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Impacts of unreliable communication and modified regret matching based anti-jamming approach in smart microgrid
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Contents lists available at ScienceDirect
Ad Hoc Networks journal homepage: www.elsevier.com/locate/adhoc 5 6
Impacts of unreliable communication and modified regret matching based anti-jamming approach in smart microgrid
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Bo Chai, Zaiyue Yang ⇑ State Key Lab. of Industrial Control Technology, Zhejiang University, Hangzhou 310027, China
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a r t i c l e
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i n f o
Article history: Received 22 November 2013 Received in revised form 25 March 2014 Accepted 13 May 2014 Available online xxxx
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Keywords: Smart grid Demand response Jamming attack Packet loss ratio Game theory
a b s t r a c t Demand response management (DRM) is one of the main features of smart grid, which is realized via bidirectional communications between the power provider and the consumers. Due to the vulnerabilities of communication channels, especially the wireless networks, communication is not perfect in practice and will be threatened by kinds of attacks, among which jamming attack is deemed as the primary one. In this paper, we consider jamming attack in the wireless communication of a smart microgrid. Firstly, the DRM performance degradation induced by jamming attack is fully studied and explicitly presented, by considering both packet loss ratio and load estimation error. Then, a modified regret matching based anti-jamming algorithm is proposed to enhance the quality of communication, and the performance of DRM. Finally, numerical results are presented to illustrate the impacts of unreliable communication on DRM, as well as the performance of the proposed anti-jamming algorithm. Ó 2014 Elsevier B.V. All rights reserved.
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1. Introduction Benefitting from information and communication technology, the smart grid is able to gather and behave on information in an automated fashion to enhance the efficiency, reliability and stability of generation, transmission, distribution and consumption of electricity power [1–4]. Demand response management (DRM) is an essential feature of current and future power grid, which is implemented by the system operator and consumers to control the power consumption at the consumers’ premises [5–7]. There exists a large body of literatures on the study of DRM, among which real-time pricing (RTP) is deemed as one of the most common tools to encourage efficient and economic power utilization [5,8–11]. Based on RTP, [5] proposes a DRM scheme aiming at maximizing the both the welfare of power provider and the welfare of
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⇑ Corresponding author. Tel.: +86 57187952994. E-mail addresses: [email protected] (B. Chai), [email protected] (Z. Yang).
consumer. As this scheme is simple and can be readily implemented in a distributed fashion, it is widely accepted and this paper is one of the most cited papers in the DRM field. However, it is noticed that [5] and most other existing DRM studies assume that the bidirectional communication in smart grid is perfect, which cannot be guaranteed in practical scenarios. Communications are solid foundation of realization of DRM in smart grid. Due to low installation cost and high flexibility, a variety of wireless communication technologies (e.g, Wi-Fi, ZigBee, WiMax, cellular networks) have been selected as the architecture of the smart grid communication network [12–15]. However, wireless communication introduces potential security vulnerabilities due to the sharing nature of wireless channels [16–18]. Among the potential security threats, jamming attack is deemed as a primary security threat to the perfect communication [16,19]. The adversaries can implement jamming attack to interrupt the system, and thus undoubtedly degrade the system performance [20].
http://dx.doi.org/10.1016/j.adhoc.2014.05.011 1570-8705/Ó 2014 Elsevier B.V. All rights reserved.
Q1 Please cite this article in press as: B. Chai, Z. Yang, Impacts of unreliable communication and modified regret matching based anti-jamming approach in smart microgrid, Ad Hoc Netw. (2014), http://dx.doi.org/10.1016/j.adhoc.2014.05.011
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The pioneering works considering jamming attack in smart grid are presented in [19,21,22]. [19] aimed to minimize message delay for timely communication in smart grid. [21] studied the jamming and anti-jamming in multichannel wireless communication systems from a remote sensor to the control center in smart grid. [22] stated that jamming attack will result in unreliable communication and performance degradation of smart grid. The impact of packet loss on the optimization of cost for power supply in smart grid was modeled and analyzed in [23]. A controltheoretic approach of deriving the fundamental conditions of RTP stability under the scaling and delay attacks was proposed in [24]. The authors in [25] considered the reliability of the smart grid data communications infrastructure and its impacts on the power consumption and supply optimizations. [26] illustrated the impact of malicious cyber-attacks on the energy efficiency of the smart grid. Nevertheless, the impacts of unreliable communication on DRM performance have not been explicitly and exhaustively analyzed, such as the welfare degradation of power provider and consumers. Meanwhile, besides the analysis of impacts, designing a proper anti-jamming strategy is also an important issue to address. To this end, this paper jointly considers impacts of unreliable communication on DRM performance and anti-jamming approach. Note that in this paper we shall focus on the smart microgrid, because it is smaller in scale and thus more sensitive to imperfect communication. In particular, unlike previous game-theoretic anti-jamming approaches in [20,21,27], the modified regret matching approach is proposed for the anti-jamming game, in which the players behave highly autonomously in a sense that each player only requires its own environmental information. Because the proposed anti-jamming approach is of low-level awareness of other players [28,29], it is effective and suitable to solve the anti-jamming problem in smart microgrid with limited information exchange. To the best of our knowledge, this is also the first study to discuss the anti-jamming approach using the modified regret matching approach. The contribution of this paper can be summarized as follows: The impacts of communication unreliability on DRM are fully analyzed. A modified regret matching based anti-jamming algorithm is proposed for the smart microgrid to achieve better performance under the jamming attack. Numerical results are illustrated to evaluate the performance degradation and the performance of the proposed algorithm.
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This paper is organized as follows. Section 2 describes the system models associated with DRM and jamming attack. In Section 3, performance degradation due to the communication unreliability is fully analyzed. In Section 4, a modified regret matching based anti-jamming algorithm is presented to improve communication quality under jamming attack. Performance evaluations are illustrated in Section 5. Finally, we draw conclusions in Section 6.
2. System model
137
We consider a smart microgrid with multiple consumers, one power provider and one Distribution Company Controller (DCC) involved in the DRM. The microgrid is supported by the bidirectional communication among the consumers, the power provider and the DCC. The role of DCC is to control the consumers’ power consumption and the power provider’s generation, and to coordinate each consumer with others and also with the power provider [30]. It is assumed that all available wireless communication channels are not overlapping and do not interfere with each other. Each consumer selects one channel to deliver the ‘‘demand’’ packet to DCC. Meanwhile, the attacker is able to select one channel between this consumer and DCC to jam the communication and hence results in packet loss, which will degrade the performance of the DRM as to be detailed later in Section 3. Let N , f1; 2; . . . Ng denote the consumer set, C , f1; 2; . . . Cg denote the available channel set, respectively.
138
2.1. Demand response management
156
In order to evaluate the impacts of unreliable communication on DRM, a DRM scheme has to be specified in advance. In this paper, we adopt the same analytical DRM framework in [5], which is one of the most cited paper about DRM issues and this framework is widely accepted. In this framework, DCC controls the energy consumption, and coordinates each user with others and the power provider. In each operation time slot, let s denote the supply of the power provider, and di denote the demand of the consumer i. The cost function CðsÞ indicates the cost of provider introduced by providing the supply s, and it is assumed to be increasing and convex. The utility function ui ðdi Þ quantifies the utility that consumer i receives when consuming di power, and it is assumed to be non-decreasing and concave [5,31]. For the power provider side, for any given power price p the welfare of the power provider is determined as follows, wðs; pÞ ¼ p s CðsÞ. Thus, the goal of the power provider is to determine the supply s in order to maximize its welfare, i.e.,
157
max wðs; pÞ ¼ max ps CðsÞ s
ð1Þ
s
On the other hand, for the consumer side, for any given power price p the welfare of the consumer is determined as follows, g i ðdi ; pÞ ¼ ui ðdi Þ pdi . Thus, the goal of the consumer is to determine the demand di to maximize its welfare, i.e.,
max g i ðdi ; pÞ ¼ max ui ðdi Þ pdi s
ð2Þ
di
From the perspective of social welfare [5], it is desirable to maximize the sum of all consumers’ welfare and the power provider’s welfare. In other words, the social welfare function can be obtained as
Uðs; fdi gÞ ¼
N X
N X
i¼1
i¼1
g i ðdi ; pÞ þ wðs; pÞ ¼
ui ðdi Þ CðsÞ
ð3Þ
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Meanwhile, the supply and total demand should satisfy the P constraint Ni¼1 di 6 s. In summary, the global optimization problem is formulated as, N X max Uðs; fdi gÞ ¼ ui ðdi Þ CðsÞ s;fdi g
s:t: 200 201 202 203 204 205 206 207
208
i¼1
ð4Þ
N X
kkþ1
Problem (4) is convex and therefore can be solved readily in a centralized manner. However, solving problem (4) in a distributed manner is more desirable to the power provider and the customers, because of the advantages of robustness and scalability. To this end, the dual decomposition approach can be applied [5]. First, let us define Lagrangian for the original problem (4) as:
!
N X ui ðdi Þ CðsÞ k di s
i¼1
¼
210
212
213
215 216
217 219
N X
i¼1 N X ui ðdi Þ CðsÞ þ ks k di
i¼1
ð5Þ
where k is the Lagrangian multiplier. Then, the dual objective function can be written as N X DðkÞ ¼ max Lðs; fdi g; kÞ ¼ W i ðkÞ þ UðkÞ; s;fdi g
þ dDðkÞ ¼ kk c dk " !#þ N X ¼ kk c di ðkk Þ s ðkk Þ
227 228 229 230 231 232
233
ð10Þ 235
i¼1 di ðkk Þ
where k is the iteration index and c is the step size. is the local optimizer of (7), s ðkt Þ is the local optimizer of (8), and ½zþ , maxfz; 0g. As depicted in Fig. 1, [5] proposed an optimization algorithm in a distributed and iterative way. The iteration will P stop until sk Ni¼1 di;k ¼ 0, i.e., the supply and the demand keep balance. The algorithm can be fully illustrated as follows.
236
1. The DCC announces the price to the consumers and the power provider. The initial power price is randomly selected that p1 > 0. 2. After receiving the power price pk at the kth iteration, the provider adjusts the supply according to Eq. (1), and the consumers adjust their demand according to Eq. (2). Then both the provider and the consumer send the information to the DCC as feedback. 3. After receiving the local optimal supply s and the local optimal demand di , the DCC updates the power price for next iteration by using a gradient projection h iþ P method, i.e., pkþ1 ¼ pk e sk Ni¼1 di;k , where
244
237 238 239 240 241 242 243
i¼1
ð6Þ
i¼1
where
W i ðkÞ ¼ max ui ðdi Þ kdi
ð7Þ
di
220
and
223
UðkÞ ¼ max ks CðsÞ
224
Consequently, the dual problem can be obtained as
221
Notice the first term defined in (6) can be decomposed into N separated sub-problems defined in (7) for the customers and another sub-problem defined in (8) for the energy provider. Specifically, k can be solved by iteratively applying the gradient projection method as follows,
i¼1
N X
ð9Þ
k>0
di 6 s
Lðs; fdi g; kÞ ¼
211
min DðkÞ
ð8Þ
s
e > 0 denotes the step size and ½zþ , maxfz; 0g.
