Stability constrained incentive mechanisms for distributed frequency ...

Stability constrained incentive mechanisms for distributed frequency control of power grid Minghui Zhu and Na Li Abstract— In this paper, we consider the problem of stability constrained incentive design for distributed frequency control in power grid. We aim to design incentive mechanisms under which control authorities are incentivized to implement distributed controllers such that the closed-loop system enforces frequency stability and state constraints. To solve the problem, we propose a reward based mechanism and determine a lower bound on the rewards such that the induced Nash equilibrium realizes the above objectives on dynamic behavior of power grid.

I. I NTRODUCTION In power grid, a fundamental objective is to reliably balance power generations and demands. The objective has been facilitated by the active engagement of end users, the high penetration of renewable energy and the wide deployment of advanced sensing, communication and control. Meanwhile, these new components present new challenges to power grid. Dynamic changes of user demands and the volatility of renewable energy introduce dynamic disturbances to power grid and threat another fundamental objective: regulating a set of services, such as frequency and voltage. In addition, distributed energy resources are managed by heterogeneous control authorities who seek for different and even partially conflicting subobjectives. Then the system operator may only have limited control of the partially deregulated power system. In our recent work [23], we tackle the above issues via a passive approach. In particular, we design a distributed access controller where the system operator disconnects subsystems; e.g., microgrids, which fail to fulfill certain stability conditions. In the current paper, we propose an active approach to complement that in [23]. Our results in [23] show that frequency stability and the enforcement of state constraints require that each subsystem maintains a sufficiently large local stability margin. Intuitively, one can also expect that the stability margin of the whole power grid is proportional to those of subsystems. Yet it is costly in control for each subsystem to maintain a large local stability margin. Given the selfishness of control authorities, it is of interest to incentivize control authorities such that the gaps of the interests of different parties are reduced. Contributions. In this paper, we study incentive design for distributed frequency control of synchronous generators in M. Zhu is with the Department of Electrical Engineering, Pennsylvania State University, 201 Old Main, University Park, PA, 16802, [email protected]. N. Li is with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, and the Department of Electrical Engineering, Harvard University, 29 Oxford Street, Cambridge, MA 02138, na [email protected].

power grid. In particular, we first formulate the problem of stability constrained incentive design where the system operator incentivizes control authorities to choose distributed controllers which can maximize the stability margin of power grid, simultaneously ensuring frequency stability and enforcing state constraints. Inspired by fixed-prize raffles in [19], we propose a reward based incentive mechanism where each control authority obtains a portion of a fixed reward and its reward is proportional to its contribution to the system stability. The contribution of each control authority is characterized by the spectrum of its own subsystem. The incentive mechanism presents a non-cooperative game among control authorities. Once receiving the rewards corresponding to a Nash equilibrium, control authorities choose distributed controllers to commit to their proposals. We identify the lower bound on the reward under which the induced Nash equilibrium can enforce the stability condition. We also provide other analytic properties of the incentive scheme. Literature review. In the literature of micro economics, there are a large number of incentive or pricing schemes which can partially mitigate selfish behavior in competitive scenarios. Theses schemes have been applied to many domains, including the demand response of power grid [4], [6], [7], [20], communication networks [14] and transportation networks [18]. However, the set of papers do not consider dynamic systems. In the framework of multi-player linear-quadraticregulator games in, e.g., [2], each player is associated with a finite-horizon or infinite-horizon performance index. In contrast, we adopt the stability margin as the performance index. This allows us to deal with state constraints, dynamic changes of grid topologies and the disturbances induced by demand response and renewable integration. The effects of external disturbances on dynamic performance of the power grid have been receiving increasing attention, including sensitivity analysis [9], [10], reachability analysis [1], [13] and set-theoretic method [5]. Distributed control of the power grid has been extensively studied. The classic distributed control includes power system stabilizer (PSS) and automatic generation control (AGC) [8], [15], [16], [21], [22]. The recent papers [11], [12] investigate AGC with wind integration. Notations. Let λmax (A) be the real part of the eigenvalue λi of matrix A where Re(λi ) ≥ Re(λj ) for any eigenvalue λj of A. Denote the supremum norm of the truncation of u(t) in [t1 , t2 ] by kuk[t1 ,t2 ] , supt1 ≤t≤t2 ku(t)k1 . The dynamic 1 In

this paper, we will use the infinite norm for the vectors and matrices.

