Robust Consensus-Based Droop Control for Multiple ... - IEEE Xplore
Robust Consensus-Based Droop Control for Multiple Power Converters in Isolated Micro-Grids Lin-Yu Lu, IEEE Student Member, and Chia-Chi Chu, IEEE Member
Abstract—In order to provide autonomous power sharing of the conventional droop-control scheme for multiple power converters in isolated micro-grids, two consensus-based droopcontrol schemes will be examined. Since the conventional uniform droop gain ratio scheme requires identical droop gain ratio settings among all converters, applications to large-scale systems are restricted. An alternative scheme with the non-uniform droop gain ratio is proposed in this paper to simplify the droop gain setting while maintaining the perfect power sharing. Also, this new scheme is more robust since it is independent of particular gain settings. Real-time simulations of a micro-grid system are conducted to validate the effectiveness of this robust consensusbased droop control method. Index Terms—Micro-Grid, Converter Droop Control, Consensus Algorithm.
I. I NTRODUCTION
I
N conventional isolated micro-grids, the real powerfrequency (𝑃 -𝑓 ) droop control and reactive-voltage magnitude (𝑄-𝑉 ) droop control have been widely utilized for decentralized power sharing operations among distributed energy interface converters (DICs) [1], [2], [3]. However, the performance of the reactive power sharing under 𝑄-𝑉 droop control may be deteriorated due to its dependence on output line impedances [4], [5], [6]. In order to provide more accurate reactive power sharing, the 𝑄-𝑉˙ droop control method has been proposed recently [6]. Although the reactive power sharing under 𝑄-𝑉˙ droop control can be indeed independent of the output line impedances, incorrect reactive power sharing under non-uniform line impedances across the network were still observed [6]. In recent years, the consensus-based distributed control have been widely investigated for microgrid applications [7]. The main objective of this consensus control algorithm is to achieve agreement among all agents in a network considering certain desirable states of the entire system. Communication infrastructure of a multi-agent network is often limited to only local information exchange available. Since the consensus algorithms require only neighbour-toneighbour interaction, marvellous scalability and robustness can be achieved. This inherently distributed characteristics makes the consensus-based control scalable and can be easily implemented. In order to obtain more accurate power sharing in microgrid applications, the consensus-based 𝑃 -𝑓 droop control with perfectly uniform droop gain ratio scheme was studied by [7] recently. The uniform droop gain ratio scheme requires that This work was supported in part by the National Science Council of Taiwan, R.O.C., under Grant NSC 102-2221-E-007-074. The authors are with Department of Electrical Engineering, National Tsing Hua University, Hsinchu, 300, Taiwan, R.O.C. (E-mail: [email protected]; [email protected])
978-1-4799-3432-4/14/$31.00 ©2014 IEEE
the droop gain ratio of all converters are identical. Thus, this scheme is not very robust under droop gain variations. This disadvantage will limit its practical applications for large-scale micro-grids. In order to overcome this difficulties, the nonuniform droop gain ratio scheme will be proposed to simplify droop gain setting while maintaining the desired power sharing under the steady state in this paper. By following the averageconsensus principle, this modified consensus-droop control with the non-uniform droop gain ratio can be proven to be independent of particular droop gain. In addition, the closedloop dynamics of this robust consensus-droop control with the non-uniform droop gain ratio can be shown to be stable by the more common transient energy function (TEF) approach. We will call this consensus-based droop control with the nonuniform droop gain ratio scheme as the robust consensusbased droop control. In order to validate the feasibility of this robust consensus-based droop control method, real-time simulations are performed on a 14-bus/6-converter microgrid. Simulation results indicate that this robust consensusdroop droop control can indeed provide larger damping while maintaining precise active and reactive power sharing. II. C ONVENTIONAL D ROOP C ONTROL Consider a micro-grid with 𝑛 buses as shown in Fig. 1. Buses 1, ⋅ ⋅ ⋅ 𝑚 are power sources embedded with power converters. Buses 𝑚+1, ⋅ ⋅ ⋅ , 𝑛 are load buses. All line impedances are lossy with the uniform 𝑅/𝑋 ratio defined by 𝑅𝑖𝑗 /𝑋𝑖𝑗 = 𝐾 = −𝐺𝑖𝑗 /𝐵𝑖𝑗 , where 𝑅𝑖𝑗 , 𝑋𝑖𝑗 , 𝐺𝑖𝑗 and 𝐵𝑖𝑗 are resistance, reactance, conductance and susceptance of each transmission line between node 𝑖 and 𝑗, respectively. Now the active and reactive power flow equations can be expressed as 𝑃𝑖 − 𝐾𝑄𝑖 = (1 + 𝐾 2 )
∑𝑛
2
𝑄𝑖 + 𝐾𝑃𝑖 = −(1 + 𝐾 )
𝑗=1
∑𝑛
𝑉𝑖 𝑉𝑗 𝐵𝑖𝑗 sin 𝜃𝑖𝑗 ,
𝑗=1
𝑉𝑖 𝑉𝑗 𝐵𝑖𝑗 cos 𝜃𝑖𝑗 .
(1a) (1b)
For 𝑖 = 1+𝑚, ⋅ ⋅ ⋅ 𝑛, load dynamics are represented as follows: ∑𝑛 𝑃𝐿𝑖 = − 𝑉𝑖 𝑉𝑗 𝐵𝑖𝑗 sin 𝜃𝑖𝑗 − 𝐷𝐿𝑝,𝑖 𝜃˙𝑖 , 𝑗=1 ∑𝑛 𝑉𝑖 𝑉𝑗 𝐵𝑖𝑗 cos 𝜃𝑖𝑗 − 𝐷𝐿𝑞,𝑖 𝑉˙ 𝑖 . 𝑄𝐿𝑖 = 𝑗=1
Bus1
Busm +1
Busm + 2
Loads
Bus2
Physical network Busm
Busn Physical connection Communication (topology dependent)
Fig. 1. Several Power converters in an isloated micro-grid.