Fig. 1. Interaction in the proposed DRM.
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4. Repeat steps 2 and 3 until the power price remains unchanged.
Remark 1. Notice that above algorithm is able to achieve the global optimal value of social welfare, under the assumption that the bi-directional communication between provider and DCC, as well as DCC and consumers is perfect. For the traditional grid, the power provider determines the price without consideration of the response of consumers. Smart grid benefits from the interaction between the DCC and the consumers. We can consider an extreme case that all the communication channels between the DCC and the consumers fail due to the serious jamming. Thus the interaction cannot be implemented successfully. The smart grid will operate as a traditional grid since the smart grid cannot receive the response of consumers. It is noteworthy that the DRM framework in [5] is proposed for smart grid. Thus the results are still valid for smart grid. With integration of the renewable energy, the DRM for smart grid has been extended [7]. Moreover, there exists a large literature of DRM schemes, which extend the work in [5]. However, the basic principle still keeps: pricing has effects on the generation and the demand. In turn, the generation and the demand also have effects on the pricing. Intuitively, when a more general DRM scheme is considered, jamming attack will impact the smart grid in such a way: the jamming attack results in poor performance of the communication system. Then the interaction between power providers and consumers has been effected. As a result, the performance of DRM degrades. We will study the jamming attack in smart grid with consideration of renewable energy in the future work. 2.2. Communication channel and jamming model A general architecture of smart grid communication consists of the following components: Home Area Network (HAN), Neighborhood Area Network (NAN), and Wide Area Network (WAN) [13]. Wireless communication technologies (e.g., ZigBee, Wifi) are deemed as potential technologies employed to realized the HAN and NAN [12,13]. For protecting the communication between DCC and the power provider, wired communication technologies (e.g., fiber optics, power line communication (PLC)) are utilized. For low cost and high scalability, wireless communication technologies are utilized for the communication between DCC and the consumers.
However, wireless communication is vulnerable to Denial-of-Service (DoS) attacks [16]. Typically, the attacker can conduct DoS attack to jam the communication between each components and degrade the system performance [27,32]. In our work, we assume the attacker can jam the communication from consumers to DCC. Another case is that the attacker jams the communication from DCC to consumers. However, if one consumer receives the real-time price successfully, it can broadcast to the neighbors, so that all consumer can receive the correct information. Thus, from the perspective of attacker, it is more reasonable to jam the communication from consumers to DCC. After receiving the packet from the consumer, DCC will send the acknowledgement (ACK) back. A rational jammer will listen during the ACK transmission interval rather than randomly jamming the ACK packets. Therefore, we also assume that the jammer knows whether it succeeds in jamming the transmitting channels for all the past time slots [27]. We consider a general jamming model as follows. Both the consumers and the attacker choose one channel only from the channel set at the same time. If the selected channel of the consumer and the selected channel of the attacker are the same, the jamming attack will be successful and vice versa. As a result, the ‘‘demand’’ packet will not be delivered to the DCC [27]. Specifically, as depicted in Fig. 2, the real demand of consumer i at iteration k is di;k . However, due to the existence of jamming attack, this information may not be successfully delivered to DCC. If the jamming attack is successfully conducted, DCC cannot receive di;k and thus ^ has to estimate its value. Let di;k denote the demand information of consumer i at iteration k finally presented at DCC, which can be written into
( ^ d i;k
¼
304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337
338 di;k di;k
þ ei;k
successful jamming; otherwise:
ð11Þ 340
where ei;k denotes the estimation error.
341
3. Theoretical analysis on performance degradation
342
The proposed DRM scheme [5] is realized based on the assumption of perfect communication between the consumers and the DCC [30]. If it is the case, as presented previously the power price will converge to the global optimal point p . In addition, the supply and demand will be balanced,
343
Fig. 2. Jamming attack model in the wireless communication between consumers and the DCC.
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365
s ¼
N X di
The corresponding global optimal social welfare is denoted as U . However, if the communication is imperfect under jamming attack, packet loss is possible and DCC has to use the ^ when it does not receive the real estimated demand d i;k demand di;k of consumer i. Denote the packet loss ratio as d. Thus there are approximately dN demand packets to be estimated at the DCC side. Under this circumstance, the proposed DRM scheme [5] cannot reach its global price ~ p ; instead, it will obtain a suboptimal price denoted as p via iterative updating similar to (10). However, the termination criterion is no longer (12), but becomes the following equation due to imperfect communication, N N N X X X ^ ¼d ~ þ e þ ð1 dÞ d ~ d d i i i i i¼1
368 369 370 371 372
373
ð12Þ
i¼1
~s ¼
367
i¼1
N N N X X X ~ ¼ dE þ ~ d d ¼ d ei þ i i i¼1
i¼1
ð13Þ
i¼1
~Þ u0i ðd i
~ Þ u0N ðd N
~¼ p
376
Similarly, ~s is computed at the power provider, in order to ~, as maximize its welfare (1) for the given price p
377
381 382 383 384 385 386 387 388 389 390 391
¼ ¼
¼ ¼
~ ¼ C 0 ð~sÞ p
ð14Þ
ð15Þ
Consequently, the corresponding suboptimal social e. welfare can be obtained as U From (13), it is clear that if dE ¼ 0, the suboptimal solution is identical to the optimal solution. On the other hand, both the packet loss ratio d and the accumulated estimation error E will influence the performance of DRM. In the following parts, we shall separately analyze the performance impacts caused by d and E, in terms of the degradation of social welfare.
396
ð16Þ
397
411
~ ~ 1 @p @p ¼Eþ @d C 00 ð~sÞ @d
N X i¼1
1 00 ~ u d
412
413
ð21Þ
i
i
415
That is, 1 C 00 ð~sÞ
416
417
E PN
ð22Þ
1 ~Þ i¼1 u00 ðd i i
419 420
421
399
~ @C ð~sÞ @p @~s ¼ ¼ C 00 ð~sÞ @d @d @d
400
and
403
~ @u0i ðd~i Þ @p @ d~i ¼ ¼ u00i ðd~i Þ @d @d @d
404
Since C 00 ð~sÞ and u00i ðd~i Þ are nonzero, we have
~ @~s 1 @p E ¼ ¼ P @d C 00 ð~sÞ @d 1 C 00 ð~sÞ Ni¼1 00 1~ ui ðdi Þ
ð23Þ 423
and
424
425
~i ~ @d 1 @p E ¼ ¼ P ~ @d u00i ðd~i Þ @d u00 d N 00 ~ i di 00 ~ ui i i¼1 C ðsÞ
ð24Þ u00i
1 ðd~i Þ
427
As it is possible E > 0 or E < 0, we have to consider both cases. Case 1: E > 0, i.e., the supply can fully satisfy the demand of all the consumers. Note the cost function CðsÞ is increasing and convex; meanwhile, the utility function ui ðdÞ is increasing and concave. Thus, we have 0
00
~ > 0; C ð~sÞ ¼ p
C ð~sÞ > 0
ð25Þ
and
u0i ðd~i Þ
ð17Þ
ð18Þ
428 429 430 431 432 433
434 436 437
438
~ > 0; ¼p
u00i ðd~i Þ
<0
ð26Þ
Taking (25) and (26) into (22)–(24) implies,
~ @p > 0; @d
@~s > 0; @d
~ @d i <0 @d
~s > s ;
440 441
442
ð27Þ
444 445
446
~
ð28Þ
448
That is, when E > 0, the jamming attack results in the increase of power price and supply, and decrease of demand. Now, we can consider the degradation of social welfare caused by jamming attack as follows,
449
N ~ @Cð~sÞ X @d @~s i ¼ ¼ u0i ðd~i Þ C 0 ð~sÞ @d @d @d @d @d i¼1 i¼1 ! N N ~ X X ~ @d @~s 1 1 @p i ~ ~ ~ 00 p p ¼p ¼ 00 ~ ~ @d @d @d C ð s Þ d ð Þ u i i¼1 i¼1 i ! N X ~E p 1 1 ~E < 0 ð29Þ ¼ 00 ¼ p P 00 ~ C ð~sÞ C 001ð~sÞ Ni¼1 00 1~ i¼1 ui ðdi Þ
~ @U
0
405
ð20Þ
Substituting (19) and (20) into (16) gives
~>p ; p
N ~i X @~s @d ¼Eþ @d @d i¼1
401
409
~ ~ @d 1 @p i ¼ ~ @d @d u00i d i
~ @p ¼ @d
407 408
Taking partial derivative of (13)–(15) to d, we can yield
394
and
In other words,
3.1. Impacts of unreliable communication on DRM performance
393
392
ð19Þ
Consequently, we have
where E denotes the accumulated estimation error. Mean~i and ~s denote, separately, the suboptimal demand while, d ~. and supply corresponding to the suboptimal price p ~i is computed at consumer i, in order to maxNote that d ~, as imize its welfare (2) for the given price p
~ Þ u01 ðd 1
~ @~s 1 @p ¼ C 00 ð~sÞ @d @d
i¼1
375
378 380
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349
450 451 452 453
454
N X @ui ðd~i Þ
ui ðdi Þ
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It implies that the social welfare will degrade due to packet loss when E > 0. Case 2: E < 0. From (22)–(24), we have
462
~ @p < 0; @d
463
In other words,
466
~ < p ; p
464
467
468
473 474 475 476 477
478
482 483
484
486 487
488 490
ð31Þ
¼
N X ui ðli Þ i¼1
ð33Þ
N X s:t: li 6 ~s i¼1
By introducing the Lagrangian multiplier g, the optimization problem (33) can be converted into the following dual form [33] N N X X DðgÞ ¼ max ui ðli Þ þ g ~s li fli g
i¼1
ð34Þ
i¼1
and the corresponding dual optimization problem is
min DðgÞ
ð35Þ
g>0
492
495
@D @D ¼ ¼ ¼0 @~l @~l
496
That is to say,
499
@uj @ui ¼ ¼ ¼g @~l @~l
493
497
500 501
502
504 505
506
i
509
510
On the other hand, since ui is an increasing function, we should also have N X ~l ¼ ~s i
ð38Þ
i¼1
which gives
@d
i¼1
¼
@~s @d
ð39Þ
N X e e U ui ~li C ð~sÞ S; f~li g ¼
N @C ð~sÞ X @~li @~s ¼ u0i ðl~i Þ C 0 ð~sÞ @d @d @d i¼1
¼
~ u0i ð~li Þ C 0 ð~sÞ u0i ð~li Þ C 0 ð~sÞ @ p ¼ 00 @d C ð~sÞ C 00 ð~sÞ ~ÞE ðg p PN 00 1 C ð~sÞ i¼1
1 C ð~sÞ 00
E PN
1 ~Þ i¼1 u00 ðd i i
1 ~Þ u00i ðd i
ð41Þ 516
~i Þ ¼ C 0 ð~sÞ ¼ p ~, According to (14) and (15), we have u0i ðd ~ Þ. Since u0 ðd ~ Þ is a ~ ¼ u0i ð~li Þ C 0 ð~sÞ ¼ u0i ð~li Þ u0i ðd thus g p i i i ~i ; k p ~ > 0. Recall concave function and ~li < di < d ~ < 0 and E < 0, we have C 00 ð~sÞ > 0; u00i d i
e @U
<0
517 518 519 520
521
ð42Þ
523
There also exists another interpretation of (42) as fol lows. Since (21) holds, the suboptimal solution e S; f~li g is
524
a feasible solution of the original optimization problem e < U for E < 0, and consequently, (4). Thus we have U e @U < 0. @d
526
Remark 2. Finally, a conclusion can be drawn from both cases.