system x˙ = f (x, u, t) is called input-to-state stable (ISS, for short) if there exist class KL function β(·, ·) and class K functions γ(·) such that for all xi (t0 ) ∈ Xii the following holds for all t ≥ t0 :

where Ni (t) ⊆ V \ {i}. In (2), the power flow between control authorities i and j is modeled by:

kx(t)k ≤ max{β(kx(t0 )k, t − t0 ), γ(kuk[t0 ,t] )},

The swing dynamics, the first equation in (2), captures the frequency evolution. The second equation in (2) and (3) describe the branch flow dynamics. The third and fourth equations in (2) stand for the dynamic system of turbinegovernor which is controlled via the reference point ∆Pref i .

(1)

where the functions of β and γ are time independent. II. D ISTRIBUTED FREQUENCY CONTROL In this section, we will investigate distributed frequency control of interconnected synchronous generators. The parameters used in this section are summarized in Table I.

∆Pij (t) = Tij sin(∆θi (t) − ∆θj (t)).

(3)

TABLE I: Generator variables w θ PM ∗ PM PL PL∗ Pv Pv∗ Pij Pref ∗ Pref v Pw Pren ∗ Pren

angular frequency phase angle mechanical power set-point of mechanical power actual load forecasted load stream valve position set-point of stream valve position tie-line power flow between control authorities i and j reference power set-point of reference power wind speed wind power actual turbine power forecasted turbine power

Fig. 1: Generation and loads of bus i.

A. Power system model We consider a power system comprised by a collection of interconnected buses. Each bus represents a control authority which may consist of a variety of generators and/or loads. It has been a common practice to lump all the generators (resp. loads) of a control authority as a single generator (resp. load). At each control authority, we assume that there is a mechanical generation PMi , wind generation Preni and load PLi . The control authorities are connected with each other through the power flow Pij . 1) Generator dynamics: We will adopt the model of synchronous generators in [16] for each control authority. The notation ∆ is used to indicate a deviation from nominal ∗ ∗ ∗ value [w∗ , θi∗ , PL∗i , PM , Pv∗i , Pren , Pref ]T . For example, i i i ∆wi represents the deviation of angular frequency from the constant set-point w∗ ; e.g., 60 Hz. At time instant t, the state-space model of control authority i is given by the following: X d∆wi 1 =− Di ∆wi + ∆Pij dt Mi j∈Ni (t)
− ∆PMi + ∆PLi − ∆Preni , d∆θi = 2π∆wi , dt
d∆PMi 1 =− ∆PMi − ∆Pvi , dt TCH i
d∆Pvi 1 1 =− ∆Pvi + ∆wi − ∆Pref i , (2) dt TGi Ri

Fig. 2: An illustration of dynamic changes of the grid topology where the generators in the left circle switch to the island mode and the generators on the right-hand side switch to the grid-connected mode. 2) Wind integration: Partial load of each control authority is fulfilled by wind energy. The power injection from a wind turbine of control authority i is given by the following relations: Pwi (t) =

1 ρπri2 vi (t)3 2

Preni (t) = Cp Pwi (t),

(4)

where vi (t) is wind speed and the quantity Cp is known as the turbine’s power coefficient. The theoretic limit of Cp is 0.59 according to Betz’s law, and it typically ranges from 0.2 to 0.4 (see [3], [17]). In the remainder of this paper, we will use Cp = 0.59. Due to the volatility of wind energy, vi (t) (resp. Preni (t)) ∗ is probably different from the nominal value vi∗ (resp. Pren ). i The discrepancy is modeled as a deterministic disturbance which is denoted by dreni (t) , ∆Preni (t) , Preni (t) − ∗ Pren . Similarly, dLi (t) = ∆PLi (t) , PLi (t) − PL∗i is the i disturbance induced by load deviation. Let di (t) , dLi (t) − dreni (t).

