer in hierarchical control, and it is related with the optimization of the dispatch of the resources available in the power system, which is also called economic dispatch ^[9], ^[10]. Economic dispatch is a highly studied problem, some approaches use convex optimization methods, principally derived from the work of Nedicét al. in ^[11], based on this work several approaches have been made ^{[12] - [14]}. Moreover, other approaches using game theoretical approaches have been made ^[15]-[18] proving the effectiveness of game theory to handle economic dispatch problems. Tertiary control can be either centralized or distributed, distributed controllers are used in large-scale systems where centralized controllers often suffer from computation and communication overheads. However, instead of several advantages of distributed controllers, centralized controllers are highly used due to their robustness and easily configured and maintained under operation.

Economical dispatch denotes the problem of determining the optimum output power of a set of generators in a power system in order to satisfy loads and fulfill power system requirements. Economic dispatch problem consists of minimizing the total generation cost ^{[18), [19]}. Here, the cost of the power supplied by each generator is minimized using an utility function based on a quadratic function ^{[12], [13], [20]} such as

where F ( ^_pi ) is the utility function of generator i, ^_pi is the power generated by generator i. Moreover, ^{_{αi, βi, ζ}} ? R_≥0 are utility coefficients. Using the utility function of all agents in the power system we can formulate the economic dispatch problem as follows

where ^_PL is the power demanded by the load in the system, N is the number of generators in the power system, ^_pi and ^_pi are the maximum and minimum power capacity respectively. Economic dispatch problem in (23) is solved when constraints (23) and (23) are fulfilled, moreover the economic dispatch criterion states that all agents should operate with the same marginal utilities as follows

where θ ? R > 0. In order to solve (23) a game theoretical method based on population games is used, specifically, replicator dynamics solve iteratively the economic dispatch problem using criterion (24). The before statement can be summarized in the following lemma

Lemma 1 A solution of problem (23), P* belonging to the feasible set ∆, is an optimal solution if and only if ? ^_iSW (Pi*) = ? ^_jSW (Pj ^*) for all i, j.

Population Games Approach

One of the main contributions of this work is to optimize, through a tertiary control, the respon- se of a microgrid against an attack. In order to reach this contribution, a centralized population dynamics control based on game theory is proposed. Population dynamics represent how a pure population strategy progress over time. In population games, replicator dynamics have been used in several engineering applications such as economic power dispatch, lighting systems, and urban drainage systems ^[18]. These applications have been made considering some implementation be- nefits, such as robustness to dynamic environmental uncertainties.

The replicator dynamics solves the economic dispatch problem as a resource allocation problem. Replicator dynamics considers G = {1, 2, 3 . . . N} finite number of generators, who adopt a i-th strategy from a finite set of pure strategies. Accordingly, to achieve an appropriate performance in the steady state, the power demanded by the load should be the sum of all power set points. The payoff function ( ^_Fi ) is associated with the chosen strategy, therefore, a continuous time implemen- tation is also used. In its general form, the replicator dynamics can be represented as

where β is a parameter that permits modify the convergence speed ^[21]. The average payoff in the population F¯(p) is given by

The results in ^[22] are used to guarantee the power balance, the choice of (26) guarantees the invariance of the constraint set ∆ defined as

This result ensures that if ^_pi (0) ? ∆, ?θυοτ;i ? G, all ^_pi (t) ? ∆ with t ³ 0. In other words, the power demanded by the loads in the system always is dispatched i.e, the replicator dynamic algorithm guarantees a perfect balance between the power demanded and the power generated. One of the characteristics of the replicator dynamics algorithm is that in the stationary state the fitness function of all agents is equal to the average fitness (F¯). This condition allows solving the problem of economic dispatch through the replicator dynamics since the fitness function is derived from the utility function. In order to include the constraints in (23), we use Lagrange Multipliers as

where µ is the Lagrange Multipliers, the fitness functions are defined as

Finally, the fitness functions for this replicator dynamics are

Notice that ? stands for the Jacobian of a function. Now, we state the following theorem in order to characterize the optimality of the proposed algorithm

Theorem 1 Assume a convergence constant β ? R ^{_{. Furthermore, let Pi}} (k), with i ? Ν , being the set points generated by executing (25), with fitness functions defined as in (31) and (32) ^{_{. Then, Pi}} (k) with i converges to the optimal solution Pi∗ that belongs to the feasible set of power, that is

Proof.

Since we have defined fitness functions as (31) and (32), by definition a potential function for the population game (25) is h(P, µ) = L = ^_SW (P) µ ^T(R P - r) and considering the form of the economic dispatch in (23), it can be shown that the game satisfies the external symmetry (24). When the optimality condition in Lemma 1 is reached then ^_Fi ( ^_pi ) = ^_Fj ( ^_pj ) for all i, j and it is noticed that in the replicator dynamics (25) we have

To reach optimality, both sides of the equation must be equal. At the optimal point, this only can be equal to zero. It is possible to see through Lemma 1, that

Then, it is possible to see that

Then, (33) is

which implies that (25) reaches an optimal point.