The coordinated fuzzy control of hierarchical power balance of new energy microgrid

2.1 Secondary control of integrated microgrid for power generation and energy storage

In general, we think that microgrid has N buses (i = 1, 2 … N). The electrical physical network of the MG can be depicted in diagrams G ^MG  = (V ^MG , E ^MG ) where nodes V ^MG represent buses and edges E ^MG  ⊆ V ^MG  × V ^MG represent line connections. If there is line connection between the i-th node and the j-th node, then Z_ij = R_ij + jX_ij, i represents the local node, and j represents the adjacent node.

In order to establish a more realistic microgrid model, this section includes the power flow model of DG and ESS microgrid, and the following assumptions are needed to simplify the power flow formula.

It is assumed that the line impedance of microgrid can be regarded as inductive and exceeds the line resistance in microgrid, that is, the inductive reactance is far greater than the impedance (Xij >> Rij). When the phase difference between two adjacent inverters θi - θj approaches 0, the sin(θi - θj) ≈ θi - θj approximately holds.

When Hypothesis 1 holds, the active power can be expressed as [12]:

P ij = ∑ j = 1 N V i V j X ij ( θ i - θ j )(1)

In the formula, Vi is the voltage amplitude of the i-th bus.

X i = 1 / ( ∑ j = 1 N X ij - 1 )(2)

ω i = ω * - m i ( P i - P i * )(3)

In the formula, ω* is the expected value of frequency and P* is the expected value of active power. ωi is actually angular frequency, and its unit is rad/s. The θ ̇ i = ω i - ω * and m_i is the sag coefficient. The sag coefficient of active power sharing is defined as follows:

m i = ω i ¯ - ω i P i ¯ - P i _(4)

The droop control rules for DG and ESS are shown in Fig. 1.

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Fig. 1

Sag control rules for DG and ESS. (a) Droop control curve of ESS. (b) Droop control curve of DG.

ω i ¯ - ω i = Δ ω is the allowable fluctuation range of the global frequency of the whole MG. Droop control curve as shown in Fig. 1. This droop control curve means that when ωi = ω^*, the power of ESS is Pi = Pi^* = 0, that is, it is not discharged or charged, and the active power of microgrid is all provided by DG.

Next, aiming at the problem of active power distribution between DG and ESS in microgrid, the contents of droop control and secondary control in formula (3) are further studied. According to Kirchhoff’s Law, the power generation of each unit can be calculated according to P i = P ij + P i L . Combined with formula (5), e can get [13]:

P i ( t ) = ∑ j = 1 N w ij ( θ i ( t ) - θ j ( t ) ) + P i L ( t )(5)

Among them, the deviation between bus voltage and expected voltage is very small when the system is stable, so w _ij can be regarded as a constant, w ij = V i V j X ij = ( V * ) 2 X ij . P i L is the total load of bus i.

The initial droop control is converted to:

θ ̇ i ( t ) = - m i P i ( t ) + ∫ ( ω i ( t ) - ω * ) + Ω i ( t ) = - m i P i ( t ) + θ i ( t ) + Ω i ( t )(6)

Then, can get the system formula of each DG:

P i ( t ) = ∑ j = 1 N w ij ( θ i ( t ) - θ j ( t ) ) + P i L , i ∈ S DG θ ̇ i ( t ) = ω i ( t ) - ω * = - m i P i ( t ) + θ i ( t ) + Ω i ( t )(7)

Similarly, considering the characteristics of ESS, can get the system formula of each ESS:

P i ( t ) = ∑ j = 1 N w ij ( θ i ( t ) - θ j ( t ) ) + P i L , i ∈ S ESS E ̇ i ( t ) = - 1 3600 C i P i ( t ) θ ̇ i ( t ) = ω i ( t ) - ω * = - m i P i ( t ) + θ i ( t ) + Ω i ( t )(8)

For the sake of simplicity of the model, the loss of charge and discharge is ignored here.

To describe this strategy, we define an auxiliary function variable Φ _i  = f _i  (P _i ) to act as a bridge for power sharing between DG and ESS.

For each ESS, it is defined as [14]:

Φ i ( t ) = - 1 3600 C i P i ( t ) , i ∈ S ESS(9)

For each DG, there is actually no energy limit for energy storage, and the power output can only be positive, but a very large virtual capacity is set for unified control. Therefore, the auxiliary variable Φ _i is defined as follows:

Φ i ( t ) = - P i ( t ) + 0.5 P ¯ i 3600 C i , i ∈ S DG(10)

This paper chooses αi = 0.5.