3) Compact model: For each control authority i, we denote its state by xi , [∆wi ∆θi ∆PMi ∆Pvi ]T and control by ui , ∆Pref i . Let χij (t) , −[0 sin(θi (t) − θj (t))−(θi (t)−θj (t)) 0 0] represent the linearization error. With this set of notations, we compactly write the model of (2) and (3) for control authority i as follows: X Aij xj (t) + Bi ui (t) + Ci di (t) x˙ i (t) = Aii xi (t) +

on Ni,max is N − 1. For (6), we have dkxi (t)k2 = xi (t)T x˙ i (t) = xi (t)T A¯i xi (t) + xi (t)T zi (t) 2dt ≤ λmax (A¯i )kxi (t)k2 + kxi (t)kkzi (t)k ≤ λmax (A¯i )kxi (t)k2 X + kxi (t)k( kAij kkxj (t)k + kCi kkdi (t)k j∈Ni (t)

j∈Ni (t)

+

X

Aij χij (t),

(5)

j∈Ni (t)

  Bi ,  

0 0 0 1 TGi

Tij Mi

0 0 0

0 0 1 TCH i 1 − TG i

 , 

    ,  

 0 0 0 0  , 0 0  0 0 



 Ci ,  

X

kAij k)kxi (t)k2

j∈Ni

i

0  0 Aij ,   0 0

2kAij k(kxi (t)k + kxj (t)k))

j∈Ni

≤ (λmax (A¯i ) + 2

where the system matrices are given by:  X Tij Di 1 −M − Mi i  Mi  j∈Ni (t)  2π 0 0 Aii ,   1 0 0 − TCH  i 1 − TG Ri 0 0 

+

X

+ kxi (t)k(

X

3kAij kkxj (t)k + kCi kkdi (t)k).

(7)

j∈Ni

So, we have ¯

P

kxi (t)k ≤ max{2e(λmax (Ai )+2 j∈Ni kAij k)(t−t0 ) kxi (t0 )k, 3(Ni,max + 1) maxj∈Ni kAij k P max kxj k[t0 ,t] , 2|λmax (A¯i )| − 4 j∈Ni kAij k j∈Ni (Ni,max + 1)kCi k P kdi k[t0 ,t] } 2|λmax (A¯i )| − 4 j∈Ni kAij k , max{βi (kxi (t0 )k, t − t0 ), γij kxj k[t0 ,t] , γid kdi k[t0 ,t] }. j∈Ni

− M1i



0 0 0

 . 

To ensure safe operation of synchronous generators, the deviations from the nominal values cannot exceed certain thresholds. So, the state xi (t) in (5) must be restricted in the constraint set Xii all the time. Remark 2.1: The readers are referred to [16] for the detailed derivation of (2) and (3). As pointed out in [12], fast and low-magnitude wind disturbances can be considered as small disturbances for today’s power grid. The paper [5] numerically verifies the fidelity of linearized dynamics subject to wind disturbances. • B. Distributed frequency control Consider dynamic system (5). One can verify that (Aii , Bi ) is controllable; i.e., one can choose Ki to arbitrarily place the poles of A¯i , Aii + Bi Ki . We now proceed to identify Ki such that system (5) under ui = Ki xi satisfies the property of input-to-state stability. Let us consider x˙ i (t) = A¯i xi (t) + zi (t), (6) P where zi (t) , j∈Ni (t) Aij xj (t) + Ci di (t) + P A χ (t). ij ij j∈Ni (t) Let Ni , ∪t≥t0 Ni (t) and Ni,max , |Ni | where the quantity Ni,max is known a priori. A trivial upper bound