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(2a) (2b)
(1a) and (1b) can be treated as modified power flow equations of a pure inductive network with equivalent power injections 𝑃𝑖 − 𝐾𝑄𝑖 and 𝑄𝑖 + 𝐾𝑃𝑖 [8]. The subscript 𝐿 in (2a) and (2b) stands for the load bus. Here, the 𝐷𝐿𝑝,𝑖 𝜃˙𝑖 and 𝐷𝐿𝑞,𝑖 𝑉˙ 𝑖 terms are introduced to represent load dependences on the bus frequency [9] and the time derivative of voltage magnitude [10]. Traditionally, the real power-frequency 𝑃 -𝑓 droop control and reactive-voltage magnitude 𝑄-𝑉 droop control have been adapted for decentralized power sharing operations. However, the performance of reactive power sharing under conventional 𝑄-𝑉 droop control may be deteriorated due to its dependence on output line impedances [4], [5], [6]. In order to provide more accurate reactive power sharing, the 𝑄-𝑉˙ droop control method has been proposed recently [6]. Specifically, if all fast dynamics of power converters are neglected, the 𝑃 -𝑓 /𝑄𝑉˙ droop control for autonomous power sharing operations of multiple power converters can be described as follows: 𝐷𝑝,𝑖 𝜃˙𝑖
=
𝑘𝑝,𝑖 𝑝˙ 𝑖 𝐷𝑞,𝑖 𝑉𝑖 𝑣˙ 𝑖 𝑘𝑞,𝑖 𝑞˙𝑖
= = =
𝑃𝑖∗ − 𝑃𝑖 − 𝑝𝑖 , 𝐷𝑝,𝑖 𝜃˙𝑖 ,
𝑄∗𝑖 − 𝑄𝑖 − 𝑞𝑖 , 𝐷𝑞,𝑖 𝑉𝑖 𝑣˙ 𝑖 ,
(3) (4) (5) (6)
where 𝑣𝑖 is the transformed bus voltage defined by 𝑣𝑖 = ln 𝑉𝑖 and 𝑣˙𝑖 = 𝑉˙𝑖 ⋅ 𝑉𝑖−1 . 𝑃𝑖∗ and 𝑄∗𝑖 are the nominal active and reactive power ratings of the DIC, respectively. 𝐷𝑝,𝑖 and 𝐷𝑞,𝑖 are the droop gains corresponding to active and reactive power control. 𝑝𝑖 and 𝑞𝑖 are the 𝜃˙ and 𝑉˙ restoration mechanism to maintain the frequency and voltage profile. The terminal voltage and the frequency of the converter will be held at nominal value under the steady-state. The stability of the closed-loop system can be proven by the transient energy function (TEF) approach. Let the state variable of each bus be 𝒙𝑖 = [𝜃𝑖 , 𝑣˙ 𝑖 , 𝑝𝑖 , 𝑞𝑖 ]𝑇 and state variables of the overall system be 𝒙𝑇 = [𝜽 𝑇 , 𝒗˙ 𝑇 , 𝒑𝑇 , 𝒒 𝑇 ], the TEF of the closed-loop system can be defined by 𝑈1 (𝒙) = 𝑊1 (𝒙)+𝑊2 (𝒙)+𝑊3 (𝒙) +𝑊4 (𝒑, 𝒒),
(7)
where
⎞ ⎛ 𝑛 𝑛 1 ⎝∑ ∑ 2 𝑊1 (𝒙) = − (1+𝐾 )𝑉𝑖 𝑉𝑗 𝐵𝑖𝑗 cos 𝜃𝑖𝑗 ⎠ , 2 𝑖=1 𝑗=1
𝑊2 (𝒙) = − 𝑊3 (𝒙) = − 𝑊4 (𝒑, 𝒒) =
𝑚 ∑
(𝑃𝑗∗ −𝐾𝑄∗𝑗 )𝜃𝑗 +
𝑗=1 𝑚 ∑
𝑛 ∑
(𝑄∗𝑗 +𝐾𝑃𝑗∗ )⋅𝑣𝑗 +
𝑗=1 𝑚 ∑
(𝑃𝐿𝑗 −𝐾𝑄𝐿𝑗 )𝜃𝑗 ,
𝑗=𝑚+1 𝑛 ∑
(𝑄𝐿𝑗 +𝐾𝑃𝐿𝑗 )⋅𝑣𝑗 ,
(8a)
(8b)
(8c)
𝑗=𝑚+1
1 (𝑝2 + 𝑞𝑗2 ). 2 𝑗=1 𝑗
(8d)
Under the this framework, the closed-loop micro-grid under the 𝑃 -𝑓 /𝑄-𝑉˙ droop control (3) and (5) can be re-written in the following compact form: 𝑫 𝑖 𝒙˙ 𝑖 = −𝑨𝑖 (∂𝑈1 / ∂𝒙𝑖 ), where
(9)
⎤ ⎡ ⎤ 𝐷𝑝,𝑖 0 −𝐾𝐷𝑞,𝑖 𝑉𝑖 0 1 1 0 −𝐾 ⎢ 0 𝑘𝑝,𝑖 −𝐾𝐷𝑞,𝑖 𝑉𝑖 0 ⎥ ⎢0 0 0 0 ⎥ . (10) 𝑫𝑖 = ⎣ , 𝑨𝑖 = ⎣ 0 −𝐾 1 1 ⎦ 𝐾𝐷𝑝,𝑖 0 𝐷𝑞,𝑖 𝑉𝑖 0 ⎦ 0 0 0 0 𝐾𝐷𝑝,𝑖 0 0 𝑘𝑞,𝑖 ⎡
Thus, the overall closed-loop system can be expressed as 𝑫 𝒙˙ = −𝑨(∂𝑈1 / ∂𝒙).
(11)
The matrix 𝑫 −1 𝑨 is positive semi-definite with 2𝑛 zeroeigenvalues. In particular, these zero-eigenvalues do not lead to a continuum of equilibrium points, but rather isolated equilibrium points as algebraic solutions of (3) to (6). Thus, (11) is indeed a quasi-gradient system [11] and every trajectory will converge to one of the isolated equilibrium points. Therefore, the 𝑃 -𝑓 /𝑄-𝑉˙ droop control can indeed ensure the complete stability of the closed-loop system. III. C ONSENSUS -BASED D ROOP C ONTROL Although quite accurate real power sharing can be ensured by (11), incorrect reactive power sharing under non-uniform line impedances across the network were still observed [6]. It is due to the existence of zero-eigenvalues for 𝑨𝑖 . Thus, the trajectory will diverge for general lossy networks and the pure 𝑃 -𝑓 and 𝑄-𝑉˙ relationship for droop control will no longer hold. The consensus-based droop control, which allows data exchange between networking neighbours by message passing algorithms, was proposed to overcome this difficulty. In the past, only the consensus-based 𝑃 -𝑓 droop control is considered [7]. In this paper, we will extend their work to consider consensus-based 𝑃 -𝑓 and 𝑄-𝑉˙ droop control simultaneously. By introducing the Laplacian matrix among all power converters, the closed-loop of the consensus-based 𝑃 -𝑓 /𝑄-𝑉˙ droop control can be constructed as follows: 𝐷𝑝,𝑖 𝜃˙𝑖 = 𝑘𝑝,𝑖 𝑝˙ 𝑖 = 𝐷𝑞,𝑖 𝑉𝑖 𝑣˙ 𝑖 = 𝑘𝑞,𝑖 𝑞˙𝑖 =
𝑃𝑖∗ − 𝑃𝑖 − 𝑝𝑖 , (12) ∑𝑚 ˙ 𝐷𝑝,𝑖 𝜃𝑖 + 𝑙𝑖𝑗 (𝑝𝑖 /𝐷𝑝,𝑖 − 𝑝𝑗 /𝐷𝑝,𝑗 ) , (13) 𝑗=1
𝑄∗𝑖 − 𝑄𝑖 − 𝑞𝑖 , (14) ∑𝑚 𝐷𝑞,𝑖 𝑉𝑖 𝑣˙ 𝑖 + 𝑙𝑖𝑗 (𝑞𝑖 /𝐷𝑞,𝑖 − 𝑞𝑗 /𝐷𝑞,𝑗 ).(15) 𝑗=1