529
The jamming attack in bidirectional communications reduces the demand of the consumer. The jamming attack may not always increase the supply of the power provider. The social welfare of DRM drops with an increasing of the packet loss ratio.
531
@d
525
527 528
530
ð40Þ
532 533 534 535 536 537
3.2. Impacts of estimation error on DRM performance
538
In this part, we shall consider the impacts of estimation error E on demand, supply and welfare, for a fixed given packet loss ratio d. This investigation is important because E relies on the estimation algorithm being applied. Therefore, different estimation algorithms will lead to different E. Even for the same estimation algorithm, if E is timedependent it will vary across different time slots. In other words, even under the same d, the demand, supply and welfare will also be influenced by the time varying E. From (13), it is clear that the analysis of E is parallel to that of d. That is, similar to (16)–(18) we have
539
N ~ X @~s @d i ¼dþ @E @E i¼1
541 542 543 544 545 546 547 548 549
552 553
554
ð44Þ
and
i¼1
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540
550
ð43Þ
and
~ @C 0 ð~sÞ @p @~s ¼ ¼ C 00 ð~sÞ @E @E @E
Now, the social welfare is given by
512
ð37Þ
j
N X @~li
508
ð36Þ
j
i
i¼1
@d
514
!
Considering the optimal solution f~li g, it should satisfy the following condition,
491
¼
ð32Þ
That is, the supply cannot fully satisfy the demand of all the consumers. Let li denote the actual load allocated at consumer i, and we shall consider the optimal allocation of fli g, denoted by f~li g. For a given ~s, this problem becomes the maximization of the accumulated welfare of all consumers under the P constraint of Ni¼1 li 6 ~s, i.e.,
N @ui ~ li X
513
N X @~li @~s @~s ¼ u0i ðl~i Þ C 0 ð~sÞ ¼ u0i ðl~i Þ C 0 ð~sÞ @d @d @d i¼1
i¼1
fli g
481
~ > d d i
i
max
480
~s < s ;
ð30Þ
N X ~ d
470
472
@~s < 0; @d
e @U ¼ @d
By recalling (12), it is clear that
~s < 471
~ @d i >0 @d
Its partial derivative to d can be expressed as
556 557
558
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560
~i Þ ~ ~ @u0i ðd @p ~ Þ @ di ¼ ¼ u00i ðd i @E @E @E
561
Solving above equations gives
562
564
~ @p ¼ @E
1 C 00 ð~sÞ
ð45Þ
d PN
ð46Þ
1 ~Þ i¼1 u00 ðd i i
565 567
~ @~s 1 @p ¼ 00 ~ @E C ðsÞ @E
568
and
569
571 572
573
575 576
577
ð47Þ
~ ~ @d 1 @p i ¼ ~ Þ @E @E u00 ðd i i
ð48Þ
When E > 0, we have N N ~ Þ @Cð~sÞ X ~ e X @U @ui ðd i ~ Þ @ di C 0 ð~sÞ @~s ¼ ¼ u0i ðd i @E @E @E @E @E i¼1 i¼1 ! N X ~ 1 1 @p ~ ~d < 0: ¼p 00 ¼p 00 ~ ~ @E C ð s Þ d ð Þ u i i¼1
ð49Þ
i
When E < 0, we have
e @U ¼ @E
N X i¼1
N @ui ð~li Þ @Cð~sÞ X @~li @~s ¼ u0i ð~li Þ C 0 ð~sÞ @E @E @E @E i¼1
@~s XN @~li @~s 0 ~ ¼ u0i ð~li Þ i¼1 C 0 ð~sÞ ¼ ui ðli Þ C 0 ð~sÞ @E @E @E ~ u0i ð~li Þ C 0 ð~sÞ u0i ð~li Þ C 0 ð~sÞ @ p d ¼ ¼ P 1 @E C 00 ð~sÞ C 00 ð~sÞ Ni¼1 u00 1ðd~ Þ C 00 ð~sÞ i
¼ 579 580
~Þd ðg p PN 1 C 00 ð~sÞ i¼1
1 ~Þ u00i ðd i
Note that since E < 0;
@e U @E
i
>0
ð50Þ
e < U . > 0 still implies U
7
4. Modified regret matching based anti-jamming policy
581
Above analysis clearly indicates that jamming attack leads to packet loss and hence results in the degradation of social welfare. The degradation is more severe for larger packet loss ratio d, under any specified demand estimation algorithm. Therefore, in order to maintain the optimal social welfare by depressing d, an anti-jamming approach is proposed in this section.
582
4.1. Formulation of the anti-jamming game
589
Here, we consider a scenario that there are multiple communication channels available between the consumers and DCC, and each consumer has to select one channel for information delivery each time; on the other hand, the attacker only has the ability of jamming one channel each time. Thus, for the anti-jamming side, i.e., the consumers and the DCC, this problem becomes the design of optimal channel selection policy under jamming attacks. Due to the hostile and strategic nature of jamming attacks, game theory emerges as a natural tool to determine the optimal policies of the consumers under jamming attack. Let the jamming/anti-jamming behavior in microgrid be modelled as a repeated game. The consumer set N , f1; 2; . . . Ng and the DCC are formed as a sendersreceiver pair, which is deemed as a defender in the game in opposite to the attacker. The game is formulated as follows [34],
590
Player: The defender a and the attacker b. Strategy: The available channel set C , f1; 2; . . . Cg. The strategy for defender a and attacker b at stage h are denoted as sha ; shb , respectively. Note that sh ¼ sha ; shb
608
denotes the strategy of the game at stage h.
Fig. 3. Supply, demand and welfare vs power price.
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Utility function: uha sha ; shb and uhb shb ; sha represent the
uha sha ; shb ¼ 1 dðhÞ uhb shb ; sha ¼ dðhÞ
utility function of the defender and the attacker, respectively.
ð51Þ ð52Þ
624
616 617 618 619 620 621
e , or The defender aims to maximize the social welfare U equivalently, minimize the packet loss ratio d; meanwhile, e , or equivalently, maxithe attacker aims at minimizing U and mizing d. Thus, the utility functions uha sha ; shb uhb shb ; sha can be described as follows,
where dðhÞ represents the packet loss ratio at stage h, and is calculated as 1 minus the receiving packet number to the number of the consumers N. In addition, dðhÞ can be independently calculated by the defender without the requirements of the information of the attacker.
20
24
18
23
16
22
14
Amount
Social welfare
21 12 10
20 19
8 suboptimal social welfare optimal social welfare Welfare of the power provdier Welfare of the consumers
6 4
18
16
2 0
Supply Demand
17
0
0.2
0.4
0.6
15
0.8
0
0.2
Packet loss ratio
0.4
0.6
0.8
Packet loss ratio
Fig. 4. Supply, demand and welfare vs packet loss ratio when E ¼ 10.
622
20
22.8
18
22.7 Supply Demand
16 22.6
12 suboptimal social welfare optimal social welfare Welfare of the power provdier Welfare of the consumers
10 8
Amount
Social welfare
14 22.5
22.4
22.3
6 22.2
4 2
0
0.2
0.4
0.6
Packet loss ratio
0.8
22.1
0
0.2
0.4
0.6
0.8
Packet loss ratio
Fig. 5. Supply, demand and welfare vs packet loss ratio when E ¼ 10.
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Then, a couple of important definitions in repeated game are presented below [28]. Empirical distribution: The joint strategy space for the defender and the attacker is C C. The empirical distribution zh is a distribution on the joint strategy space to stage h, and is computed as,
636 638 639 640 641 642 643 644 645 646
647
zh ðsÞ ¼
1 ft 6 h : st ¼ sg for every s 2 C C h
ð53Þ
where j:j represents the number of elements of a finite set. Note that s is a strategy in the joint strategy space C C, and zh ðsÞ in fact represents the probability of selecting the strategy s in all stages before h. Correlated equilibrium: A probability distribution wðsÞ on joint strategy space C C is a correlated equilibrium of the game, if for player a, for x; y 2 C, the following condition is satisfied,
X
wðsÞ½ua ðx; sb Þ ua ðsÞ 6 0
650
In addition, for the player b, the similar condition should also be satisfied. wðsÞ is the probability for choosing strategy s, in which sa ¼ y. Since the correlated equilibrium is a stable condition, the superscript h can be canceled. (54) implies that the expected utility of any defender strategy w s0a ; sb is not greater than the expected utility of the strategy wðsÞ. In other words, neither of the defender and the attacker will derive from the stable state if the other does not derive. That is to say, the probability wðsÞ is a correlated equilibrium.
653 654 655 656 657 658 659
2 3þ h h i X 1 h t t t t Ra ðx; yÞ ¼ 4 u y; sb ua ðs Þ 5 h t6h:st ¼x a
665 666 667 668 669 670 671 672 673 674 675 676 677 678
679
ð55Þ 681
ð54Þ
s2CC;sa ¼y
652
The modified regret matching approach is different, since the player only needs to know its set of actions and the stream of utilities that it has received in the past [28]. Due to the hostile relationship between the attacker and the defender, the players in the anti-jamming game have no incentive to share its information to the other side. Thus it is reasonable and necessary to adopt the modified regret matching approach in the anti-jamming game. The procedure of the modified regret matching approach for the defender consists of two steps: first, compute the average regret at stage h; then, update the channel selection policy for stage h þ 1. The details are presented below. The average regret of the defender from x 2 C to y 2 C at stage h is defined as follows,
a
649
651
9
B. Chai, Z. Yang / Ad Hoc Networks xxx (2014) xxx–xxx
which is clearly interpreted as the measure of average ‘‘regret’’ of the defender for not selecting channel y every time that channel x is selected. However, the defender knows neither the action of the attacker stb nor its explicit utility function, hence uta y; stb
682
cannot be directly computed. Instead, the average regret can be estimated as follows,
687
2P 3þ P pta ðxÞ t h ua ðst Þ ht6h:sta ¼x uta ðst Þ t ¼y t t6h:s p ðyÞ a a e h ðx; yÞ ¼ 4 5 R a h
683 684 685 686 688
689
ð56Þ 691
660
4.2. Modified regret matching based anti-jamming algorithm
661
The implementation of regret-matching by a player requires that player to observe the strategies of all players in the past. Moreover, the player should also be able to compute its utility when its strategy in the past changes.