Then the gain matrix Ki is chosen such that X − λmax (A¯i ) > ai , 2 kAij k j∈Ni

1 + (Ni,max + 1) max{max 3kAij k, kCi k}. j∈Ni 2

(8)

As a result, control authority i is ISS with all the gain functions being linear and contractive. Let ∆ij ∈ R>0 such that {xi ∈ Rni | kxi k ≤ γij (∆ij )} ⊆ Xii , δi ∈ R>0 such that {xi ∈ Rni | kxi k ≤ ˆ i and the scalar γid (δi )} ⊆ Xii . We further define the set X ˆ δi as follows: ˆ i , {xi ∈ Xii | βi (kxi k, 0) < min ∆ji }, X j∈Ni

δˆi < δi is such that ∀s ∈ [0 δˆi ), it holds that max γij ◦ γjd (s) < min ∆ji ,

j∈Ni

j∈Ni

∀t ≥ t0 .

The following theorem summarizes the stability of the closed-loop system. ˆ i , kdi k∞ ≤ δˆi and (8) is Theorem 2.1: If xi (t0 ) ∈ X satisfied, then the closed-loop system is ISS with respect to ˆ i for all i ∈ V and t ≥ t0 . d. In addition, xi (t) ∈ X Proof: This is a direct result of [23]. Theorem 2.1 characterizes the stability and constraint enforcement of power grid. To further quantify the stability margin and thus convergence rate, let us compactly rewrite the second inequality in (7) as follows: dkx(t)k2 ≤ x(t)T Π(t)x(t) + Π0 (t)kx(t)kkd(t)k, dt

where the matrix Π(t) is given by: Πii (t) = λmax (A¯i ),

Πij (t) = 1,

j ∈ Ni (t).

By the Gershgorin circle theorem, the spectrum of Π(t) is contained in ∪i∈V B(λmax (A¯i ), Ni,max ). Thus, the stability margin of power grid is determined by ∪i∈V B(λmax (A¯i ), Ni,max ). III. S TABILITY CONSTRAINED INCENTIVE MECHANISMS In this section, we will introduce the problem of stability constrained incentive design. To solve the problem, we will propose a reward based mechanism where the award amount received by each control authority is proportional to one’s contribution to the system stability. Finally, we will formally analyze the proposed incentive scheme. A. Stability constrained incentive design From the perspective of the system operator, there are two objectives: (1) ensuring the frequency stability and constraint enforcement; (2) maximizing the stability margin of power grid. The first goal can be achieved by the distributed controller u∗i (t) = Ki xi (t) such that the spectrum of A¯i satisfies (8). However, the control authorities are selfinterested and may seek for different and even partially conflicting subobjectives. So, u∗i (t) may not capture the best interest of control authority i. Assume that distributed ˜ i xi (t) represents the best interest of controller u ˜i (t) = K control authority i. However, the sufficient condition (8) may not be satisfied under u ˜i (t). Consequently, the closed-loop system may not be stable and/or the state constraints may be violated. In addition, let us assume that the sufficient condition (8) holds under u ˜i (t). Then the system operator aims to further ˜ i such that incentivize control authority i to choose certain K the stability margin is maximized. With these observations, one can view (8) as a hard constraint and the stability margin of power grid as a soft constraint or performance index. The system operator aims to incentivize self-interested control authorities to achieve two objectives. This introduces the problem of stability constrained incentive design. B. Stability constrained incentive mechanism In what follows, we will propose a reward based scheme built on fixed-prize raffles in [19] to address the problem of stability constrained incentive design. The reward based scheme introduces a non-cooperative game among the control authorities and parameterized by the reward. After that, we will identify a lower bound of the reward such that the induced Nash equilibrium enforces the stability condition (8). 1) Low level decision making - Nash equilibrium: The system operator adopts si = λmax (A˜i )−λmax (A¯i ) with si ≥ 0 as the contribution made by control authority i to increase the stability margin. In order to stimulate contribution, the system operator provides a reward with some fixed amount R > 0. The value R together with some constant δ ∈ (0, R) are publicized to all the control authorities. The players