In the above formulation, 𝑙𝑖𝑗 is the [𝑖, 𝑗] entry of the graph Laplacian 𝑳𝑐𝑜𝑚𝑚 of interconnected power converters. Since the off-diagonal entries of 𝑳𝑐𝑜𝑚𝑚 indicate the physical connectivity between vertices of the network, the communication between power converters is then established between physical neighbours of each converter only. Unlike the conventional droop control scheme, additional terms in (13) and (15) will provide the power sharing adjustment mechanism among neighbouring converters. It is worthy to pointing out that if the consensus control is inactivated by setting 𝑙𝑖𝑗 = 0 for all 𝑖, 𝑗, the consensus-based droop control will be reduced to 𝑃 -𝑓 /𝑄-𝑉˙ droop control. All trajectories will still converge to one of the equilibrium point. Thus, the consensus-based droop control can be considered as a supplementary control actions from neighbouring converters. This implies that the overall consensus-based droop control algorithm is composed of two phases: the primary local 𝑃 𝑓 /𝑄-𝑉˙ droop control and the supplementary consensus-based droop control. The local 𝑃 -𝑓 /𝑄-𝑉˙ droop control will dominate dynamical behaviours of the closed-loop system in the early stage. Thus, in the first phase, the frequency restoration and synchronization mechanism governed by (12) and (14) will reach the steady-state soon. Then the system will move to the
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consensus-based droop control phase. Under this framework, the initial state of the consensus-based droop control under steady-state of ∑𝑚(12) can be represented approximately by 𝜃˙𝑎𝑣𝑔 (0) = ( 𝑗=1 𝑃𝑗∗ − 𝑃𝑗 (0) − 𝑝𝑗 (0))/𝐷𝑇 , where 𝐷𝑇 = ∑𝑚 ˙ 𝑗=1 𝐷𝑝,𝑗 . Since 𝜃𝑎𝑣𝑔 (0) is nearly zero under the frequency restoration and synchronization mechanisms, the dynamics of (13) become the first order consensus algorithm with ∑𝑚 𝑙𝑖𝑗 (𝑝𝑖 /𝐷𝑝,𝑖 − 𝑝𝑗 /𝐷𝑝,𝑗 ) . (16) 𝑘𝑝,𝑖 𝑝˙ 𝑖 =
verters, the precise settings (19) and (21) are required. This condition will restrict the applications for practical large-scale micro-grid since damping variations will destroy the perfect power sharing. In order to avoid the precise uniform droop gain ration setting and provide more flexibility for individual droop gain settings, the following consensus-based droop control with uniform droop gain is proposed: −1 ∗ 𝑃𝑖 𝑝 𝑖 , 𝐷𝑝,𝑖 𝜃˙𝑖 = 𝑃𝑖∗ − 𝑃𝑖 − 𝐷𝑝,𝑖 ∑𝑚 𝑘𝑝,𝑖 𝑝˙ 𝑖 = 𝐷𝑝,𝑖 𝜃˙𝑖 + 𝑙𝑖𝑗 (𝑝𝑖 /𝐷𝑝,𝑖 − 𝑝𝑗 /𝐷𝑝,𝑗 ) ,
𝑗=1
Following the average-consensus principle, (16) would drive all 𝑝𝑖 /𝐷𝑝,𝑖 to be identical as ∑𝑚 ∑𝑚 𝑝𝑖 /𝐷𝑝,𝑖 = 𝑝𝑗 (0)/𝐷𝑇 = (𝑃𝑗∗ −𝑃𝑗 (0))/𝐷𝑇 , (17) 𝑗=1
−1 ∗ 𝐷𝑞,𝑖 𝑉𝑖 𝑣˙ 𝑖 = − 𝑄𝑖 − 𝐷𝑞,𝑖 𝑄 𝑖 𝑞𝑖 , ∑𝑚 𝑘𝑞,𝑖 𝑞˙𝑖 = 𝐷𝑞,𝑖 𝑉𝑖 𝑣˙ 𝑖 + 𝑙𝑖𝑗 (𝑞𝑖 /𝐷𝑞,𝑖 − 𝑞𝑗 /𝐷𝑞,𝑗 ) . 𝑗=1 ∑𝑚 (𝑃 ∗ −𝑃𝑗 (0)−𝐷−1 𝑃 ∗ 𝑝𝑗 (0)) is Likewise, 𝜃˙𝑎𝑣𝑔 (0) = 𝑗=1 𝑗 ∑𝑚 𝐷𝑝,𝑗𝑝,𝑗 𝑗 𝑗=1
𝑗=1
which is exactly the average frequency deviation among power converters. If we substitute (17) into (12), ∑𝑚 𝑃𝑗∗ − 𝑃𝑗 (0)/(𝑃𝑖∗ 𝐷𝑇 ). (18) 𝑃𝑖 /𝑃𝑖∗ = 1 − 𝐷𝑝,𝑖 ⋅
𝐷𝑝,1 /𝑃1∗
=
𝐷𝑝,2 /𝑃2∗
=
∗ ⋅ ⋅ ⋅ 𝐷𝑝,𝑚 /𝑃𝑚 .
(19)
Obviously, appending the consensus term would drive the real power outputs of power converters toward ∗ 𝑃1 /𝑃1∗ = 𝑃2 /𝑃2∗ = ⋅ ⋅ ⋅ 𝑃𝑚 /𝑃𝑚 .
(20)
In other words, active power outputs of all power converters share system loads in the micro-grid proportional to their nominal ratings. Similar conclusions about reactive power can be made by considering (14) and (15). When the perfect uniform droop gain ratio is chosen as follows: 𝐷𝑞,1 /𝑄∗1 = 𝐷𝑞,2 /𝑄∗2 = ⋅ ⋅ ⋅ 𝐷𝑞,𝑚 /𝑄∗𝑚 ,
(21)
the following reactive power sharing will be reached: 𝑄1 /𝑄∗1 = 𝑄2 /𝑄∗2 = ⋅ ⋅ ⋅ 𝑄𝑚 /𝑄∗𝑚 .
𝑫 𝑐𝑜𝑛 𝒙˙ = −𝑨𝑐𝑜𝑛 (∂𝑈1 /∂𝒙).
𝑝𝑖/𝐷𝑝,𝑖 =
(23)
It can be verified that 𝑫 𝑐𝑜𝑛 = 𝑫 which is of full rank. ˜ 𝑐𝑜𝑛 ), where 𝑨 ˜ 𝑐𝑜𝑛 = 𝑑𝑖𝑎𝑔(0, −𝑳𝑐𝑜𝑚𝑚𝑫 −1 , 0, 𝑨𝑐𝑜𝑛 = (𝑨+𝑨 𝑝 −1 −𝑳𝑐𝑜𝑚𝑚𝑫 𝑞 ). 2𝑛 zero-eigenvalues of 𝑨𝑐𝑜𝑛 are eliminated by the consensus-based droop control. Only two inherent zeroeigenvalues of the 𝑳𝑐𝑜𝑚𝑚 term are left. By eigenvector analysis of 𝑨𝑖 , it can be shown that these two zero-eigenvalues of 𝑨𝑐𝑜𝑛 are indeed an isolated equilibrium point. Since each positive term of 𝑳𝑐𝑜𝑚𝑚 will appear on the diagonal entry of 𝑨𝑐𝑜𝑛 , these additional negative terms will appear on the analytical expression of 𝑑𝑈1 /𝑑𝑡. Thus, the closed-loop system under the consensus-based droop control will become more stable since more damping terms are included. IV. ROBUST C ONSENSUS -BASED D ROOP C ONTROL Although the consensus-based droop control described by (23) can achieve accurate power sharing among power con-
(24c) (24d)
∑𝑚
𝑗=1
−1 ∗ 𝐷𝑝,𝑗 𝑃𝑗 𝑝𝑗 (0)/𝐷𝑇 =
∑𝑚
(𝑃𝑗∗ −𝑃𝑗 (0))/𝐷𝑇, (25)
𝑗=1
which is exactly the average frequency deviation among power converters. By substituting it into (24a), we have 𝑃𝑖 /𝑃𝑖∗ = 1 −
∑𝑚
𝑗=1
(𝑃𝑗∗ − 𝑃𝑗 (0))/𝐷𝑇 , 𝑖 = 1, ⋅ ⋅ ⋅ 𝑚.