664
20
24
18 23 16 14 22 12
Amount
663
Social welfare
662
where pta ðxÞ represents the probability of the defender for selecting channel x at stage t. After calculating the estimated average regret, the defender adaptively updates the probability of selecting channels to achieve higher utility. If the defender selects channel x at stage h, the probability of selecting channel
10 8 6
suboptimal social welfare optimal social welfare Welfare of the power provdier Welfare of the consumers
21 Supply Demand
20
4 19 2 0 −10
−5
0
Estimation error
5
10
18 −10
−5
0
5
10
Estimation error
Fig. 6. Supply, demand and welfare vs estimation error when d ¼ 0:5.
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y at stage h þ 1 is approximately proportional to the average regret from x to y, i.e.,
phþ1 a ðyÞ ¼ 1
m h
c
( min
e h ðx; yÞ R a
l
)
;
1 C1
þ
m Ch
ð57Þ
c
where l is a sufficiently large constant to keep the sum of the probability not exceeding 1. m and c are constants, and c should be less than 0.25 [28]. On the other hand, the probability of selecting channel x is the remaining probability,
2
quadratic cost function CðsÞ ¼ as2 þ bs þ c and a quadratic ai
utility function ui ðxi Þ ¼ v i xi 2 ðxi Þ2 ; 0 6 xi 6 xi;max , which have been widely used in literatures [5,30]. The parameters are set as a ¼ 0:01; b ¼ 0:001 and c ¼ 0. v i and ai are randomly selected from ½1; 2 and ½0:01; 0:02, respectively. Algorithm 1. Modified Regret Matching based Anti-Jamming Algorithm for Defender a
710 711 712 713 714 715 716 717 718 719 720 721 722
phþ1 a ðxÞ ¼ 1
phþ1 a ðyÞ
ð58Þ
y¼1;y–x
Based on the above presentation, we summarize the proposed modified regret matching based anti-jamming algorithm in Algorithm 1. Referring to Theorem 2 in [28], the convergence and performance of the Algorithm 1 can be guaranteed as follows. Theorem 1. If the defender adopts the modified regret matching anti-jamming algorithm, then 1. For both a and b and every x; y 2 C; Rha ðx; yÞ ! 0 almost surely as h ! 1. 2. The empirical distribution zh converges almost surely to the set of correlated equilibrium of the game as h ! 1.
gðhÞ ¼ 1 rec
13:
733 734 735 736 737 738
packet ; N
hþ1
compute P a according to (56)–(58); according to the probability select channel shþ1 a
14: 15:
723 724
732
1: randomly select one channel as the initial channel s1a ; 2: set p1a ðxÞ ¼ C1 ; 8x 2 C; 3: for each stage h ¼ 1; 2 . . . do 4: rec packet ¼ 0; 5: while the stage h has not finished do 6: The DCC receives the packet at channel sha ; 7: for each consumer i 2 N do 8: if the packet of consumer i is received by the DCC then 9: rec packet þ þ; 10: end if 11: end for 12: end while
708 C X
731
Phþ1 a ; 16:
5. Performance evaluation
h if shþ1 a ! ¼ sa then
to the consumers the DCC will broadcast shþ1 a via a security channel at stage h; 18: end if 19: end for 17:
725
5.1. Communication quality impacts on DRM performance
726
In this section, a group of simulations have been conducted to evaluate the communication quality impacts on DRM performance. Without loss of any generality, we consider a scenario with one power provider, one DCC and 100 consumers in a smart microgrid. We assume a
728 729 730
763
As shown in Fig. 3, we have fully investigate the relationship among price and supply, demand, the customers’
45.2
45
44.8
Amount
727
44.6
44.4
44.2 optimal supply or demand suboptimal supply suboptimal demand
44
43.8
0
10
20
30
40
50
60
70
80
90
100
Simulation index Fig. 7. Supply and demand under unreliable communication.
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welfare, the power provider’s welfare and the social welfare. It guarantees that at the optimal point, the social welfare is maximized while the supply and the demand keep in balance, i.e., the balance between the supply and the demand and the social welfare maximization are achieved at the optimal price point. It is clear that the social welfare and the welfare of power provider increases with the price firstly and drops dramatically when the price exceeds the optimal price. In addition, the welfare of the consumers increases with the price firstly and drops before reaching the optimal price.
Fig. 4 presents the impacts of packet loss ratio on the welfare when E ¼ 10. With the increasing of the packet loss ratio, the supply will increase slowly and the demand will drop fast. In addition, both the welfare of the power provider and the welfare of consumers decrease. As a consequence, the social welfare drops with the increase of the packet loss ratio. Similarly, Fig. 5 present the impacts of packet loss ratio on the welfare when E ¼ 10. With the increasing of the packet loss ratio, both the supply and the demand will drop. Differently, the welfare of the power provider decreases and the welfare of consumers increases.
30
25
20
Amount
767
Optimal social welfare Suboptiaml social welare Welfare of the power provider Welfare of the consuemrs
15
10
5
0
20
40
60
80
100
Simulation index Fig. 8. Social welfare under unreliable communication.
0.9
proposed anti−jamming alogrithm random anti−jamming algorithm
0.8 0.7
Packet loss ratio
766
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0.6 0.5 0.4 0.3 0.2 0.1
0
20
40
60
80
100
Experiment index Fig. 9. Comparison between the proposed algorithm and random anti-jamming algorithm.
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225 220
Social welfare
215 210 205 200 195 proposed anti−jamming alogrithm random anti−jamming algorithm
190 185
0
10
20
30
40
50
60
70
80
90
100
Experiment index Fig. 10. Social welfare under two different algorithm.
26.2
0.6 proposed anti−jamming alogrithm random anti−jamming algorithm
26
0.5
25.8
0.45
25.6
Social welfare
Packet loss ratio
0.55
0.4 0.35 0.3
25.4 25.2 25
0.25
24.8
0.2
24.6
0.15
24.4
0.1
3
4
5
6
Number of channels
7
proposed anti−jamming alogrithm random anti−jamming algorithm
3
4
5
6
7
Number of channels
Fig. 11. Packet loss ratio and social welfare vs number of channels under two algorithms.
788 789 790 791 792 793 794 795 796 797
In total, the social welfare still drops with the increase of the packet loss ratio. We also investigate the impacts of estimation errors on the performance. We observe the supply, the demand, the welfare of the power provider, the welfare of the consumers and the social welfare jointly when E increases form 10 to 10, for a given packet loss ratio d ¼ 0:5. The social welfare is maximized when E ¼ 0. All the simulation results in Figs. 4–6 coincide with the theoretical analysis in Section 3.
100 Monte Carlo trials have been implemented to compare the optimal supply and demand under perfect communications with those suboptimal ones under unreliable communication. The simulation is performed for a given packet loss ratio d ¼ 0:2 and assuming the estimation error ei following a uniform distribution in ½0:5; 0:5. As shown in Fig. 7, the suboptimal demand is always smaller than the optimal demand. Meanwhile, the suboptimal supply fluctuates around the optimal supply, which verifies the theoretical analysis.
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Fig. 8 shows the optimal social welfare under perfect communication. Meanwhile, it also depicts the suboptimal ones under jamming attack. It is clear that the suboptimal social welfare will drop correspondingly to the larger gap between the suboptimal supply and the optimal supply. 5.2. Modified regret matching based on anti-jamming algorithm
837
To evaluate the performance of the proposed anti-jamming algorithm, we perform the second group of simulations. In order to have a baseline being compared with, a random anti-jamming algorithm is presented: the defender randomly selects one channel for transmission, i.e., the probability of selecting a certain channel for transmission is random. For the attacker, we assume that the attacker selects the certain channel for jamming, which is with the highest probability for the transmission of the defender. 100 Monte Carlo trails have been implemented to compare the packet loss ratio under the two algorithms with 4 available channels. As shown in Fig. 9, the packet loss ratio under the proposed algorithm is always lower than that under the random anti-jamming algorithm. Similarly, as shown in Fig. 10, the social welfare under the proposed algorithm is always higher than that under the random anti-jamming algorithm. Fig. 11 shows that with more available communication channels, the packet loss ratio, as well as the social welfare, can be further improved. Meanwhile, it is constantly observed that the proposed algorithm achieves better performance than the random anti-jamming algorithm in terms of packet loss ratio and social welfare.
838
6. Conclusion
839
858
In this paper, we consider jamming attack in communication networks in smart microgrid. Based on the aforementioned research results, the relationship between DRM performance (i.e., social welfare) and communication quality is well analyzed. With larger packet loss ratio, the smart microgrid will suffer worse DRM performance. Thus, it is important and necessary to enhance the communication quality under jamming attack. A modified regret matching based anti-jamming algorithm is proposed to solve the challenge. Through theoretical analysis, the algorithm will drive the defender converging to the correlated equilibrium. Simulation results have shown that our proposed algorithm achieves good performance in terms of packet loss ratio and social welfare. Furthermore, state-of-the-art communication technology can be plugged into smart microgrid to further improve the communication quality, although it will involve more expensive implementation cost. For the future work, we will study the tradeoff between the communication cost and DRM performance in smart microgrid.
859
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815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836
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860 861 862 863
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[28] S. Hart, A. Mas-Colell, A reinforcement procedure leading to correlated equilibrium, Econ. Essays: A Festschrift Werner Hildenbrand (2001) 181–200. [29] Q. Yu, J. Chen, Y. Sun, Y. Fan, X. Shen, Regret matching based channel assignment for wireless sensor networks, in: Proc. IEEE ICC, 2010, pp. 1–5. [30] R. Deng, J. Chen, X. Cao, Y. Zhang, S. Maharjan, S. Gjessing, Sensingperformance tradeoff in cognitive radio enabled smart grid, IEEE Trans. Smart Grid 4 (1) (2013) 302–310. [31] R. Deng, Z. Yang, J. Chen, Load scheduling with price uncertainty and coupling constraints, in: Proc. IEEE Power & Energy Society General Meeting (PES-GM), 2013. [32] G. Carl, G. Kesidis, R. Brooks, S. Rai, Denial-of-service attackdetection techniques, IEEE Internet Comput. 10 (1) (2006) 82–89. [33] S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. [34] A.B. MacKenzie, L. Dasilva, W. Tranter, Game Theory for Wireless Engineers, Morgan and Claypool Publishers, 2006.
Zaiyue Yang received his B.S. and M.S. degrees from Department of Automation, University of Science and Technology of China, Hefei, China, in 2001 and 2004, respectively, and Ph.D. degree from Department of Mechanical Engineering, University of Hong Kong, in 2008. Then, he worked as postdoctoral fellow and research associate in Dept. Applied Mathematics, Hong Kong Polytechnic University before joining Zhejiang University, Hangzhou, China, in 2010. He is currently an associate professor there. His current research interests include smart grid, signal processing and control theory.