are control authorities whose decision variables are si = λmax (A¯i ) − λmax (A˜i ) ≥ 0. Then the system operator allocates the rewards to the control authorities according to their contributions. In particular, the system operator allocates the portion ss¯i R to control authority i where s¯ , 1T s. That is, each control authority can get a larger portion of the reward if he makes a bigger contribution to the system stability. The reward allocation scheme introduces competitiveness among the control authorities. In order to incentivize the control authorities to increase their contributions, the system operator pays for the additional reward s¯ − R if s¯ ≥ R. If the total contribution exceeds the reward value; i.e., s¯ ≥ R, then s − R) from the control authority receives payoff R ss¯i + hi (¯ system operator. If the total contribution s¯ does not reach R − δ; i.e., s¯ < R − δ, then control authorities receive zero award from the system operator. As a result, the utility of control authority i is given by the following: Ui (s) = R

si + hi (¯ s − R) − si , s¯

if s¯ ≥ R − δ; otherwise, Ui (s) = 0. In the utility Ui , the function hi satisfies the following assumption: Assumption 3.1: The function hi : R → R with hi (0) = 0 is twice differentiable, and concave. There P non-decreasing i (s) is ξ > 0 such that i∈V dhdν < 1 for all s ≥ ξ. Each control authority aims to maximize its own utility. This induces a non-cooperative game among control authorities. We will use Nash equilibrium as the solution notion of the game. Definition 3.1: The joint decision s∗ is a Nash equilibrium if Ui (si , s∗−i ) ≤ Ui (s∗ ) for any si ≥ 0. We denote Nash equilibrium as s∗ (R) , {s∗i (R)}i∈V where the dependency of Nash equilibrium on R is highlighted. We will investigate the existence and uniqueness of Nash equilibrium. After computing a Nash equilibrium s∗ (R), each control ˜ i such that authority i commits to s∗i (R) by choosing K ∗ ¯ ˜ λmax (Ai ) − λmax (Aii + Bi Ki ) = si (R). The reward based incentive mechanism is summarized as follows and delineated in Figure 3: 1) The system operator chooses R and δ; 2) The control authorities collectively determine a Nash equilibrium s∗ (R); ˜ i such that 3) Each control authority i chooses K ˜ i ) = s∗ (R); λmax (A¯i ) − λmax (Aii + Bi K i 4) Each control authority implements distributed con˜ i xi (t). troller ui (t) = K 2) High level decision making - Social optimum: In Theorem 3.1, we will show that for any R ≥ Rmin , ∗ ˆ ¯ ˆ R(max i∈V (ai + λmax (Ai ))) with RL in (12), s (R) induces distributed controllers which enforces (8). Combining this with maximizing the stability margin, the goals of the system operator are formalized as follows: max Uop (R) , 1T s∗ (R) − R.

R≥Rmin

(9)

Let Θmax (R) ⊆ V be the set of i ∈ V such that s∗i (R) ≥ for all j 6= i. Let Θmin (R) ⊆ V be the set of i ∈ V such that s∗i (R) ≤ s∗j (R) for all j 6= i. Let Bi (R) > 0 such that R+hi (Bi (R)−R)−Bi (R) < 0.

s∗j (R)

D. Analysis

Fig. 3: The framework of stability constrained mechanism design.