(26)
In comparison with (18), it is obvious that 𝑃𝑖 /𝑃𝑖∗ is now independent from each individual droop gain ratio 𝐷𝑝,𝑖 /𝑃𝑖∗ . Therefore, the above formula suggests that individual droop gain setting and precise power sharing can be achieved simultaneously. Analogy between (24a) and (24c) implies that accurate reactive power sharing can also be made. We will call this consensus-based droop control with the non-uniform droop gain ratio scheme as the robust consensus-based droop control. The stability of the closed-loop system described by the robust consensus-based control can also be proven the TEF. (24a)-(24d) can be written in compact form as ¯ 𝑐𝑜𝑛 (∂𝑈1 /∂𝒙), 𝑫 𝑐𝑜𝑛 𝒙˙ = −𝑨
(22)
The stability of the closed-loop dynamics can also be established by the TEF. (12)-(15) can be reformulated as the following compact form:
(24b)
nearly zero. By following the average-consensus principle, (24b) will drive all 𝑝𝑖 /𝐷𝑝,𝑖 to be identical as
𝑗=1
Since in conventional droop control, the droop gain ratio is chosen to be uniform as follows:
𝑗=1
𝑄∗𝑖
(24a)
(27)
¯ 𝑨 ˜ 𝑐𝑜𝑛 ). Entries in 𝑨 ¯ are only different form ¯ 𝑐𝑜𝑛 = (𝑨+ where 𝑨 ¯ 𝑨 in ∂𝑈1 /∂𝒑 and ∂𝑈1 /∂𝒒. Since 𝑫 −1 𝑐𝑜𝑛 𝑨𝑐𝑜𝑛 is indeed positive semi-definite, this robust consensus-based droop control scheme is also completely stable. V. R EAL -T IME S IMULATIONS In order to validate the performance of modified consensusbased 𝑃 -𝑓 /𝑄-𝑉˙ droop controller, simulations of two 14bus/6-converter micro-grids have been conducted by real-time simulators OPAL-RT [12]. The micro-grid is actually part of the IEEE-69 bus distribution system [13] with 𝑅/𝑋 ratio of all impedances adjusted to be uniform. Control block diagrams of this robust consensus-based 𝑃 -𝑓 /𝑄-𝑉˙ droop controller for each individual converter 𝑖 are shown in Fig. 2. Real-time simulations are performed by Opal-RT Technologies. Parameters of power converters are listed in [14]. The corresponding droop coefficients of the base case are listed in TABLE I. Since the heaviest load of the micro-grid is located at 𝐵𝑢𝑠14 , the droop gain of the nearest power converter 𝐷𝐼𝐶13 is used to provide the extra damping. Three test scenarios are carried out as follows:
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Droop controller Vi , abc ii , abc
abc → dq e abc → dq e
ii , dq
P− f droop
Pi
Vi , dq
LPF
Pi & Qi calculation
LPF Qi
θi Voltage & Current controller Vi*,dq
Q − V droop
Vi , dq
Predictive current controller
ii , dq
Pi
1 θi D p ,i
*
Pi
Pi* D p ,i
1 k p ,i s f restoration
n
ωb
θi
dq e → abc
Consensus ⎛ p pj ⎞ − ⎟ ⎟ ⎝ p ,i D p , j ⎠
Qi* Dq ,i
Qi
Q -V droop
(b)
Q13
P13
θi
Case I
Case II
1 Vi 1 ΔVi Dq ,i s
Qi*
∑ Lij ⎜⎜ D i j =1
P - f droop
1 s
(a)
busi
PWM modulator
PI
1 k q ,i s V restoration
n
Vi
Consensus Vi ,0 ⎛ q qj ⎞ − ⎟ ⎟ ⎝ q ,i Dq , j ⎠
ωnom
500 var/div 2 s/div
(d)
ω13
Fig. 2. Robust consensus-based droop control for each power converter 𝑖. 8
Case II
500 W/div 2 s/div
(c)
7
Case III
Case III
∑ Lij ⎜⎜ D i j =1
Case I
*
V13
Case I Case I Case III
9
Vnom
Case II
Case II
Case III 50 m rad ⋅ s −1 /div 2 s/div
DIC7 S8
1
3
2
10
DIC3
14
13
S14
DIC10
DIC4
DIC13
S = S6 + S8 + S11 + S14
S11 6
5
4
Fig. 4. (a) Active power output 𝑃13 ; (b) Reactive power output 𝑄13 ; (c) Power frequency 𝜔13 ; (d) Bus voltage 𝑉13 .
12
11
DIC5
S6
⎧⎪ S heavy = 6.68 kW + j 4.79 kvar ⎨ ⎪⎩ Slight = 2.23 kW + j1.59 kvar
Fig. 3. One-line diagram of the 14-bus/6-converter micro-grid.
Case I: Consensus-based droop control with perfectly uniform droop gain settings; ∙ Case II: Consensus-based droop control with imperfectly −1 = 4 × 10−5 rad⋅s−1 /W and droop gain settings: 𝐷𝑝,13 −1 −3 −1 𝐷𝑞,13 = 6 × 10 V⋅s /var; ∙ Case III: Robust consensus-based droop control with −1 = 4 × 10−5 imperfectly droop gain settings: 𝐷𝑝,13 −1 −1 −3 −1 rad⋅s /W and 𝐷𝑞,13 = 6 × 10 V⋅s /var. For all of the three cases, the system loads are varied from 𝑆𝑙𝑖𝑔ℎ𝑡 to 𝑆ℎ𝑒𝑎𝑣𝑦 at 𝑡 = 4𝑠, and the simulation results are shown in Fig. 4. For sake of better observations, only the power outputs and states of 𝐷𝐼𝐶13 are illustrated. Case I shows the converging rate and desired power sharing under steadystate with perfectly uniform droop gain settings. Though with faster converging rate, the original consensus-based droop results in undesired power sharing under the steady-state when imperfectly droop setting is applied in Case II. This is then resolved by the proposed robust consensus-based droop control, which provides much faster power output responses and achieves more accurate power. ∙
VI. C ONCLUSION In this paper, the robust consensus-based droop control is proposed to simplify the droop gain setting while maintaining the perfect power sharing when the steady state reaches.
TABLE I D ROOP C OEFFICIENTS , 𝑖 = 3, 4, 5, 7, 10, 13.
1/𝐷𝑝,𝑖
2 × 10−5 rad⋅s−1 /W
1/𝐷𝑞,𝑖
6 × 10−3 V⋅s−1 /var 1 × 10−2 s
𝑘𝑝,𝑖 = 𝑘𝑞,𝑖 𝑳𝑐𝑜𝑚,𝑝 , 𝑳𝑐𝑜𝑚,𝑞
5
1 × 10 , 1 × 104 𝑳𝑐𝑜𝑚 VA⋅s
2.5 V/div 2 s/div
By following the average-consensus theorem, this consensusdroop control can be proven to be more robust and be independent of particular droop gain settings. Real-time simulations for a 14-bus/6-consenter micro-grid are then developed on real-time simulator OPAL-RT. This robust consensus-based 𝑃 -𝑓 /𝑄-𝑉˙ droop controllers are validated in them with corresponding impedance conditions. Compared with consensusbased droop control strategies with the uniform droop gain ratio scheme, the proposed control method obviously alleviate the effects of non-ideal line impedances with better dynamical performances. Accurate power sharing is also achieved along with restorations of power frequency and terminal voltages of power converters. R EFERENCES [1] R. H. Lasseter, J. H. Eto, B. Schenkman, J. Stevens, H. Vollkommer, D. Klapp, E. Linton, H. Hurtado, and J. Roy, “Certs microgrid laboratory test bed,” IEEE Trans. Power Del., vol. 26, pp. 325–332, Jan. 2011. [2] M. C. Chandorkar, D. M. Divan, and B. Banerjee, “Control of distributed ups systems,” in Proc. IEEE PESC., 1994, pp. 197–204. [3] A. Engler and N. Soultanis, “Droop control in lv-grids,” in Proc. IEEE FPS, 2005, pp. 1–6. [4] C. K. Sao and P. W. Lehn, “Autonomous load sharing of voltage source converters,” IEEE Trans. Power Del., vol. 20, pp. 1009–1016, Apr. 2005. [5] K. D. Brabandere, B. Bolsens, J. V. d. Keybus, A. Woyte, J. Driesen, and R. Belmans, “A voltage and frequency droop control method for parallel inverters,” IEEE Trans. Power Electron., vol. 22, pp. 1107–1115, Jul. 2007. [6] C. T. Lee, C. C. Chu, and P. T. Cheng, “A new droop control method for the autonomous operation of distributed energy resource interface converters,” IEEE Trans. Power Electron., vol. 28, pp. 1980–1993, Apr. 2013. [7] J. W. S. Porco, F. Dofler, and F. Bullo, “Synchronization and power sharing for droop-controlled inverters in islanded microgrids,” Automatica, vol. 49, pp. 2603–2611, Sept. 2013. [8] Y. H. Moon, B. H. Choi, T. H. Rho, and B. K. Choi, “The development of equivalent system technique for deriving an energy function reflecting transfer conductances,” IEEE Trans. Power Syst., vol. 14, pp. 1335– 1341, Nov. 1999. [9] A. R. Bergen and D. J. Hill, “A structure preserving model for power system stability analysis,” IEEE Trans. Power App. Syst., vol. PAS-100, pp. 25–35, Jan. 1981. [10] K. Walve, “Modeling of power system components at severe disturbance,” CIGRE Report, 38-18, 1986. [11] H. D. Chiang and C. C. Chu, “A systematic search method for obtaining multiple local optimal solutions of nonlinear programming problems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 83, pp. 1497– 1529, Nov. 1995. [12] C. Dufour and J. Belanger, “A PC-based real-time parallel simulator of electric systems and drives,” in Proc. IEEE PCEE, 2004, pp. 105–113. [13] H. D. Chiang and R. Jean-Jumeau, “Optimal network reconfigurations in distribution systems: part 2 : solution algorithms and numerical results,” IEEE Trans. Power Del., vol. 5, pp. 1568–1574, Jul. 1990. [14] L. Y. Lu and C. C. Chu, “Autonomous power management and load sharing in isolated micro-grids by consensus-based droop control of power converters,” in Proc. IEEE IFEEC, 2013, to be published.