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Bo Chai received the B.Sc degree in control science and engineering from Zhejiang University, Hangzhou, China, in 2010. He is a member of the Group of Networked Sensing and Control (IIPC-nesC) in the State Key Laboratory of Industrial Control Technology, Zhejiang University. His research interests include game theoretic approaches, smart grid and cognitive radio.
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Contents lists available at ScienceDirect
Ad Hoc Networks journal homepage: www.elsevier.com/locate/adhoc 5 6
Impacts of unreliable communication and modified regret matching based anti-jamming approach in smart microgrid
3 4 7
Q1
8
Bo Chai, Zaiyue Yang ⇑ State Key Lab. of Industrial Control Technology, Zhejiang University, Hangzhou 310027, China
9
a r t i c l e
1 2 1 4 12 13 14 15 16
i n f o
Article history: Received 22 November 2013 Received in revised form 25 March 2014 Accepted 13 May 2014 Available online xxxx
17 18 19 20 21 22 23
Keywords: Smart grid Demand response Jamming attack Packet loss ratio Game theory
a b s t r a c t Demand response management (DRM) is one of the main features of smart grid, which is realized via bidirectional communications between the power provider and the consumers. Due to the vulnerabilities of communication channels, especially the wireless networks, communication is not perfect in practice and will be threatened by kinds of attacks, among which jamming attack is deemed as the primary one. In this paper, we consider jamming attack in the wireless communication of a smart microgrid. Firstly, the DRM performance degradation induced by jamming attack is fully studied and explicitly presented, by considering both packet loss ratio and load estimation error. Then, a modified regret matching based anti-jamming algorithm is proposed to enhance the quality of communication, and the performance of DRM. Finally, numerical results are presented to illustrate the impacts of unreliable communication on DRM, as well as the performance of the proposed anti-jamming algorithm. Ó 2014 Elsevier B.V. All rights reserved.
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40 41
1. Introduction Benefitting from information and communication technology, the smart grid is able to gather and behave on information in an automated fashion to enhance the efficiency, reliability and stability of generation, transmission, distribution and consumption of electricity power [1–4]. Demand response management (DRM) is an essential feature of current and future power grid, which is implemented by the system operator and consumers to control the power consumption at the consumers’ premises [5–7]. There exists a large body of literatures on the study of DRM, among which real-time pricing (RTP) is deemed as one of the most common tools to encourage efficient and economic power utilization [5,8–11]. Based on RTP, [5] proposes a DRM scheme aiming at maximizing the both the welfare of power provider and the welfare of
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⇑ Corresponding author. Tel.: +86 57187952994. E-mail addresses: [email protected] (B. Chai), [email protected] (Z. Yang).
consumer. As this scheme is simple and can be readily implemented in a distributed fashion, it is widely accepted and this paper is one of the most cited papers in the DRM field. However, it is noticed that [5] and most other existing DRM studies assume that the bidirectional communication in smart grid is perfect, which cannot be guaranteed in practical scenarios. Communications are solid foundation of realization of DRM in smart grid. Due to low installation cost and high flexibility, a variety of wireless communication technologies (e.g, Wi-Fi, ZigBee, WiMax, cellular networks) have been selected as the architecture of the smart grid communication network [12–15]. However, wireless communication introduces potential security vulnerabilities due to the sharing nature of wireless channels [16–18]. Among the potential security threats, jamming attack is deemed as a primary security threat to the perfect communication [16,19]. The adversaries can implement jamming attack to interrupt the system, and thus undoubtedly degrade the system performance [20].
http://dx.doi.org/10.1016/j.adhoc.2014.05.011 1570-8705/Ó 2014 Elsevier B.V. All rights reserved.
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The pioneering works considering jamming attack in smart grid are presented in [19,21,22]. [19] aimed to minimize message delay for timely communication in smart grid. [21] studied the jamming and anti-jamming in multichannel wireless communication systems from a remote sensor to the control center in smart grid. [22] stated that jamming attack will result in unreliable communication and performance degradation of smart grid. The impact of packet loss on the optimization of cost for power supply in smart grid was modeled and analyzed in [23]. A controltheoretic approach of deriving the fundamental conditions of RTP stability under the scaling and delay attacks was proposed in [24]. The authors in [25] considered the reliability of the smart grid data communications infrastructure and its impacts on the power consumption and supply optimizations. [26] illustrated the impact of malicious cyber-attacks on the energy efficiency of the smart grid. Nevertheless, the impacts of unreliable communication on DRM performance have not been explicitly and exhaustively analyzed, such as the welfare degradation of power provider and consumers. Meanwhile, besides the analysis of impacts, designing a proper anti-jamming strategy is also an important issue to address. To this end, this paper jointly considers impacts of unreliable communication on DRM performance and anti-jamming approach. Note that in this paper we shall focus on the smart microgrid, because it is smaller in scale and thus more sensitive to imperfect communication. In particular, unlike previous game-theoretic anti-jamming approaches in [20,21,27], the modified regret matching approach is proposed for the anti-jamming game, in which the players behave highly autonomously in a sense that each player only requires its own environmental information. Because the proposed anti-jamming approach is of low-level awareness of other players [28,29], it is effective and suitable to solve the anti-jamming problem in smart microgrid with limited information exchange. To the best of our knowledge, this is also the first study to discuss the anti-jamming approach using the modified regret matching approach. The contribution of this paper can be summarized as follows: The impacts of communication unreliability on DRM are fully analyzed. A modified regret matching based anti-jamming algorithm is proposed for the smart microgrid to achieve better performance under the jamming attack. Numerical results are illustrated to evaluate the performance degradation and the performance of the proposed algorithm.
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This paper is organized as follows. Section 2 describes the system models associated with DRM and jamming attack. In Section 3, performance degradation due to the communication unreliability is fully analyzed. In Section 4, a modified regret matching based anti-jamming algorithm is presented to improve communication quality under jamming attack. Performance evaluations are illustrated in Section 5. Finally, we draw conclusions in Section 6.
2. System model
137
We consider a smart microgrid with multiple consumers, one power provider and one Distribution Company Controller (DCC) involved in the DRM. The microgrid is supported by the bidirectional communication among the consumers, the power provider and the DCC. The role of DCC is to control the consumers’ power consumption and the power provider’s generation, and to coordinate each consumer with others and also with the power provider [30]. It is assumed that all available wireless communication channels are not overlapping and do not interfere with each other. Each consumer selects one channel to deliver the ‘‘demand’’ packet to DCC. Meanwhile, the attacker is able to select one channel between this consumer and DCC to jam the communication and hence results in packet loss, which will degrade the performance of the DRM as to be detailed later in Section 3. Let N , f1; 2; . . . Ng denote the consumer set, C , f1; 2; . . . Cg denote the available channel set, respectively.
138
2.1. Demand response management
156
In order to evaluate the impacts of unreliable communication on DRM, a DRM scheme has to be specified in advance. In this paper, we adopt the same analytical DRM framework in [5], which is one of the most cited paper about DRM issues and this framework is widely accepted. In this framework, DCC controls the energy consumption, and coordinates each user with others and the power provider. In each operation time slot, let s denote the supply of the power provider, and di denote the demand of the consumer i. The cost function CðsÞ indicates the cost of provider introduced by providing the supply s, and it is assumed to be increasing and convex. The utility function ui ðdi Þ quantifies the utility that consumer i receives when consuming di power, and it is assumed to be non-decreasing and concave [5,31]. For the power provider side, for any given power price p the welfare of the power provider is determined as follows, wðs; pÞ ¼ p s CðsÞ. Thus, the goal of the power provider is to determine the supply s in order to maximize its welfare, i.e.,
157
max wðs; pÞ ¼ max ps CðsÞ s
ð1Þ
s
On the other hand, for the consumer side, for any given power price p the welfare of the consumer is determined as follows, g i ðdi ; pÞ ¼ ui ðdi Þ pdi . Thus, the goal of the consumer is to determine the demand di to maximize its welfare, i.e.,
max g i ðdi ; pÞ ¼ max ui ðdi Þ pdi s
ð2Þ
di
From the perspective of social welfare [5], it is desirable to maximize the sum of all consumers’ welfare and the power provider’s welfare. In other words, the social welfare function can be obtained as
Uðs; fdi gÞ ¼
N X
N X
i¼1
i¼1
g i ðdi ; pÞ þ wðs; pÞ ¼
ui ðdi Þ CðsÞ
ð3Þ
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Meanwhile, the supply and total demand should satisfy the P constraint Ni¼1 di 6 s. In summary, the global optimization problem is formulated as, N X max Uðs; fdi gÞ ¼ ui ðdi Þ CðsÞ s;fdi g
s:t: 200 201 202 203 204 205 206 207
208
i¼1
ð4Þ
N X
kkþ1
Problem (4) is convex and therefore can be solved readily in a centralized manner. However, solving problem (4) in a distributed manner is more desirable to the power provider and the customers, because of the advantages of robustness and scalability. To this end, the dual decomposition approach can be applied [5]. First, let us define Lagrangian for the original problem (4) as:
!
N X ui ðdi Þ CðsÞ k di s
i¼1
¼
210
212
213
215 216
217 219
N X
i¼1 N X ui ðdi Þ CðsÞ þ ks k di
i¼1
ð5Þ
where k is the Lagrangian multiplier. Then, the dual objective function can be written as N X DðkÞ ¼ max Lðs; fdi g; kÞ ¼ W i ðkÞ þ UðkÞ; s;fdi g
þ dDðkÞ ¼ kk c dk " !#þ N X ¼ kk c di ðkk Þ s ðkk Þ
227 228 229 230 231 232
233
ð10Þ 235
i¼1 di ðkk Þ
where k is the iteration index and c is the step size. is the local optimizer of (7), s ðkt Þ is the local optimizer of (8), and ½zþ , maxfz; 0g. As depicted in Fig. 1, [5] proposed an optimization algorithm in a distributed and iterative way. The iteration will P stop until sk Ni¼1 di;k ¼ 0, i.e., the supply and the demand keep balance. The algorithm can be fully illustrated as follows.
236
1. The DCC announces the price to the consumers and the power provider. The initial power price is randomly selected that p1 > 0. 2. After receiving the power price pk at the kth iteration, the provider adjusts the supply according to Eq. (1), and the consumers adjust their demand according to Eq. (2). Then both the provider and the consumer send the information to the DCC as feedback. 3. After receiving the local optimal supply s and the local optimal demand di , the DCC updates the power price for next iteration by using a gradient projection h iþ P method, i.e., pkþ1 ¼ pk e sk Ni¼1 di;k , where
244
237 238 239 240 241 242 243
i¼1
ð6Þ
i¼1
where
W i ðkÞ ¼ max ui ðdi Þ kdi
ð7Þ
di
220
and
223
UðkÞ ¼ max ks CðsÞ
224
Consequently, the dual problem can be obtained as
221
Notice the first term defined in (6) can be decomposed into N separated sub-problems defined in (7) for the customers and another sub-problem defined in (8) for the energy provider. Specifically, k can be solved by iteratively applying the gradient projection method as follows,
i¼1
N X
ð9Þ
k>0
di 6 s
Lðs; fdi g; kÞ ¼
211
min DðkÞ
ð8Þ
s
e > 0 denotes the step size and ½zþ , maxfz; 0g.