The optimal solution R∗ of (9) is referred to as social optimum. C. Discussion There are a couple of reasons for us to study the proposed reward based scheme. First, a similar scheme was experimented in Indian to reduce traffic congestion and see [18]. Secondly, our scheme can be viewed as a non-cooperative game parameterized by the reward R. The parametric structure allows us to derive a simple lower bound for R in Theorem 3.1 such that the induced Nash equilibrium can ensure the stability condition (8). This provides a guideline for the system operator to choose the reward value. Our reward based incentive scheme is built on fixedprize raffles in [19]. However, the decision variables in our scheme are not upper bounded. This allows us to perform the sensitivity analysis of Nash equilibrium with respect to the reward. This set of analysis is novel and necessary to ensure (8). We would like to introduce a set of notations for next section. Let G∗ be the optimal solution to the following optimization problem: max G≥0

N X

hi (G) − G.

(10)

i=1

PN By Assumption 3.1, there is G0 > 0 such that i=1 hi (G)− P N G < 0 for all G ≥ G0 . Recall that i=1 hi (0) = 0. This ∗ implies that G exists. The quantity RL ≥ δ is sufficiently large such that −1 +

RL dhi (G∗ ) + > 0, RL + G∗ dν

∀i ∈ V.

(11)

Given ∆ > 0, we let ˆ L (∆) , max{RL , (1 R δ+

∆ dhi (G∗ ) dν

+

∆2 ∗

(G ) 2 ( dhidν )

+

Ui (s∗ (R)) ≤ R + hi (Bi (R) − R) − Bi (R) < 0. Consider s¯ where s¯−i = s∗−i (R) and s¯i = 0. Since Ui (¯ s) ≥ 0, then we reach a contradiction and thus s∗i (R) < Bi (R). Hence, s∗ (R) is identical to the following game: maxsi Ui (s) s.t. si ∈ [0, Bi (R)]. In this game, the utility functions are concave and the decision variables lie in compact sets. Hence, s∗ (R) exists. The uniqueness of Nash equilibrium can be proven by following similar arguments of Lemma 3 in [19]. 2) Analysis of Nash equilibrium: Theorem 3.1 summarizes a set of properties of Nash equilibrium for the low level. In particular, (P1) means that the total contribution is within a constant distance to the reward. This indicates that the competitiveness created by the incentive mechanism promotes the total contribution. (P2) further examines the positive externality of s∗i with respect to R. Notice that Li (R) is strictly increasing in R and Figure (4) is an example of Li (R). (P3) then indicates that s∗i (R) could be beyond any ˆ L (∆). The given ∆ > 0 by choosing a reward larger than R property has been used to determine the minimum reward Rmin . (P4) shows that some s∗i (R) increases at any R. The relation (13) in (P5) indicates that each control authority receives a non-trivial portion of the reward at any Nash equilibrium, demonstrating partial fairness of the incentive scheme. Theorem 3.1: The following properties hold for Nash equilibrium s∗ (R): T ∗ ∗ • (P1) Given any R > 0, R + G ≥ 1 s (R) ≥ R − δ; ∗ • (P2) Given any R ≥ RL , si (R) > Li (R) , (R−δ)2 (G∗ ) R (−1 + dhidν + R+G ∗ ) > 0; R ∗ ˆ L (∆) • (P3) Given any ∆ > 0, si (R) ≥ ∆ for any R ≥ R and i ∈ V ; • (P4) Given any R ≥ RL , there is some i ∈ V such that ds∗ i (R) > 0; dR • (P5) It holds that s∗i (R) R T ∗ 1 s (R)

Li (R) > 0, R + G∗ s∗ (R) dhi (G∗ ) lim inf ∗i ≥ . R→+∞ s ¯ (R) dν

∗

(G ) − 12 dhidν )G∗ , ∗ 1 dhi (G ) 2 dν

s

1) Existence and uniqueness of Nash equilibrium: The following lemma shows the existence and uniqueness of Nash equilibrium. Lemma 3.1: Given any R > 0, there is a unique Nash equilibrium. Proof: Assume s∗i (R) ≥ Bi (R) for some i. Then we have

2δ∆ dhi (G∗ ) dν

}.