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Abstract—In order to provide autonomous power sharing of the conventional droop-control scheme for multiple power converters in isolated micro-grids, two consensus-based droopcontrol schemes will be examined. Since the conventional uniform droop gain ratio scheme requires identical droop gain ratio settings among all converters, applications to large-scale systems are restricted. An alternative scheme with the non-uniform droop gain ratio is proposed in this paper to simplify the droop gain setting while maintaining the perfect power sharing. Also, this new scheme is more robust since it is independent of particular gain settings. Real-time simulations of a micro-grid system are conducted to validate the effectiveness of this robust consensusbased droop control method. Index Terms—Micro-Grid, Converter Droop Control, Consensus Algorithm.
I. I NTRODUCTION
I
N conventional isolated micro-grids, the real powerfrequency (𝑃 -𝑓 ) droop control and reactive-voltage magnitude (𝑄-𝑉 ) droop control have been widely utilized for decentralized power sharing operations among distributed energy interface converters (DICs) [1], [2], [3]. However, the performance of the reactive power sharing under 𝑄-𝑉 droop control may be deteriorated due to its dependence on output line impedances [4], [5], [6]. In order to provide more accurate reactive power sharing, the 𝑄-𝑉˙ droop control method has been proposed recently [6]. Although the reactive power sharing under 𝑄-𝑉˙ droop control can be indeed independent of the output line impedances, incorrect reactive power sharing under non-uniform line impedances across the network were still observed [6]. In recent years, the consensus-based distributed control have been widely investigated for microgrid applications [7]. The main objective of this consensus control algorithm is to achieve agreement among all agents in a network considering certain desirable states of the entire system. Communication infrastructure of a multi-agent network is often limited to only local information exchange available. Since the consensus algorithms require only neighbour-toneighbour interaction, marvellous scalability and robustness can be achieved. This inherently distributed characteristics makes the consensus-based control scalable and can be easily implemented. In order to obtain more accurate power sharing in microgrid applications, the consensus-based 𝑃 -𝑓 droop control with perfectly uniform droop gain ratio scheme was studied by [7] recently. The uniform droop gain ratio scheme requires that This work was supported in part by the National Science Council of Taiwan, R.O.C., under Grant NSC 102-2221-E-007-074. The authors are with Department of Electrical Engineering, National Tsing Hua University, Hsinchu, 300, Taiwan, R.O.C. (E-mail: [email protected]; [email protected])
978-1-4799-3432-4/14/$31.00 ©2014 IEEE
the droop gain ratio of all converters are identical. Thus, this scheme is not very robust under droop gain variations. This disadvantage will limit its practical applications for large-scale micro-grids. In order to overcome this difficulties, the nonuniform droop gain ratio scheme will be proposed to simplify droop gain setting while maintaining the desired power sharing under the steady state in this paper. By following the averageconsensus principle, this modified consensus-droop control with the non-uniform droop gain ratio can be proven to be independent of particular droop gain. In addition, the closedloop dynamics of this robust consensus-droop control with the non-uniform droop gain ratio can be shown to be stable by the more common transient energy function (TEF) approach. We will call this consensus-based droop control with the nonuniform droop gain ratio scheme as the robust consensusbased droop control. In order to validate the feasibility of this robust consensus-based droop control method, real-time simulations are performed on a 14-bus/6-converter microgrid. Simulation results indicate that this robust consensusdroop droop control can indeed provide larger damping while maintaining precise active and reactive power sharing. II. C ONVENTIONAL D ROOP C ONTROL Consider a micro-grid with 𝑛 buses as shown in Fig. 1. Buses 1, ⋅ ⋅ ⋅ 𝑚 are power sources embedded with power converters. Buses 𝑚+1, ⋅ ⋅ ⋅ , 𝑛 are load buses. All line impedances are lossy with the uniform 𝑅/𝑋 ratio defined by 𝑅𝑖𝑗 /𝑋𝑖𝑗 = 𝐾 = −𝐺𝑖𝑗 /𝐵𝑖𝑗 , where 𝑅𝑖𝑗 , 𝑋𝑖𝑗 , 𝐺𝑖𝑗 and 𝐵𝑖𝑗 are resistance, reactance, conductance and susceptance of each transmission line between node 𝑖 and 𝑗, respectively. Now the active and reactive power flow equations can be expressed as 𝑃𝑖 − 𝐾𝑄𝑖 = (1 + 𝐾 2 )
∑𝑛
2
𝑄𝑖 + 𝐾𝑃𝑖 = −(1 + 𝐾 )
𝑗=1
∑𝑛
𝑉𝑖 𝑉𝑗 𝐵𝑖𝑗 sin 𝜃𝑖𝑗 ,
𝑗=1
𝑉𝑖 𝑉𝑗 𝐵𝑖𝑗 cos 𝜃𝑖𝑗 .
(1a) (1b)
For 𝑖 = 1+𝑚, ⋅ ⋅ ⋅ 𝑛, load dynamics are represented as follows: ∑𝑛 𝑃𝐿𝑖 = − 𝑉𝑖 𝑉𝑗 𝐵𝑖𝑗 sin 𝜃𝑖𝑗 − 𝐷𝐿𝑝,𝑖 𝜃˙𝑖 , 𝑗=1 ∑𝑛 𝑉𝑖 𝑉𝑗 𝐵𝑖𝑗 cos 𝜃𝑖𝑗 − 𝐷𝐿𝑞,𝑖 𝑉˙ 𝑖 . 𝑄𝐿𝑖 = 𝑗=1
Bus1
Busm +1
Busm + 2
Loads
Bus2
Physical network Busm
Busn Physical connection Communication (topology dependent)
Fig. 1. Several Power converters in an isloated micro-grid.