In the second equation of formulas (8) and (9), use E ̇ i ( t ) = Φ i ( t ) , i ∈ S ESS to estimate the energy stored in each ESS. In order to satisfy the purpose of integrated control of DG and ESS, formula (10) is designed here, which can be regarded as virtual energy stored in DG, and use F ̇ i ( t ) = Φ i ( t ) , i ∈ S DG to express it.

When each DG system is linearized, the DG system in formula (7) can be rewritten as:

{ x ̇ i ( t ) = f i ( x i ) + g i ( x i ) u i ( t ) y i ( t ) = h i ( x i ) = Φ i ( t ) , i ∈ S DG(11)

Among them, In order to correlate the control input u _i  (t) with the output y _i  (t), according to the input and output feedback linearization method, we obtain the following formula by finding the second derivative of y _i  (t):

y ̈ i ( t ) = L f i 2 h i ( x i ) + L g 1 L f 1 h i ( x i ) u i ( t ) = ɛ i ( t )(12)

Among them, _ɛi (t)  is the newly defined auxiliary control variable.

Next, we define that y i ( t ) = Φ i ( t ) , y ̇ i ( t ) = Γ i ( t ) , then the nonlinear DG system shown in formula (11) can be transformed into a set of second-order linear systems as follows:

{ Φ ̇ i ( t ) = Γ i ( t ) Γ ̇ i ( t ) = ɛ i ( t ) , i ∈ S DG(13)

At the same time, the control input u _i  (t) in the original system can be calculated by the following formula:

u i ( t ) = ɛ i ( t ) - L f 1 2 h i ( x i ) L g 1 L f 1 h i ( x i )(14)

By a similar process, the ESS system represented by formula (8) can also be converted into a set of third-order systems as represented by formula (15):

{ E ̇ i ( t ) = Φ i ( t ) Φ ̇ i ( t ) = Γ i ( t ) Γ ̇ i ( t ) = ɛ i ( t ) , i ∈ S ESS(15)

Next, the detailed calculation method of feedback linearization is given.

The calculation method of DG system shown in formula (11) is as follows [15]:

First, y _i  (t) derives from x _i  (t):

y ̇ i = L f i h i ( x i ) + L g i h i ( x i ) u i(16)

Among them,

L f i h i ( x i ) = ∂ h i ∂ x i f i ( x i ) = [ ∂ h i ∂ θ i , ∂ h i ∂ Ω i ] f i ( x i ) = β i ( - m i P i + θ i + Ω i )(17)

L g i h i ( x i ) = ∂ h i ∂ x i g i ( x i ) = [ ∂ h i ∂ θ i , ∂ h i ∂ Ω i ] g i ( x i ) = 0(18)

b i = - 0.5 ∑ i = 1 N w ij 3600 C i(19)

When the second derivative is found, u _i appears, the derivative is end, and we directly set ɛ i ( t ) = y ̈ i . Thus, the substitution relation shown in formula (16) is obtained.

The calculation process for feedback linearization of ESS system represented by formula (15) is as follows:

x ̇ i = f i ( x i ) + g i ( x i ) u i y i = h i ( x i )(20)

In the same step, y _i  (t) derives from x _i  (t):

y ̇ i = L f i h i ( x i ) + L g i h i ( x i ) u i(21)

Among them,

L f i h i ( x i ) = ∂ h i ∂ x i f i ( x i ) = Φ i L g i h i ( x i ) = ∂ h i ∂ x i g i ( x i ) = 0(22)

Since the coefficient of u _i is still 0, it has been eliminated, so we continue to derive y ̇ i :

y ⃛ i = L f i 3 h i ( x i ) + L g i L f i 2 h i ( x i ) u i(23)

Among them,

L f i 2 h i ( x i ) = ∂ L f i h i ( x i ) ∂ x i f i ( x i ) = b i ′ ( - m i P i + θ i + Ω i ) L f i L g i h i ( x i ) = ∂ L f i h i ( x i ) ∂ x i g i ( x i ) = 0 b i ′ = ∂ h i ∂ x i = - ∑ i = 1 N w ij 3600 C i(24)

Since the coefficient of u _i is still 0, we continue to derive y ̈ i again:

y ⃛ i = L f i 3 h i ( x i ) + L g i L f i 2 h i ( x i ) u i(25)

Among them,

L f i 3 h i ( x i ) = ∂ L f i 2 h i ( x i ) ∂ x i f i ( x i ) = γ i ′ ( - m i P i + θ i + Ω i ) L g i L f i 2 ( x i ) = ∂ L f i 2 h i ( x i ) ∂ x i g i ( x i ) = b i ′ γ i ′ = b i ′ ( - m i ∑ i = 1 N w ij + 1 )(26)