(12)

≥

(13) (14)

Proof: In the proof, we will drop the dependency of s∗ on R. We will use the notation s¯∗ , 1T s∗ .

ˆL ≥ δ + Since R ≥ R

∆

dhi (G∗ ) dν

+

r

∆2

(

dhi (G∗ ) 2 ) dν

+

2δ∆

dhi (G∗ ) dν

, the

right-hand side of (19) is greater than or equal to ∆ and so is s∗i (R) ≥ ∆. This completes the proof of (P3). By (P2) and (15), the first-order partial derivative of Ui with respect to si vanishes at s∗ (R). That is, dhi (¯ s∗ − R) s¯∗ − s∗ dUi (s∗ ) = −1 + R ∗ 2 i + = 0. dsi (¯ s ) dν We now proceed Notice that d2 Ui (s∗ ) ds2i d2 Ui (s∗ ) dsi dsj d2 Ui (s∗ ) dsi dR

Fig. 4: An example of Li (R).

The property (P1) is a direct result of Lemma 4 and the arguments after Corollary 1 in [19]. Given any R ≥ 0, the Nash equilibrium s∗ (R) must satisfy the first-order condition:

dUi (s ) = −1 + dsi Since that

dhi dν

s¯ − s∗ R ∗ 2i (¯ s )

.. .

 = 

(16)

d2 Ui (s) dsi dR ∗

dUi (s∗ ) dhi (G∗ ) s¯∗ − s∗ ≥ −1 + R ∗ 2 i + dsi (¯ s ) dν s∗i dhi (G∗ ) 1 = −1 + R( ∗ − ∗ 2 ) + s¯ (¯ s ) dν ∗ 1 si dhi (G∗ ) ≥ −1 + R( − )+ , ∗ 2 R+G (R − δ) dν (17) where in the last inequality we use (P1). If s∗i ≤ Li (R), it dUi (s∗ ) follows from (17) that dsi > 0, contradicting the firstorder condition (15). Hence, s∗i (R) > Li (R). This completes the proof of (P2). ˆ L (∆), we We now proceed ∗to show (P3). Since R ≥ R

−1 +

and then

dhi (G∗ ) R 1 dhi (G∗ ) + ≥ . dν R + G∗ 2 dν

(18)

(R − δ)2 1 dhi (G∗ ) . R 2 dν

(19)

< 0 and

> 0. Pick any i ∈ Θmin (R). Then 2s∗i0 (R) − d2 Ui0 (s∗ ) dsi0 dsj

< 0. Hence, we have

X d2 Ui0 (s∗ ) ds∗j (R) dUi0 (s∗ ) = > 0. dsi0 dsj dR dsi0 dR

(23)

j∈V

The relation∗ (23) implies that there is at least one i ∈ V dsi (R) such that dR > 0. This completes the proof of (P4). By using (P1) and (P2), we have (13). Take the limit on R at both sides of (13), we reach (14). 3) Analysis of social optimum: The following theorem examines the properties of Uop . (P6) verifies the uniform boundedness of Uop . Figure 5 shows an example of such bounds on Uop . Theorem 3.2: The following properties hold for Uop : •

(P6)

max{−δ,

min{G∗ ,

dhi (G∗ ) −1 dν dh (G∗ ) N − idν

dhi (0) −1 dν dhi (0) N − dν

≤

R}

Uop (R)

R}.