Fig. 1. Interaction in the proposed DRM.
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4. Repeat steps 2 and 3 until the power price remains unchanged.
Remark 1. Notice that above algorithm is able to achieve the global optimal value of social welfare, under the assumption that the bi-directional communication between provider and DCC, as well as DCC and consumers is perfect. For the traditional grid, the power provider determines the price without consideration of the response of consumers. Smart grid benefits from the interaction between the DCC and the consumers. We can consider an extreme case that all the communication channels between the DCC and the consumers fail due to the serious jamming. Thus the interaction cannot be implemented successfully. The smart grid will operate as a traditional grid since the smart grid cannot receive the response of consumers. It is noteworthy that the DRM framework in [5] is proposed for smart grid. Thus the results are still valid for smart grid. With integration of the renewable energy, the DRM for smart grid has been extended [7]. Moreover, there exists a large literature of DRM schemes, which extend the work in [5]. However, the basic principle still keeps: pricing has effects on the generation and the demand. In turn, the generation and the demand also have effects on the pricing. Intuitively, when a more general DRM scheme is considered, jamming attack will impact the smart grid in such a way: the jamming attack results in poor performance of the communication system. Then the interaction between power providers and consumers has been effected. As a result, the performance of DRM degrades. We will study the jamming attack in smart grid with consideration of renewable energy in the future work. 2.2. Communication channel and jamming model A general architecture of smart grid communication consists of the following components: Home Area Network (HAN), Neighborhood Area Network (NAN), and Wide Area Network (WAN) [13]. Wireless communication technologies (e.g., ZigBee, Wifi) are deemed as potential technologies employed to realized the HAN and NAN [12,13]. For protecting the communication between DCC and the power provider, wired communication technologies (e.g., fiber optics, power line communication (PLC)) are utilized. For low cost and high scalability, wireless communication technologies are utilized for the communication between DCC and the consumers.
However, wireless communication is vulnerable to Denial-of-Service (DoS) attacks [16]. Typically, the attacker can conduct DoS attack to jam the communication between each components and degrade the system performance [27,32]. In our work, we assume the attacker can jam the communication from consumers to DCC. Another case is that the attacker jams the communication from DCC to consumers. However, if one consumer receives the real-time price successfully, it can broadcast to the neighbors, so that all consumer can receive the correct information. Thus, from the perspective of attacker, it is more reasonable to jam the communication from consumers to DCC. After receiving the packet from the consumer, DCC will send the acknowledgement (ACK) back. A rational jammer will listen during the ACK transmission interval rather than randomly jamming the ACK packets. Therefore, we also assume that the jammer knows whether it succeeds in jamming the transmitting channels for all the past time slots [27]. We consider a general jamming model as follows. Both the consumers and the attacker choose one channel only from the channel set at the same time. If the selected channel of the consumer and the selected channel of the attacker are the same, the jamming attack will be successful and vice versa. As a result, the ‘‘demand’’ packet will not be delivered to the DCC [27]. Specifically, as depicted in Fig. 2, the real demand of consumer i at iteration k is di;k . However, due to the existence of jamming attack, this information may not be successfully delivered to DCC. If the jamming attack is successfully conducted, DCC cannot receive di;k and thus ^ has to estimate its value. Let di;k denote the demand information of consumer i at iteration k finally presented at DCC, which can be written into
( ^ d i;k
¼
304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337
338 di;k di;k
þ ei;k
successful jamming; otherwise:
ð11Þ 340
where ei;k denotes the estimation error.
341
3. Theoretical analysis on performance degradation
342
The proposed DRM scheme [5] is realized based on the assumption of perfect communication between the consumers and the DCC [30]. If it is the case, as presented previously the power price will converge to the global optimal point p . In addition, the supply and demand will be balanced,
343
Fig. 2. Jamming attack model in the wireless communication between consumers and the DCC.
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365
s ¼
N X di
The corresponding global optimal social welfare is denoted as U . However, if the communication is imperfect under jamming attack, packet loss is possible and DCC has to use the ^ when it does not receive the real estimated demand d i;k demand di;k of consumer i. Denote the packet loss ratio as d. Thus there are approximately dN demand packets to be estimated at the DCC side. Under this circumstance, the proposed DRM scheme [5] cannot reach its global price ~ p ; instead, it will obtain a suboptimal price denoted as p via iterative updating similar to (10). However, the termination criterion is no longer (12), but becomes the following equation due to imperfect communication, N N N X X X ^ ¼d ~ þ e þ ð1 dÞ d ~ d d i i i i i¼1
368 369 370 371 372
373
ð12Þ
i¼1
~s ¼
367
i¼1
N N N X X X ~ ¼ dE þ ~ d d ¼ d ei þ i i i¼1
i¼1
ð13Þ
i¼1
~Þ u0i ðd i
~ Þ u0N ðd N
~¼ p
376
Similarly, ~s is computed at the power provider, in order to ~, as maximize its welfare (1) for the given price p
377
381 382 383 384 385 386 387 388 389 390 391
¼ ¼
¼ ¼
~ ¼ C 0 ð~sÞ p
ð14Þ
ð15Þ
Consequently, the corresponding suboptimal social e. welfare can be obtained as U From (13), it is clear that if dE ¼ 0, the suboptimal solution is identical to the optimal solution. On the other hand, both the packet loss ratio d and the accumulated estimation error E will influence the performance of DRM. In the following parts, we shall separately analyze the performance impacts caused by d and E, in terms of the degradation of social welfare.
396
ð16Þ
397
411
~ ~ 1 @p @p ¼Eþ @d C 00 ð~sÞ @d
N X i¼1
1 00 ~ u d
412
413
ð21Þ
i
i
415
That is, 1 C 00 ð~sÞ
416
417
E PN
ð22Þ
1 ~Þ i¼1 u00 ðd i i
419 420
421
399
~ @C ð~sÞ @p @~s ¼ ¼ C 00 ð~sÞ @d @d @d
400
and
403
~ @u0i ðd~i Þ @p @ d~i ¼ ¼ u00i ðd~i Þ @d @d @d
404
Since C 00 ð~sÞ and u00i ðd~i Þ are nonzero, we have
~ @~s 1 @p E ¼ ¼ P @d C 00 ð~sÞ @d 1 C 00 ð~sÞ Ni¼1 00 1~ ui ðdi Þ
ð23Þ 423
and
424
425
~i ~ @d 1 @p E ¼ ¼ P ~ @d u00i ðd~i Þ @d u00 d N 00 ~ i di 00 ~ ui i i¼1 C ðsÞ
ð24Þ u00i
1 ðd~i Þ
427
As it is possible E > 0 or E < 0, we have to consider both cases. Case 1: E > 0, i.e., the supply can fully satisfy the demand of all the consumers. Note the cost function CðsÞ is increasing and convex; meanwhile, the utility function ui ðdÞ is increasing and concave. Thus, we have 0
00
~ > 0; C ð~sÞ ¼ p
C ð~sÞ > 0
ð25Þ
and
u0i ðd~i Þ
ð17Þ
ð18Þ
428 429 430 431 432 433
434 436 437
438
~ > 0; ¼p
u00i ðd~i Þ
<0
ð26Þ
Taking (25) and (26) into (22)–(24) implies,
~ @p > 0; @d
@~s > 0; @d
~ @d i <0 @d
~s > s ;
440 441
442
ð27Þ
444 445
446
~
ð28Þ
448
That is, when E > 0, the jamming attack results in the increase of power price and supply, and decrease of demand. Now, we can consider the degradation of social welfare caused by jamming attack as follows,
449
N ~ @Cð~sÞ X @d @~s i ¼ ¼ u0i ðd~i Þ C 0 ð~sÞ @d @d @d @d @d i¼1 i¼1 ! N N ~ X X ~ @d @~s 1 1 @p i ~ ~ ~ 00 p p ¼p ¼ 00 ~ ~ @d @d @d C ð s Þ d ð Þ u i i¼1 i¼1 i ! N X ~E p 1 1 ~E < 0 ð29Þ ¼ 00 ¼ p P 00 ~ C ð~sÞ C 001ð~sÞ Ni¼1 00 1~ i¼1 ui ðdi Þ
~ @U
0
405
ð20Þ
Substituting (19) and (20) into (16) gives
~>p ; p
N ~i X @~s @d ¼Eþ @d @d i¼1
401
409
~ ~ @d 1 @p i ¼ ~ @d @d u00i d i
~ @p ¼ @d
407 408
Taking partial derivative of (13)–(15) to d, we can yield
394
and
In other words,
3.1. Impacts of unreliable communication on DRM performance
393
392
ð19Þ
Consequently, we have
where E denotes the accumulated estimation error. Mean~i and ~s denote, separately, the suboptimal demand while, d ~. and supply corresponding to the suboptimal price p ~i is computed at consumer i, in order to maxNote that d ~, as imize its welfare (2) for the given price p
~ Þ u01 ðd 1
~ @~s 1 @p ¼ C 00 ð~sÞ @d @d
i¼1
375
378 380
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349
450 451 452 453
454
N X @ui ðd~i Þ
ui ðdi Þ
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It implies that the social welfare will degrade due to packet loss when E > 0. Case 2: E < 0. From (22)–(24), we have
462
~ @p < 0; @d
463
In other words,
466
~ < p ; p
464
467
468
473 474 475 476 477
478
482 483
484
486 487
488 490
ð31Þ
¼
N X ui ðli Þ i¼1
ð33Þ
N X s:t: li 6 ~s i¼1
By introducing the Lagrangian multiplier g, the optimization problem (33) can be converted into the following dual form [33] N N X X DðgÞ ¼ max ui ðli Þ þ g ~s li fli g
i¼1
ð34Þ
i¼1
and the corresponding dual optimization problem is
min DðgÞ
ð35Þ
g>0
492
495
@D @D ¼ ¼ ¼0 @~l @~l
496
That is to say,
499
@uj @ui ¼ ¼ ¼g @~l @~l
493
497
500 501
502
504 505
506
i
509
510
On the other hand, since ui is an increasing function, we should also have N X ~l ¼ ~s i
ð38Þ
i¼1
which gives
@d
i¼1
¼
@~s @d
ð39Þ
N X e e U ui ~li C ð~sÞ S; f~li g ¼
N @C ð~sÞ X @~li @~s ¼ u0i ðl~i Þ C 0 ð~sÞ @d @d @d i¼1
¼
~ u0i ð~li Þ C 0 ð~sÞ u0i ð~li Þ C 0 ð~sÞ @ p ¼ 00 @d C ð~sÞ C 00 ð~sÞ ~ÞE ðg p PN 00 1 C ð~sÞ i¼1
1 C ð~sÞ 00
E PN
1 ~Þ i¼1 u00 ðd i i
1 ~Þ u00i ðd i
ð41Þ 516
~i Þ ¼ C 0 ð~sÞ ¼ p ~, According to (14) and (15), we have u0i ðd ~ Þ. Since u0 ðd ~ Þ is a ~ ¼ u0i ð~li Þ C 0 ð~sÞ ¼ u0i ð~li Þ u0i ðd thus g p i i i ~i ; k p ~ > 0. Recall concave function and ~li < di < d ~ < 0 and E < 0, we have C 00 ð~sÞ > 0; u00i d i
e @U
<0
517 518 519 520
521
ð42Þ
523
There also exists another interpretation of (42) as fol lows. Since (21) holds, the suboptimal solution e S; f~li g is
524
a feasible solution of the original optimization problem e < U for E < 0, and consequently, (4). Thus we have U e @U < 0. @d
526
Remark 2. Finally, a conclusion can be drawn from both cases.