At this point, after the third derivative is obtained, u _i appears, and the derivative is end, and we set ɛ i ( t ) = y ⃛ i . Thus, the conversion relationship of the initial control variables of the system shown in formula (27) is obtained:

u i ( t ) = ɛ i ( t ) - L f i 3 h i ( x i ) L g i L f i 2 h i ( x i )(27)

The detailed control objectives is proposed:

lim t → ∞ ( E i ( t ) - E j ( t ) ) = 0 , ∀ i , j ∈ S ESS lim t → ∞ ( Φ i ( t ) - Φ j ( t ) ) = 0 , ∀ i , j ∈ S ESS ∪ S DG(28)

In the formula, the first formula indicates that SOC in ESS is balanced in steady state. The second condition means that DG and ESS share power fairly according to their rated power and capacity.

It can be seen from formulas (13) and (15) that the state space orders of DG and ESS systems are different. In order to solve the problems of power distribution and SOC balance caused by different characteristics of DG and ESS, we introduce an auxiliary variable Fi (t) for each DG distributed generation to supplement the orders, so as to design distributed controllers conveniently. The dynamic formula of Fi (t) is given by the following formula [16]:

F ̇ i ( t ) = Φ i ( t ) , i ∈ S DG(29)

In this way, the two systems of different orders can be unified as:

ξ i ( t ) = { [ F i ( t ) Φ i ( t ) Γ i ( t ) ] T , i ∈ S DG [ E i ( t ) Φ i ( t ) Γ i ( t ) ] T , i ∈ S ESS(30)

By conjunction, can get:

ξ ̇ i ( t ) = A ξ i ( t ) + B ɛ i ( t )(31)

Among them, a = [010; 001; 000] and B = [0; 0; 1].

At this point, the framework of the whole system is very clear, as shown in Fig. 2.

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Fig. 2

System frame diagram.

First of all, DG and ESS have the same sampling interval h, so for i ∈ S ^DG  ∪ S ^ESS , the time sequences of each sampling by the sampler is { mh } ( m ∈ ℕ ) , the sampled data is ξ _i  (mh).

After the original data is specified, we will specify the data sequence { t k i h } k ∈ ℕ obtained after the event trigger mechanism, which is defined by the following dynamic trigger mechanism:

t k + 1 i h = inf m ∈ ℕ { mh > t k i he i T ( t ) Σ e i ( t ) ⩾ δ κ i T ( t k i h ) Σ κ i ( t k i h ) + v i β i ( t ) }(32)

Among them, t 0 i = 0 , e i ( t ) = ξ i ( t k i h ) - ξ i ( mh ) ( t ∈ [ mh , mh + h ) , κ i ( t k i h ) = ∑ j = 1 N a ij [ ξ i ( t k i h ) - ξ j ( t k j h ) ] , t k ′ j = max k ∈ ℕ { t k j t k j h ⩽ t }

The dynamic items β _i  (t) is defined by the following formula [17]: β ̇ i ( t ) = - β i ( t ) + κ i T ( t k i h ) W κ i ( t k i h )

Scalars δ > 0 and matrices Σ > 0, W > 0 in trend terms will be found by LMI toolbox.

In this way, can design the event-triggered controller as follows to achieve the purpose of active power sharing and SOC equalization.

ɛ i ( t ) = K ∑ j = 1 N a ij [ ξ i ( t k i h ) - ξ j ( t k j h ) ](33)

The controller gain K ∈ R^1×3 will also be solved through LMI toolbox.

Lemma 1: If the communication graph G is connected, then there is a western matrix U satisfying U ^T U = I _N , so that U T L U = Λ = diag { λ 1 , ⋯ , λ N } . The eigenvalue λ₁ of Laplace matrix L satisfies λ _N  ⩾ . . . ⩾ λ₃ ⩾ λ₂ > λ₁ = 0.

Lemma 2: For a vector η (t) = col{ η₁ (t) , …, η _N  (t) }, if there are matrices R and S that can make [ R S S T R ] > 0 , then the following formula holds:

- h ∫ t - h t η ̇ T ( s ) ( L ⊗ R ) η ̇ ( s ) ds ⩽ - φ T ( t ) [ L ⊗ R L ⊗ S L ⊗ S T L ⊗ R ] φ ( t )(34)

In order to describe more concisely, we sort out the system model and make ξ (t) = col{ ξ₁ (t) , ⋯ , ξ _N  (t) }. For t ∈ [mh, mh + h), there is d (t) = t-mh, so d (t) ∈ [0, h). By substituting formula (38) into formula (37), can get the final model of the system:

ξ ̇ ( t ) = ( I N ⊗ A ) ξ ( t ) + ( L ⊗ BK ) ( ξ ( t - d ( t ) ) + e ( t ) )(35)

Among them, e (t) = col{ e₁ (t) , ⋯ , e _N  (t) }. We define η ( t ) = ( N ⊗ I ) ξ ( t ) , N = I N - 1 N 11 T .