Proof: Sum (16) over i and we have R(N − 1) X dhi (¯ s∗ − R) −N + + = 0. s¯∗ dν i∈V

This implies the following relation:

Combining (P2) and (18) renders s∗i (R) ≥

d2 Ui (s) d2 si

0

s¯ (R) < 0 and thus

dhi (G ) )G∗ dν ∗ 1 dhi (G ) 2 dν

(21)

(22)

For any s ≥ 0 and R ≥ 0, we have

∗

dhi (¯ s − R) + . dν

  > 0. 

d2 UN (s∗ (R)) dsN dR

−

have R ≥

s¯∗ − s∗i d2 hi (¯ s − R) + , (¯ s∗ )3 d2 ν d2 hi (¯ s∗ − R) 2s∗ − s¯∗ = −2R i ∗ 3 + , (¯ s ) d2 ν d2 hi (¯ s∗ − R) s¯∗ − s∗i − = . ∗ 2 (¯ s ) d2 ν = −2R

ds1 dR

(15)

is non-increasing, it follows from (P1) and (16)

(1− 12

from (20).

It follows from the implicit function theorem and (21) that    d2 U (s∗ (R)) 2 ∗ ds∗ i (s (R)) 1 1 (R) ··· d U d2 s1 ds1 dsN dR    .. .. .. ..   − . . . .    ∗ dsN (R) d2 UN (s∗ (R)) d2 UN (s∗ (R)) ··· dR ds ds d2 sN  d2 U 1(s∗N(R)) 

Let us proceed to show (P2). The first-order partial derivative of Ui with respect to si at s∗ = s∗ (R) is given by: ∗

ds∗ i dR

1

dUi (s∗ ) ≤ 0. dsi

∗

to derive an expression for

(20)

s¯∗ =

N−

R(N − 1) . P dhi (¯ s∗ −R) i∈V

dν

≤

Fig. 5: An example of Uop (R). Since

dhi (¯ s∗ −R) dν

is non-increasing, then we have

∗

dhi (G ) −1 dν dhi (G∗ ) N − dν

R ≤ Uop (R) ≤

dhi (0) −1 dν dhi (0) N − dν

R.

(24)

The combination of (24) and (P1) reaches the desired result (P6). IV. C ONCLUSIONS In this paper, we have formulated the problem of stability constrained incentive design. We have proposed a reward based incentive scheme and determined the lower bound of the reward such that the induced Nash equilibrium can ensure the system stability and constraint enforcement. R EFERENCES [1] M. Althoff, M. Cvetkovic, and M. Illic. Transient stability analysis by reachable set computation. In IEEE PES International Conference and Exhibition on Innovative Smart Grid Technologies, pages 1 – 8, 2012. [2] T. Basar and G. Olsder. Dynamic noncooperative game theory. SIAM Classics in Applied Mathematics, 1999. [3] A. Bertz. Introduction to the theory of flow machines. Oxford: Permagon press, 1966. [4] L. Chen, N. Li, L. Jiang, and S. Low. Optimal Demand Response: Problem Formulation and Deterministic Case, pages 63–85. Control and Optimization Methods for Electric Smart Grids. Springer, 2011. [5] Y.C. Chen and A.D. Dominguez-Garcia. A method to study the effect of renewable resource variability on power system dynamics. IEEE Transactions on Power Systems, 27(4):1978 – 1989, 2012. [6] C.W. Gellings. The Smart Grid: Enabling Energy Efficiency and Demand Response. U.K.: CRC Press, 2009. [7] C.W. Gellings and J.H. Chamberlin. Demand side management: Concepts and methods. The Fairmont Press, 1988. [8] D.J. Glover, M.S. Sarma, and T.J. Overbye. Power system analysis and design. Cengage Learning, 2012. [9] I.A. Hiskens and J. Alseddiqui. Sensitivity, approximation and uncertainty in power system dynamic simulation. IEEE Transactions on Power Systems, 21(4):1808 – 1820, 2006. [10] I.A. Hiskens and M.A. Pai. Trajectory sensitivity analysis of hybrid systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 47(2):204 – 220, 2000. [11] M. Ilic and Q. Liu. Toward Sensing, Communications and Control Architectures for Frequency Regulation in Systems with Highly Variable Resources, pages 3–33. Control and Optimization Methods for Electric Smart Grids. Springer, 2011.

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