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(2a) (2b)
(1a) and (1b) can be treated as modified power flow equations of a pure inductive network with equivalent power injections 𝑃𝑖 − 𝐾𝑄𝑖 and 𝑄𝑖 + 𝐾𝑃𝑖 [8]. The subscript 𝐿 in (2a) and (2b) stands for the load bus. Here, the 𝐷𝐿𝑝,𝑖 𝜃˙𝑖 and 𝐷𝐿𝑞,𝑖 𝑉˙ 𝑖 terms are introduced to represent load dependences on the bus frequency [9] and the time derivative of voltage magnitude [10]. Traditionally, the real power-frequency 𝑃 -𝑓 droop control and reactive-voltage magnitude 𝑄-𝑉 droop control have been adapted for decentralized power sharing operations. However, the performance of reactive power sharing under conventional 𝑄-𝑉 droop control may be deteriorated due to its dependence on output line impedances [4], [5], [6]. In order to provide more accurate reactive power sharing, the 𝑄-𝑉˙ droop control method has been proposed recently [6]. Specifically, if all fast dynamics of power converters are neglected, the 𝑃 -𝑓 /𝑄𝑉˙ droop control for autonomous power sharing operations of multiple power converters can be described as follows: 𝐷𝑝,𝑖 𝜃˙𝑖
=
𝑘𝑝,𝑖 𝑝˙ 𝑖 𝐷𝑞,𝑖 𝑉𝑖 𝑣˙ 𝑖 𝑘𝑞,𝑖 𝑞˙𝑖
= = =
𝑃𝑖∗ − 𝑃𝑖 − 𝑝𝑖 , 𝐷𝑝,𝑖 𝜃˙𝑖 ,
𝑄∗𝑖 − 𝑄𝑖 − 𝑞𝑖 , 𝐷𝑞,𝑖 𝑉𝑖 𝑣˙ 𝑖 ,
(3) (4) (5) (6)
where 𝑣𝑖 is the transformed bus voltage defined by 𝑣𝑖 = ln 𝑉𝑖 and 𝑣˙𝑖 = 𝑉˙𝑖 ⋅ 𝑉𝑖−1 . 𝑃𝑖∗ and 𝑄∗𝑖 are the nominal active and reactive power ratings of the DIC, respectively. 𝐷𝑝,𝑖 and 𝐷𝑞,𝑖 are the droop gains corresponding to active and reactive power control. 𝑝𝑖 and 𝑞𝑖 are the 𝜃˙ and 𝑉˙ restoration mechanism to maintain the frequency and voltage profile. The terminal voltage and the frequency of the converter will be held at nominal value under the steady-state. The stability of the closed-loop system can be proven by the transient energy function (TEF) approach. Let the state variable of each bus be 𝒙𝑖 = [𝜃𝑖 , 𝑣˙ 𝑖 , 𝑝𝑖 , 𝑞𝑖 ]𝑇 and state variables of the overall system be 𝒙𝑇 = [𝜽 𝑇 , 𝒗˙ 𝑇 , 𝒑𝑇 , 𝒒 𝑇 ], the TEF of the closed-loop system can be defined by 𝑈1 (𝒙) = 𝑊1 (𝒙)+𝑊2 (𝒙)+𝑊3 (𝒙) +𝑊4 (𝒑, 𝒒),
(7)
where
⎞ ⎛ 𝑛 𝑛 1 ⎝∑ ∑ 2 𝑊1 (𝒙) = − (1+𝐾 )𝑉𝑖 𝑉𝑗 𝐵𝑖𝑗 cos 𝜃𝑖𝑗 ⎠ , 2 𝑖=1 𝑗=1
𝑊2 (𝒙) = − 𝑊3 (𝒙) = − 𝑊4 (𝒑, 𝒒) =
𝑚 ∑
(𝑃𝑗∗ −𝐾𝑄∗𝑗 )𝜃𝑗 +
𝑗=1 𝑚 ∑
𝑛 ∑
(𝑄∗𝑗 +𝐾𝑃𝑗∗ )⋅𝑣𝑗 +
𝑗=1 𝑚 ∑
(𝑃𝐿𝑗 −𝐾𝑄𝐿𝑗 )𝜃𝑗 ,
𝑗=𝑚+1 𝑛 ∑
(𝑄𝐿𝑗 +𝐾𝑃𝐿𝑗 )⋅𝑣𝑗 ,
(8a)
(8b)
(8c)
𝑗=𝑚+1
1 (𝑝2 + 𝑞𝑗2 ). 2 𝑗=1 𝑗
(8d)
Under the this framework, the closed-loop micro-grid under the 𝑃 -𝑓 /𝑄-𝑉˙ droop control (3) and (5) can be re-written in the following compact form: 𝑫 𝑖 𝒙˙ 𝑖 = −𝑨𝑖 (∂𝑈1 / ∂𝒙𝑖 ), where
(9)
⎤ ⎡ ⎤ 𝐷𝑝,𝑖 0 −𝐾𝐷𝑞,𝑖 𝑉𝑖 0 1 1 0 −𝐾 ⎢ 0 𝑘𝑝,𝑖 −𝐾𝐷𝑞,𝑖 𝑉𝑖 0 ⎥ ⎢0 0 0 0 ⎥ . (10) 𝑫𝑖 = ⎣ , 𝑨𝑖 = ⎣ 0 −𝐾 1 1 ⎦ 𝐾𝐷𝑝,𝑖 0 𝐷𝑞,𝑖 𝑉𝑖 0 ⎦ 0 0 0 0 𝐾𝐷𝑝,𝑖 0 0 𝑘𝑞,𝑖 ⎡
Thus, the overall closed-loop system can be expressed as 𝑫 𝒙˙ = −𝑨(∂𝑈1 / ∂𝒙).
(11)
The matrix 𝑫 −1 𝑨 is positive semi-definite with 2𝑛 zeroeigenvalues. In particular, these zero-eigenvalues do not lead to a continuum of equilibrium points, but rather isolated equilibrium points as algebraic solutions of (3) to (6). Thus, (11) is indeed a quasi-gradient system [11] and every trajectory will converge to one of the isolated equilibrium points. Therefore, the 𝑃 -𝑓 /𝑄-𝑉˙ droop control can indeed ensure the complete stability of the closed-loop system. III. C ONSENSUS -BASED D ROOP C ONTROL Although quite accurate real power sharing can be ensured by (11), incorrect reactive power sharing under non-uniform line impedances across the network were still observed [6]. It is due to the existence of zero-eigenvalues for 𝑨𝑖 . Thus, the trajectory will diverge for general lossy networks and the pure 𝑃 -𝑓 and 𝑄-𝑉˙ relationship for droop control will no longer hold. The consensus-based droop control, which allows data exchange between networking neighbours by message passing algorithms, was proposed to overcome this difficulty. In the past, only the consensus-based 𝑃 -𝑓 droop control is considered [7]. In this paper, we will extend their work to consider consensus-based 𝑃 -𝑓 and 𝑄-𝑉˙ droop control simultaneously. By introducing the Laplacian matrix among all power converters, the closed-loop of the consensus-based 𝑃 -𝑓 /𝑄-𝑉˙ droop control can be constructed as follows: 𝐷𝑝,𝑖 𝜃˙𝑖 = 𝑘𝑝,𝑖 𝑝˙ 𝑖 = 𝐷𝑞,𝑖 𝑉𝑖 𝑣˙ 𝑖 = 𝑘𝑞,𝑖 𝑞˙𝑖 =
𝑃𝑖∗ − 𝑃𝑖 − 𝑝𝑖 , (12) ∑𝑚 ˙ 𝐷𝑝,𝑖 𝜃𝑖 + 𝑙𝑖𝑗 (𝑝𝑖 /𝐷𝑝,𝑖 − 𝑝𝑗 /𝐷𝑝,𝑗 ) , (13) 𝑗=1
𝑄∗𝑖 − 𝑄𝑖 − 𝑞𝑖 , (14) ∑𝑚 𝐷𝑞,𝑖 𝑉𝑖 𝑣˙ 𝑖 + 𝑙𝑖𝑗 (𝑞𝑖 /𝐷𝑞,𝑖 − 𝑞𝑗 /𝐷𝑞,𝑗 ).(15) 𝑗=1