By Lemma 1, we know that L 1 = 0 . Therefore, η (t) =0 is equivalent to ξ₁ (t) = … = ξ _N  (t). By L N = N L = L , we obtain:

η ̇ ( t ) = ( I N ⊗ A ) η ( t ) + ( L ⊗ BK ) η ( t - d ( t ) ) + ( L ⊗ BK ) e ( t ) .(36)

The active power sharing and SOC balancing problems designed in the control objective are equivalent to the controller gain K in the design (37) and the event triggering parameters in formula (32), making the system (36) asymptotically stable.

Next, we prove the stability:

First, the positive definite Lyapunov function is constructed as the following formula, and its derivative is negatively definite, so that the system is stable.

W ( t ) = η T ( t ) ℘ η ( t ) + ∫ t - h t η T ( s ) Q η ( s ) ds + h ∫ - h 0 ∫ t + τ t η ̇ T ( s ) R η ̇ ( s ) dsd τ + ∑ i = 1 N β i ( t )(37)

For -h ⩽ s ⩽ 0W (t), η (s) = η (0) holds. Then, we find the derivative of W (t):

W ̇ ( t ) = 2 η T ( t ) ℘ η ̇ ( t ) + η T ( t ) Q η ( t ) - η T ( t - h ) Q η ( t - h ) + h 2 η ̇ T ( t ) R η ̇ ( t ) - h ∫ t - h t η ̇ T ( s ) R η ̇ ( s ) ds - ∑ i = 1 N β i ( t ) - ∑ i = 1 N κ i T ( t k i h ) W κ i ( t k i h )(38)

By using the event triggering mechanism (32), we know that

e T ( t ) ( I N ⊗ Σ ) e ( t ) < δ ( η ( t - d ( t ) ) + e ( t ) ) T Σ ˜ ( η ( t - d ( t ) ) + e ( t ) ) + ∑ i = 1 N v i β i ( t )(39)

Among them, Σ ¯ = L 2 ⊗ Σ .

Applying Lemma 2 and the above inequalities, can rewrite (38) as:

W ̇ ( t ) = δ T ( t ) Π δ ( t )(40)

Among them, δ ( t ) = col { η ( t ) , η ( t - d ( t ) ) , η ( t - h ) , e ( t ) , η ̇ ( t ) } , and Π = [ Π 11 * * * * Π 21 Π 22 * * * L ⊗ S T Π 32 Π 33 * * Π 41 δ L 2 ⊗ Σ 0 Π 44 * γ L ⊗ PA Π 52 0 Π 54 Π 55 ] ,

Among them, Π 11 = L ⊗ ( PA + A T P + Q - R ) Π 21 = L ⊗ ( R - S T ) + L 2 ⊗ K T B T P Π 22 = L ⊗ ( - 2 R + S + S T ) + δ L 2 ⊗ Σ Π 32 = L ⊗ ( R - S T ) , Π 33 = - L ⊗ ( Q + R ) Π 41 = L 2 ⊗ K T B T P , Π 44 = - I N ⊗ Σ + L 2 ⊗ ( δ Σ + W ) Π 52 = Π 54 = γ L 2 ⊗ PBK , Π 55 = L ⊗ ( h 2 R - 2 γ P )

F = [ f 1 f 2 ⋯ f N ] is defined, in which f 1 = [ f 11 f 12 ⋯ f 1 N ] , ⋯ , f N = [ f N 1 f N 2 ⋯ f NN ] · f ij ∈ ℝ N 2 means that the N (i - 1) + jth element is 1 and the other elements are 0. We make δ ( t ) = diag { U , ⋯ , U ︸ N } F ζ ( t ) , and by Lemma 2, we get:

δ T ( t ) Π δ ( t ) = ζ T ( t ) diag { Ψ 0 , λ 2 Ψ ( λ 2 ) , … , λ N Ψ ( λ N ) } ζ ( t )(41)

Among them, Ψ⁰ = [Ψ⁰]  _ij and

Ψ ij 0 = { - Σ , i = 4 and j = 4 0 , others(42)

Therefore, Ψ⁰ ⩽ 0, and when Ψ (λ _i ) < 0, there is W ̇ ( t ) ⩽ 0 and the derivative of W (t)is negative definite, and the proof ends.