529
The jamming attack in bidirectional communications reduces the demand of the consumer. The jamming attack may not always increase the supply of the power provider. The social welfare of DRM drops with an increasing of the packet loss ratio.
531
@d
525
527 528
530
ð40Þ
532 533 534 535 536 537
3.2. Impacts of estimation error on DRM performance
538
In this part, we shall consider the impacts of estimation error E on demand, supply and welfare, for a fixed given packet loss ratio d. This investigation is important because E relies on the estimation algorithm being applied. Therefore, different estimation algorithms will lead to different E. Even for the same estimation algorithm, if E is timedependent it will vary across different time slots. In other words, even under the same d, the demand, supply and welfare will also be influenced by the time varying E. From (13), it is clear that the analysis of E is parallel to that of d. That is, similar to (16)–(18) we have
539
N ~ X @~s @d i ¼dþ @E @E i¼1
541 542 543 544 545 546 547 548 549
552 553
554
ð44Þ
and
i¼1
Q1 Please cite this article in press as: B. Chai, Z. Yang, Impacts of unreliable communication and modified regret matching based anti-jamming approach in smart microgrid, Ad Hoc Netw. (2014), http://dx.doi.org/10.1016/j.adhoc.2014.05.011
540
550
ð43Þ
and
~ @C 0 ð~sÞ @p @~s ¼ ¼ C 00 ð~sÞ @E @E @E
Now, the social welfare is given by
512
ð37Þ
j
N X @~li
508
ð36Þ
j
i
i¼1
@d
514
!
Considering the optimal solution f~li g, it should satisfy the following condition,
491
¼
ð32Þ
That is, the supply cannot fully satisfy the demand of all the consumers. Let li denote the actual load allocated at consumer i, and we shall consider the optimal allocation of fli g, denoted by f~li g. For a given ~s, this problem becomes the maximization of the accumulated welfare of all consumers under the P constraint of Ni¼1 li 6 ~s, i.e.,
N @ui ~ li X
513
N X @~li @~s @~s ¼ u0i ðl~i Þ C 0 ð~sÞ ¼ u0i ðl~i Þ C 0 ð~sÞ @d @d @d i¼1
i¼1
fli g
481
~ > d d i
i
max
480
~s < s ;
ð30Þ
N X ~ d
470
472
@~s < 0; @d
e @U ¼ @d
By recalling (12), it is clear that
~s < 471
~ @d i >0 @d
Its partial derivative to d can be expressed as
556 557
558
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560
~i Þ ~ ~ @u0i ðd @p ~ Þ @ di ¼ ¼ u00i ðd i @E @E @E
561
Solving above equations gives
562
564
~ @p ¼ @E
1 C 00 ð~sÞ
ð45Þ
d PN
ð46Þ
1 ~Þ i¼1 u00 ðd i i
565 567
~ @~s 1 @p ¼ 00 ~ @E C ðsÞ @E
568
and
569
571 572
573
575 576
577
ð47Þ
~ ~ @d 1 @p i ¼ ~ Þ @E @E u00 ðd i i
ð48Þ
When E > 0, we have N N ~ Þ @Cð~sÞ X ~ e X @U @ui ðd i ~ Þ @ di C 0 ð~sÞ @~s ¼ ¼ u0i ðd i @E @E @E @E @E i¼1 i¼1 ! N X ~ 1 1 @p ~ ~d < 0: ¼p 00 ¼p 00 ~ ~ @E C ð s Þ d ð Þ u i i¼1
ð49Þ
i
When E < 0, we have
e @U ¼ @E
N X i¼1
N @ui ð~li Þ @Cð~sÞ X @~li @~s ¼ u0i ð~li Þ C 0 ð~sÞ @E @E @E @E i¼1
@~s XN @~li @~s 0 ~ ¼ u0i ð~li Þ i¼1 C 0 ð~sÞ ¼ ui ðli Þ C 0 ð~sÞ @E @E @E ~ u0i ð~li Þ C 0 ð~sÞ u0i ð~li Þ C 0 ð~sÞ @ p d ¼ ¼ P 1 @E C 00 ð~sÞ C 00 ð~sÞ Ni¼1 u00 1ðd~ Þ C 00 ð~sÞ i
¼ 579 580
~Þd ðg p PN 1 C 00 ð~sÞ i¼1
1 ~Þ u00i ðd i
Note that since E < 0;
@e U @E
i
>0
ð50Þ
e < U . > 0 still implies U
7
4. Modified regret matching based anti-jamming policy
581
Above analysis clearly indicates that jamming attack leads to packet loss and hence results in the degradation of social welfare. The degradation is more severe for larger packet loss ratio d, under any specified demand estimation algorithm. Therefore, in order to maintain the optimal social welfare by depressing d, an anti-jamming approach is proposed in this section.
582
4.1. Formulation of the anti-jamming game
589
Here, we consider a scenario that there are multiple communication channels available between the consumers and DCC, and each consumer has to select one channel for information delivery each time; on the other hand, the attacker only has the ability of jamming one channel each time. Thus, for the anti-jamming side, i.e., the consumers and the DCC, this problem becomes the design of optimal channel selection policy under jamming attacks. Due to the hostile and strategic nature of jamming attacks, game theory emerges as a natural tool to determine the optimal policies of the consumers under jamming attack. Let the jamming/anti-jamming behavior in microgrid be modelled as a repeated game. The consumer set N , f1; 2; . . . Ng and the DCC are formed as a sendersreceiver pair, which is deemed as a defender in the game in opposite to the attacker. The game is formulated as follows [34],
590
Player: The defender a and the attacker b. Strategy: The available channel set C , f1; 2; . . . Cg. The strategy for defender a and attacker b at stage h are denoted as sha ; shb , respectively. Note that sh ¼ sha ; shb
608
denotes the strategy of the game at stage h.
Fig. 3. Supply, demand and welfare vs power price.
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Utility function: uha sha ; shb and uhb shb ; sha represent the
uha sha ; shb ¼ 1 dðhÞ uhb shb ; sha ¼ dðhÞ
utility function of the defender and the attacker, respectively.
ð51Þ ð52Þ
624
616 617 618 619 620 621
e , or The defender aims to maximize the social welfare U equivalently, minimize the packet loss ratio d; meanwhile, e , or equivalently, maxithe attacker aims at minimizing U and mizing d. Thus, the utility functions uha sha ; shb uhb shb ; sha can be described as follows,
where dðhÞ represents the packet loss ratio at stage h, and is calculated as 1 minus the receiving packet number to the number of the consumers N. In addition, dðhÞ can be independently calculated by the defender without the requirements of the information of the attacker.
20
24
18
23
16
22
14
Amount
Social welfare
21 12 10
20 19
8 suboptimal social welfare optimal social welfare Welfare of the power provdier Welfare of the consumers
6 4
18
16
2 0
Supply Demand
17
0
0.2
0.4
0.6
15
0.8
0
0.2
Packet loss ratio
0.4
0.6
0.8
Packet loss ratio
Fig. 4. Supply, demand and welfare vs packet loss ratio when E ¼ 10.
622
20
22.8
18
22.7 Supply Demand
16 22.6
12 suboptimal social welfare optimal social welfare Welfare of the power provdier Welfare of the consumers
10 8
Amount
Social welfare
14 22.5
22.4
22.3
6 22.2
4 2
0
0.2
0.4
0.6
Packet loss ratio
0.8
22.1
0
0.2
0.4
0.6
0.8
Packet loss ratio
Fig. 5. Supply, demand and welfare vs packet loss ratio when E ¼ 10.
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Then, a couple of important definitions in repeated game are presented below [28]. Empirical distribution: The joint strategy space for the defender and the attacker is C C. The empirical distribution zh is a distribution on the joint strategy space to stage h, and is computed as,
636 638 639 640 641 642 643 644 645 646
647
zh ðsÞ ¼
1 ft 6 h : st ¼ sg for every s 2 C C h
ð53Þ
where j:j represents the number of elements of a finite set. Note that s is a strategy in the joint strategy space C C, and zh ðsÞ in fact represents the probability of selecting the strategy s in all stages before h. Correlated equilibrium: A probability distribution wðsÞ on joint strategy space C C is a correlated equilibrium of the game, if for player a, for x; y 2 C, the following condition is satisfied,
X
wðsÞ½ua ðx; sb Þ ua ðsÞ 6 0
650
In addition, for the player b, the similar condition should also be satisfied. wðsÞ is the probability for choosing strategy s, in which sa ¼ y. Since the correlated equilibrium is a stable condition, the superscript h can be canceled. (54) implies that the expected utility of any defender strategy w s0a ; sb is not greater than the expected utility of the strategy wðsÞ. In other words, neither of the defender and the attacker will derive from the stable state if the other does not derive. That is to say, the probability wðsÞ is a correlated equilibrium.
653 654 655 656 657 658 659
2 3þ h h i X 1 h t t t t Ra ðx; yÞ ¼ 4 u y; sb ua ðs Þ 5 h t6h:st ¼x a
665 666 667 668 669 670 671 672 673 674 675 676 677 678
679
ð55Þ 681
ð54Þ
s2CC;sa ¼y
652
The modified regret matching approach is different, since the player only needs to know its set of actions and the stream of utilities that it has received in the past [28]. Due to the hostile relationship between the attacker and the defender, the players in the anti-jamming game have no incentive to share its information to the other side. Thus it is reasonable and necessary to adopt the modified regret matching approach in the anti-jamming game. The procedure of the modified regret matching approach for the defender consists of two steps: first, compute the average regret at stage h; then, update the channel selection policy for stage h þ 1. The details are presented below. The average regret of the defender from x 2 C to y 2 C at stage h is defined as follows,
a
649
651
9
B. Chai, Z. Yang / Ad Hoc Networks xxx (2014) xxx–xxx
which is clearly interpreted as the measure of average ‘‘regret’’ of the defender for not selecting channel y every time that channel x is selected. However, the defender knows neither the action of the attacker stb nor its explicit utility function, hence uta y; stb
682
cannot be directly computed. Instead, the average regret can be estimated as follows,
687
2P 3þ P pta ðxÞ t h ua ðst Þ ht6h:sta ¼x uta ðst Þ t ¼y t t6h:s p ðyÞ a a e h ðx; yÞ ¼ 4 5 R a h
683 684 685 686 688
689
ð56Þ 691
660
4.2. Modified regret matching based anti-jamming algorithm
661
The implementation of regret-matching by a player requires that player to observe the strategies of all players in the past. Moreover, the player should also be able to compute its utility when its strategy in the past changes.