In the above formulation, 𝑙𝑖𝑗 is the [𝑖, 𝑗] entry of the graph Laplacian 𝑳𝑐𝑜𝑚𝑚 of interconnected power converters. Since the off-diagonal entries of 𝑳𝑐𝑜𝑚𝑚 indicate the physical connectivity between vertices of the network, the communication between power converters is then established between physical neighbours of each converter only. Unlike the conventional droop control scheme, additional terms in (13) and (15) will provide the power sharing adjustment mechanism among neighbouring converters. It is worthy to pointing out that if the consensus control is inactivated by setting 𝑙𝑖𝑗 = 0 for all 𝑖, 𝑗, the consensus-based droop control will be reduced to 𝑃 -𝑓 /𝑄-𝑉˙ droop control. All trajectories will still converge to one of the equilibrium point. Thus, the consensus-based droop control can be considered as a supplementary control actions from neighbouring converters. This implies that the overall consensus-based droop control algorithm is composed of two phases: the primary local 𝑃 𝑓 /𝑄-𝑉˙ droop control and the supplementary consensus-based droop control. The local 𝑃 -𝑓 /𝑄-𝑉˙ droop control will dominate dynamical behaviours of the closed-loop system in the early stage. Thus, in the first phase, the frequency restoration and synchronization mechanism governed by (12) and (14) will reach the steady-state soon. Then the system will move to the
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consensus-based droop control phase. Under this framework, the initial state of the consensus-based droop control under steady-state of ∑𝑚(12) can be represented approximately by 𝜃˙𝑎𝑣𝑔 (0) = ( 𝑗=1 𝑃𝑗∗ − 𝑃𝑗 (0) − 𝑝𝑗 (0))/𝐷𝑇 , where 𝐷𝑇 = ∑𝑚 ˙ 𝑗=1 𝐷𝑝,𝑗 . Since 𝜃𝑎𝑣𝑔 (0) is nearly zero under the frequency restoration and synchronization mechanisms, the dynamics of (13) become the first order consensus algorithm with ∑𝑚 𝑙𝑖𝑗 (𝑝𝑖 /𝐷𝑝,𝑖 − 𝑝𝑗 /𝐷𝑝,𝑗 ) . (16) 𝑘𝑝,𝑖 𝑝˙ 𝑖 =
verters, the precise settings (19) and (21) are required. This condition will restrict the applications for practical large-scale micro-grid since damping variations will destroy the perfect power sharing. In order to avoid the precise uniform droop gain ration setting and provide more flexibility for individual droop gain settings, the following consensus-based droop control with uniform droop gain is proposed: −1 ∗ 𝑃𝑖 𝑝 𝑖 , 𝐷𝑝,𝑖 𝜃˙𝑖 = 𝑃𝑖∗ − 𝑃𝑖 − 𝐷𝑝,𝑖 ∑𝑚 𝑘𝑝,𝑖 𝑝˙ 𝑖 = 𝐷𝑝,𝑖 𝜃˙𝑖 + 𝑙𝑖𝑗 (𝑝𝑖 /𝐷𝑝,𝑖 − 𝑝𝑗 /𝐷𝑝,𝑗 ) ,
𝑗=1
Following the average-consensus principle, (16) would drive all 𝑝𝑖 /𝐷𝑝,𝑖 to be identical as ∑𝑚 ∑𝑚 𝑝𝑖 /𝐷𝑝,𝑖 = 𝑝𝑗 (0)/𝐷𝑇 = (𝑃𝑗∗ −𝑃𝑗 (0))/𝐷𝑇 , (17) 𝑗=1
−1 ∗ 𝐷𝑞,𝑖 𝑉𝑖 𝑣˙ 𝑖 = − 𝑄𝑖 − 𝐷𝑞,𝑖 𝑄 𝑖 𝑞𝑖 , ∑𝑚 𝑘𝑞,𝑖 𝑞˙𝑖 = 𝐷𝑞,𝑖 𝑉𝑖 𝑣˙ 𝑖 + 𝑙𝑖𝑗 (𝑞𝑖 /𝐷𝑞,𝑖 − 𝑞𝑗 /𝐷𝑞,𝑗 ) . 𝑗=1 ∑𝑚 (𝑃 ∗ −𝑃𝑗 (0)−𝐷−1 𝑃 ∗ 𝑝𝑗 (0)) is Likewise, 𝜃˙𝑎𝑣𝑔 (0) = 𝑗=1 𝑗 ∑𝑚 𝐷𝑝,𝑗𝑝,𝑗 𝑗 𝑗=1
𝑗=1
which is exactly the average frequency deviation among power converters. If we substitute (17) into (12), ∑𝑚 𝑃𝑗∗ − 𝑃𝑗 (0)/(𝑃𝑖∗ 𝐷𝑇 ). (18) 𝑃𝑖 /𝑃𝑖∗ = 1 − 𝐷𝑝,𝑖 ⋅
𝐷𝑝,1 /𝑃1∗
=
𝐷𝑝,2 /𝑃2∗
=
∗ ⋅ ⋅ ⋅ 𝐷𝑝,𝑚 /𝑃𝑚 .
(19)
Obviously, appending the consensus term would drive the real power outputs of power converters toward ∗ 𝑃1 /𝑃1∗ = 𝑃2 /𝑃2∗ = ⋅ ⋅ ⋅ 𝑃𝑚 /𝑃𝑚 .
(20)
In other words, active power outputs of all power converters share system loads in the micro-grid proportional to their nominal ratings. Similar conclusions about reactive power can be made by considering (14) and (15). When the perfect uniform droop gain ratio is chosen as follows: 𝐷𝑞,1 /𝑄∗1 = 𝐷𝑞,2 /𝑄∗2 = ⋅ ⋅ ⋅ 𝐷𝑞,𝑚 /𝑄∗𝑚 ,
(21)
the following reactive power sharing will be reached: 𝑄1 /𝑄∗1 = 𝑄2 /𝑄∗2 = ⋅ ⋅ ⋅ 𝑄𝑚 /𝑄∗𝑚 .
𝑫 𝑐𝑜𝑛 𝒙˙ = −𝑨𝑐𝑜𝑛 (∂𝑈1 /∂𝒙).
𝑝𝑖/𝐷𝑝,𝑖 =
(23)
It can be verified that 𝑫 𝑐𝑜𝑛 = 𝑫 which is of full rank. ˜ 𝑐𝑜𝑛 ), where 𝑨 ˜ 𝑐𝑜𝑛 = 𝑑𝑖𝑎𝑔(0, −𝑳𝑐𝑜𝑚𝑚𝑫 −1 , 0, 𝑨𝑐𝑜𝑛 = (𝑨+𝑨 𝑝 −1 −𝑳𝑐𝑜𝑚𝑚𝑫 𝑞 ). 2𝑛 zero-eigenvalues of 𝑨𝑐𝑜𝑛 are eliminated by the consensus-based droop control. Only two inherent zeroeigenvalues of the 𝑳𝑐𝑜𝑚𝑚 term are left. By eigenvector analysis of 𝑨𝑖 , it can be shown that these two zero-eigenvalues of 𝑨𝑐𝑜𝑛 are indeed an isolated equilibrium point. Since each positive term of 𝑳𝑐𝑜𝑚𝑚 will appear on the diagonal entry of 𝑨𝑐𝑜𝑛 , these additional negative terms will appear on the analytical expression of 𝑑𝑈1 /𝑑𝑡. Thus, the closed-loop system under the consensus-based droop control will become more stable since more damping terms are included. IV. ROBUST C ONSENSUS -BASED D ROOP C ONTROL Although the consensus-based droop control described by (23) can achieve accurate power sharing among power con-
(24c) (24d)
∑𝑚
𝑗=1
−1 ∗ 𝐷𝑝,𝑗 𝑃𝑗 𝑝𝑗 (0)/𝐷𝑇 =
∑𝑚
(𝑃𝑗∗ −𝑃𝑗 (0))/𝐷𝑇, (25)
𝑗=1
which is exactly the average frequency deviation among power converters. By substituting it into (24a), we have 𝑃𝑖 /𝑃𝑖∗ = 1 −
∑𝑚
𝑗=1
(𝑃𝑗∗ − 𝑃𝑗 (0))/𝐷𝑇 , 𝑖 = 1, ⋅ ⋅ ⋅ 𝑚.
(26)
In comparison with (18), it is obvious that 𝑃𝑖 /𝑃𝑖∗ is now independent from each individual droop gain ratio 𝐷𝑝,𝑖 /𝑃𝑖∗ . Therefore, the above formula suggests that individual droop gain setting and precise power sharing can be achieved simultaneously. Analogy between (24a) and (24c) implies that accurate reactive power sharing can also be made. We will call this consensus-based droop control with the non-uniform droop gain ratio scheme as the robust consensus-based droop control. The stability of the closed-loop system described by the robust consensus-based control can also be proven the TEF. (24a)-(24d) can be written in compact form as ¯ 𝑐𝑜𝑛 (∂𝑈1 /∂𝒙), 𝑫 𝑐𝑜𝑛 𝒙˙ = −𝑨
(22)
The stability of the closed-loop dynamics can also be established by the TEF. (12)-(15) can be reformulated as the following compact form:
(24b)
nearly zero. By following the average-consensus principle, (24b) will drive all 𝑝𝑖 /𝐷𝑝,𝑖 to be identical as
𝑗=1
Since in conventional droop control, the droop gain ratio is chosen to be uniform as follows:
𝑗=1
𝑄∗𝑖
(24a)
(27)
¯ 𝑨 ˜ 𝑐𝑜𝑛 ). Entries in 𝑨 ¯ are only different form ¯ 𝑐𝑜𝑛 = (𝑨+ where 𝑨 ¯ 𝑨 in ∂𝑈1 /∂𝒑 and ∂𝑈1 /∂𝒒. Since 𝑫 −1 𝑐𝑜𝑛 𝑨𝑐𝑜𝑛 is indeed positive semi-definite, this robust consensus-based droop control scheme is also completely stable. V. R EAL -T IME S IMULATIONS In order to validate the performance of modified consensusbased 𝑃 -𝑓 /𝑄-𝑉˙ droop controller, simulations of two 14bus/6-converter micro-grids have been conducted by real-time simulators OPAL-RT [12]. The micro-grid is actually part of the IEEE-69 bus distribution system [13] with 𝑅/𝑋 ratio of all impedances adjusted to be uniform. Control block diagrams of this robust consensus-based 𝑃 -𝑓 /𝑄-𝑉˙ droop controller for each individual converter 𝑖 are shown in Fig. 2. Real-time simulations are performed by Opal-RT Technologies. Parameters of power converters are listed in [14]. The corresponding droop coefficients of the base case are listed in TABLE I. Since the heaviest load of the micro-grid is located at 𝐵𝑢𝑠14 , the droop gain of the nearest power converter 𝐷𝐼𝐶13 is used to provide the extra damping. Three test scenarios are carried out as follows:
1822
Droop controller Vi , abc ii , abc
abc → dq e abc → dq e
ii , dq
P− f droop
Pi
Vi , dq
LPF
Pi & Qi calculation
LPF Qi
θi Voltage & Current controller Vi*,dq
Q − V droop
Vi , dq
Predictive current controller
ii , dq
Pi
1 θi D p ,i
*
Pi
Pi* D p ,i
1 k p ,i s f restoration
n
ωb
θi
dq e → abc
Consensus ⎛ p pj ⎞ − ⎟ ⎟ ⎝ p ,i D p , j ⎠
Qi* Dq ,i
Qi
Q -V droop
(b)
Q13
P13
θi
Case I
Case II
1 Vi 1 ΔVi Dq ,i s
Qi*
∑ Lij ⎜⎜ D i j =1
P - f droop
1 s
(a)
busi
PWM modulator
PI
1 k q ,i s V restoration
n
Vi
Consensus Vi ,0 ⎛ q qj ⎞ − ⎟ ⎟ ⎝ q ,i Dq , j ⎠
ωnom
500 var/div 2 s/div
(d)
ω13
Fig. 2. Robust consensus-based droop control for each power converter 𝑖. 8
Case II
500 W/div 2 s/div
(c)
7
Case III
Case III
∑ Lij ⎜⎜ D i j =1
Case I
*
V13
Case I Case I Case III
9
Vnom
Case II
Case II
Case III 50 m rad ⋅ s −1 /div 2 s/div
DIC7 S8
1
3
2
10
DIC3
14
13
S14
DIC10
DIC4
DIC13
S = S6 + S8 + S11 + S14
S11 6
5
4
Fig. 4. (a) Active power output 𝑃13 ; (b) Reactive power output 𝑄13 ; (c) Power frequency 𝜔13 ; (d) Bus voltage 𝑉13 .
12
11
DIC5
S6
⎧⎪ S heavy = 6.68 kW + j 4.79 kvar ⎨ ⎪⎩ Slight = 2.23 kW + j1.59 kvar
Fig. 3. One-line diagram of the 14-bus/6-converter micro-grid.
Case I: Consensus-based droop control with perfectly uniform droop gain settings; ∙ Case II: Consensus-based droop control with imperfectly −1 = 4 × 10−5 rad⋅s−1 /W and droop gain settings: 𝐷𝑝,13 −1 −3 −1 𝐷𝑞,13 = 6 × 10 V⋅s /var; ∙ Case III: Robust consensus-based droop control with −1 = 4 × 10−5 imperfectly droop gain settings: 𝐷𝑝,13 −1 −1 −3 −1 rad⋅s /W and 𝐷𝑞,13 = 6 × 10 V⋅s /var. For all of the three cases, the system loads are varied from 𝑆𝑙𝑖𝑔ℎ𝑡 to 𝑆ℎ𝑒𝑎𝑣𝑦 at 𝑡 = 4𝑠, and the simulation results are shown in Fig. 4. For sake of better observations, only the power outputs and states of 𝐷𝐼𝐶13 are illustrated. Case I shows the converging rate and desired power sharing under steadystate with perfectly uniform droop gain settings. Though with faster converging rate, the original consensus-based droop results in undesired power sharing under the steady-state when imperfectly droop setting is applied in Case II. This is then resolved by the proposed robust consensus-based droop control, which provides much faster power output responses and achieves more accurate power. ∙
VI. C ONCLUSION In this paper, the robust consensus-based droop control is proposed to simplify the droop gain setting while maintaining the perfect power sharing when the steady state reaches.
TABLE I D ROOP C OEFFICIENTS , 𝑖 = 3, 4, 5, 7, 10, 13.
1/𝐷𝑝,𝑖
2 × 10−5 rad⋅s−1 /W
1/𝐷𝑞,𝑖
6 × 10−3 V⋅s−1 /var 1 × 10−2 s
𝑘𝑝,𝑖 = 𝑘𝑞,𝑖 𝑳𝑐𝑜𝑚,𝑝 , 𝑳𝑐𝑜𝑚,𝑞
5
1 × 10 , 1 × 104 𝑳𝑐𝑜𝑚 VA⋅s
2.5 V/div 2 s/div
By following the average-consensus theorem, this consensusdroop control can be proven to be more robust and be independent of particular droop gain settings. Real-time simulations for a 14-bus/6-consenter micro-grid are then developed on real-time simulator OPAL-RT. This robust consensus-based 𝑃 -𝑓 /𝑄-𝑉˙ droop controllers are validated in them with corresponding impedance conditions. Compared with consensusbased droop control strategies with the uniform droop gain ratio scheme, the proposed control method obviously alleviate the effects of non-ideal line impedances with better dynamical performances. Accurate power sharing is also achieved along with restorations of power frequency and terminal voltages of power converters. R EFERENCES [1] R. H. Lasseter, J. H. Eto, B. Schenkman, J. Stevens, H. Vollkommer, D. Klapp, E. Linton, H. Hurtado, and J. Roy, “Certs microgrid laboratory test bed,” IEEE Trans. Power Del., vol. 26, pp. 325–332, Jan. 2011. [2] M. C. Chandorkar, D. M. Divan, and B. Banerjee, “Control of distributed ups systems,” in Proc. IEEE PESC., 1994, pp. 197–204. [3] A. Engler and N. Soultanis, “Droop control in lv-grids,” in Proc. IEEE FPS, 2005, pp. 1–6. [4] C. K. Sao and P. W. Lehn, “Autonomous load sharing of voltage source converters,” IEEE Trans. Power Del., vol. 20, pp. 1009–1016, Apr. 2005. [5] K. D. Brabandere, B. Bolsens, J. V. d. Keybus, A. Woyte, J. Driesen, and R. Belmans, “A voltage and frequency droop control method for parallel inverters,” IEEE Trans. Power Electron., vol. 22, pp. 1107–1115, Jul. 2007. [6] C. T. Lee, C. C. Chu, and P. T. Cheng, “A new droop control method for the autonomous operation of distributed energy resource interface converters,” IEEE Trans. Power Electron., vol. 28, pp. 1980–1993, Apr. 2013. [7] J. W. S. Porco, F. Dofler, and F. Bullo, “Synchronization and power sharing for droop-controlled inverters in islanded microgrids,” Automatica, vol. 49, pp. 2603–2611, Sept. 2013. [8] Y. H. Moon, B. H. Choi, T. H. Rho, and B. K. Choi, “The development of equivalent system technique for deriving an energy function reflecting transfer conductances,” IEEE Trans. Power Syst., vol. 14, pp. 1335– 1341, Nov. 1999. [9] A. R. Bergen and D. J. Hill, “A structure preserving model for power system stability analysis,” IEEE Trans. Power App. Syst., vol. PAS-100, pp. 25–35, Jan. 1981. [10] K. Walve, “Modeling of power system components at severe disturbance,” CIGRE Report, 38-18, 1986. [11] H. D. Chiang and C. C. Chu, “A systematic search method for obtaining multiple local optimal solutions of nonlinear programming problems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 83, pp. 1497– 1529, Nov. 1995. [12] C. Dufour and J. Belanger, “A PC-based real-time parallel simulator of electric systems and drives,” in Proc. IEEE PCEE, 2004, pp. 105–113. [13] H. D. Chiang and R. Jean-Jumeau, “Optimal network reconfigurations in distribution systems: part 2 : solution algorithms and numerical results,” IEEE Trans. Power Del., vol. 5, pp. 1568–1574, Jul. 1990. [14] L. Y. Lu and C. C. Chu, “Autonomous power management and load sharing in isolated micro-grids by consensus-based droop control of power converters,” in Proc. IEEE IFEEC, 2013, to be published.
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