664
20
24
18 23 16 14 22 12
Amount
663
Social welfare
662
where pta ðxÞ represents the probability of the defender for selecting channel x at stage t. After calculating the estimated average regret, the defender adaptively updates the probability of selecting channels to achieve higher utility. If the defender selects channel x at stage h, the probability of selecting channel
10 8 6
suboptimal social welfare optimal social welfare Welfare of the power provdier Welfare of the consumers
21 Supply Demand
20
4 19 2 0 −10
−5
0
Estimation error
5
10
18 −10
−5
0
5
10
Estimation error
Fig. 6. Supply, demand and welfare vs estimation error when d ¼ 0:5.
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y at stage h þ 1 is approximately proportional to the average regret from x to y, i.e.,
phþ1 a ðyÞ ¼ 1
m h
c
( min
e h ðx; yÞ R a
l
)
;
1 C1
þ
m Ch
ð57Þ
c
where l is a sufficiently large constant to keep the sum of the probability not exceeding 1. m and c are constants, and c should be less than 0.25 [28]. On the other hand, the probability of selecting channel x is the remaining probability,
2
quadratic cost function CðsÞ ¼ as2 þ bs þ c and a quadratic ai
utility function ui ðxi Þ ¼ v i xi 2 ðxi Þ2 ; 0 6 xi 6 xi;max , which have been widely used in literatures [5,30]. The parameters are set as a ¼ 0:01; b ¼ 0:001 and c ¼ 0. v i and ai are randomly selected from ½1; 2 and ½0:01; 0:02, respectively. Algorithm 1. Modified Regret Matching based Anti-Jamming Algorithm for Defender a
710 711 712 713 714 715 716 717 718 719 720 721 722
phþ1 a ðxÞ ¼ 1
phþ1 a ðyÞ
ð58Þ
y¼1;y–x
Based on the above presentation, we summarize the proposed modified regret matching based anti-jamming algorithm in Algorithm 1. Referring to Theorem 2 in [28], the convergence and performance of the Algorithm 1 can be guaranteed as follows. Theorem 1. If the defender adopts the modified regret matching anti-jamming algorithm, then 1. For both a and b and every x; y 2 C; Rha ðx; yÞ ! 0 almost surely as h ! 1. 2. The empirical distribution zh converges almost surely to the set of correlated equilibrium of the game as h ! 1.
gðhÞ ¼ 1 rec
13:
733 734 735 736 737 738
packet ; N
hþ1
compute P a according to (56)–(58); according to the probability select channel shþ1 a
14: 15:
723 724
732
1: randomly select one channel as the initial channel s1a ; 2: set p1a ðxÞ ¼ C1 ; 8x 2 C; 3: for each stage h ¼ 1; 2 . . . do 4: rec packet ¼ 0; 5: while the stage h has not finished do 6: The DCC receives the packet at channel sha ; 7: for each consumer i 2 N do 8: if the packet of consumer i is received by the DCC then 9: rec packet þ þ; 10: end if 11: end for 12: end while
708 C X
731
Phþ1 a ; 16:
5. Performance evaluation
h if shþ1 a ! ¼ sa then
to the consumers the DCC will broadcast shþ1 a via a security channel at stage h; 18: end if 19: end for 17:
725
5.1. Communication quality impacts on DRM performance
726
In this section, a group of simulations have been conducted to evaluate the communication quality impacts on DRM performance. Without loss of any generality, we consider a scenario with one power provider, one DCC and 100 consumers in a smart microgrid. We assume a
728 729 730
763
As shown in Fig. 3, we have fully investigate the relationship among price and supply, demand, the customers’
45.2
45
44.8
Amount
727
44.6
44.4
44.2 optimal supply or demand suboptimal supply suboptimal demand
44
43.8
0
10
20
30
40
50
60
70
80
90
100
Simulation index Fig. 7. Supply and demand under unreliable communication.
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welfare, the power provider’s welfare and the social welfare. It guarantees that at the optimal point, the social welfare is maximized while the supply and the demand keep in balance, i.e., the balance between the supply and the demand and the social welfare maximization are achieved at the optimal price point. It is clear that the social welfare and the welfare of power provider increases with the price firstly and drops dramatically when the price exceeds the optimal price. In addition, the welfare of the consumers increases with the price firstly and drops before reaching the optimal price.
Fig. 4 presents the impacts of packet loss ratio on the welfare when E ¼ 10. With the increasing of the packet loss ratio, the supply will increase slowly and the demand will drop fast. In addition, both the welfare of the power provider and the welfare of consumers decrease. As a consequence, the social welfare drops with the increase of the packet loss ratio. Similarly, Fig. 5 present the impacts of packet loss ratio on the welfare when E ¼ 10. With the increasing of the packet loss ratio, both the supply and the demand will drop. Differently, the welfare of the power provider decreases and the welfare of consumers increases.
30
25
20
Amount
767
Optimal social welfare Suboptiaml social welare Welfare of the power provider Welfare of the consuemrs
15
10
5
0
20
40
60
80
100
Simulation index Fig. 8. Social welfare under unreliable communication.
0.9
proposed anti−jamming alogrithm random anti−jamming algorithm
0.8 0.7
Packet loss ratio
766
11
B. Chai, Z. Yang / Ad Hoc Networks xxx (2014) xxx–xxx
0.6 0.5 0.4 0.3 0.2 0.1
0
20
40
60
80
100
Experiment index Fig. 9. Comparison between the proposed algorithm and random anti-jamming algorithm.
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225 220
Social welfare
215 210 205 200 195 proposed anti−jamming alogrithm random anti−jamming algorithm
190 185
0
10
20
30
40
50
60
70
80
90
100
Experiment index Fig. 10. Social welfare under two different algorithm.
26.2
0.6 proposed anti−jamming alogrithm random anti−jamming algorithm
26
0.5
25.8
0.45
25.6
Social welfare
Packet loss ratio
0.55
0.4 0.35 0.3
25.4 25.2 25
0.25
24.8
0.2
24.6
0.15
24.4
0.1
3
4
5
6
Number of channels
7
proposed anti−jamming alogrithm random anti−jamming algorithm
3
4
5
6
7
Number of channels
Fig. 11. Packet loss ratio and social welfare vs number of channels under two algorithms.
788 789 790 791 792 793 794 795 796 797
In total, the social welfare still drops with the increase of the packet loss ratio. We also investigate the impacts of estimation errors on the performance. We observe the supply, the demand, the welfare of the power provider, the welfare of the consumers and the social welfare jointly when E increases form 10 to 10, for a given packet loss ratio d ¼ 0:5. The social welfare is maximized when E ¼ 0. All the simulation results in Figs. 4–6 coincide with the theoretical analysis in Section 3.
100 Monte Carlo trials have been implemented to compare the optimal supply and demand under perfect communications with those suboptimal ones under unreliable communication. The simulation is performed for a given packet loss ratio d ¼ 0:2 and assuming the estimation error ei following a uniform distribution in ½0:5; 0:5. As shown in Fig. 7, the suboptimal demand is always smaller than the optimal demand. Meanwhile, the suboptimal supply fluctuates around the optimal supply, which verifies the theoretical analysis.
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Fig. 8 shows the optimal social welfare under perfect communication. Meanwhile, it also depicts the suboptimal ones under jamming attack. It is clear that the suboptimal social welfare will drop correspondingly to the larger gap between the suboptimal supply and the optimal supply. 5.2. Modified regret matching based on anti-jamming algorithm
837
To evaluate the performance of the proposed anti-jamming algorithm, we perform the second group of simulations. In order to have a baseline being compared with, a random anti-jamming algorithm is presented: the defender randomly selects one channel for transmission, i.e., the probability of selecting a certain channel for transmission is random. For the attacker, we assume that the attacker selects the certain channel for jamming, which is with the highest probability for the transmission of the defender. 100 Monte Carlo trails have been implemented to compare the packet loss ratio under the two algorithms with 4 available channels. As shown in Fig. 9, the packet loss ratio under the proposed algorithm is always lower than that under the random anti-jamming algorithm. Similarly, as shown in Fig. 10, the social welfare under the proposed algorithm is always higher than that under the random anti-jamming algorithm. Fig. 11 shows that with more available communication channels, the packet loss ratio, as well as the social welfare, can be further improved. Meanwhile, it is constantly observed that the proposed algorithm achieves better performance than the random anti-jamming algorithm in terms of packet loss ratio and social welfare.
838
6. Conclusion
839
858
In this paper, we consider jamming attack in communication networks in smart microgrid. Based on the aforementioned research results, the relationship between DRM performance (i.e., social welfare) and communication quality is well analyzed. With larger packet loss ratio, the smart microgrid will suffer worse DRM performance. Thus, it is important and necessary to enhance the communication quality under jamming attack. A modified regret matching based anti-jamming algorithm is proposed to solve the challenge. Through theoretical analysis, the algorithm will drive the defender converging to the correlated equilibrium. Simulation results have shown that our proposed algorithm achieves good performance in terms of packet loss ratio and social welfare. Furthermore, state-of-the-art communication technology can be plugged into smart microgrid to further improve the communication quality, although it will involve more expensive implementation cost. For the future work, we will study the tradeoff between the communication cost and DRM performance in smart microgrid.
859
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815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836
840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857
860 861 862 863
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Q1 Please cite this article in press as: B. Chai, Z. Yang, Impacts of unreliable communication and modified regret matching based anti-jamming approach in smart microgrid, Ad Hoc Netw. (2014), http://dx.doi.org/10.1016/j.adhoc.2014.05.011
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Zaiyue Yang received his B.S. and M.S. degrees from Department of Automation, University of Science and Technology of China, Hefei, China, in 2001 and 2004, respectively, and Ph.D. degree from Department of Mechanical Engineering, University of Hong Kong, in 2008. Then, he worked as postdoctoral fellow and research associate in Dept. Applied Mathematics, Hong Kong Polytechnic University before joining Zhejiang University, Hangzhou, China, in 2010. He is currently an associate professor there. His current research interests include smart grid, signal processing and control theory.
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Bo Chai received the B.Sc degree in control science and engineering from Zhejiang University, Hangzhou, China, in 2010. He is a member of the Group of Networked Sensing and Control (IIPC-nesC) in the State Key Laboratory of Industrial Control Technology, Zhejiang University. His research interests include game theoretic approaches, smart grid and cognitive radio.
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Q1 Please cite this article in press as: B. Chai, Z. Yang, Impacts of unreliable communication and modified regret matching based anti-jamming approach in smart microgrid, Ad Hoc Netw. (2014), http://dx.doi.org/10.1016/j.adhoc.2014.05.011
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