# Stability of the Stochastic Model for Power Markets with Interval Parameters

Authors: Zhanhui Lu; Mengfan Ji; Weijuan Wang; Gengyin Li
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

Pertaining to the random nature of demand sides and the range of demand elasticity with suppliers and consumers, a stochastic model for power markets with interval parameters is described to illustrate uncertain external disturbances, which is a generalization of the Alvarado dynamic model, stochastic model, and interval model. The interval stochastic stability criteria of the provided model are investigated by the theory of economics, interval dynamical system, and the theory stability of stochastic differential equations. The conclusions indicate that the demand elasticity stable interval can be calculated and the random excitation intensity does not impact the system stability. Some numerical examples are given to show the applicability and validity of the obtained results from a statistical perspective.


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## Body

## 1. Introduction

In March 2015, Several Opinions on Further Deepening the Reform of Power System, issued by the Central Committee of the Communist Party of China and the State Council, pointed out the need to further improve the level of guaranteeing supply and demand balance mainly on demand side management. Government departments must maintain the overall balance of power supply, in the direction of marketization, from the two aspects of the demand side and supply side. Effective competition on both power generation and demand side, as well as independent market subjects, is needed, focusing on building a diversified-subjects, orderly competition electricity trading pattern, in order to form an electricity price mechanism adapting to the market and stimulate the vitality of the enterprise, finally making market a decisive role in the allocation of resources.The power market is the generic terms of the exchange relationships between the supply and demand sides, based on fairness, competition, voluntary, and mutual benefits. Power market stability is essential to ensure that the supply of electricity is economically reliable, so the research on it makes theoretical and practical significance on regulating the market supply and demand, setting up market rules, and supervising market operation [1, 2]. Study on power market stability uses knowledge involving mathematics, economics, and power system, applying mathematical modeling method, to analyze models dynamically and determine the economy stability. In recent years, a variety of competitive game models are widely applied in power market balancing research [3–6], mainly including Cournot model [3], Stackelberg model [4], and the supply function model [5]. In 1990, Beavis and Dobbs began to study economic system stability [7]. And the power market stability was first proposed by Alvarado using differential algebraic equations and eigenvalue method, combining market dynamics, setting up continuous power market dynamics models [8, 9]. Based on models above, a series of conditions to determine the power market and practical stability are provided in [10, 11]. Considering the instantaneously clearing of power market, the cyclic pattern of load demand and discrete biding strategy, the power market is described as a periodic difference system in [12]. In [13], according to the different demand utility, supply functions, and price forecasting models, a series of power market discrete difference models depicting autonomy, heteronomy continuous, or discrete linear periodical systems are provided, and the Lyapunov stability of these models is systematically analyzed. In fact, discussions about stability mentioned above are about Lyapunov asymptotical stability of equilibrium in the deterministic differential equation theory.Stochastic factors have been taken into consideration in the research of models for financial market. References [14, 15] study some generalizations of the Heston model with stochastic volatility in stock market, analyzing the statistical properties of the escape times and the effect role on the system. In [16], several models describing the stock market and their statistical properties are studied and compared. In [17, 18], hitting time and escape time distributions of stock price returns are analyzed, and then the distributions obtained from different models are compared with those obtained from real market data. These papers about financial market researching nonlinear model focus more on the numerical stimulation to compare different models.At present, some scholars have attributed specific systems to stochastic or interval dynamical systems in order to discuss system stability [19, 20]. In power system, due to effects from measurement error, model simplifying, and external interference, some system parameters cannot be measured accurately. An electricity model under Gauss type random excitation has been structured in [20], theoretically proving the power system mean stable and mean-square stable under small Gauss type random excitation. In [21], p-moment stability of power system under small Gauss type random excitation is verified when p is greater than or equal to 2 using Lyapunov method, Ito isometry formula and matrix theory. In [22, 23], p-moment stability of power system is also discussed. In the power market, considering that demand elasticity is uncertain and ranging within interval, a power market interval model is described, and the conditions to determine the interval stability are provided in [24]. In [25], a class of power market stochastic dynamic model is presented, and the conditions to determine the mean stability are provided considering the randomness of the electricity demanding. Factors including economy increasing, industrial structure, and energy consumption structure, such as the uncertainty of wind power fluctuation and interval, jointly become a factor that makes demand elasticity of both sides and the power demand uncertain and influence the stability of power market. Uncertainty includes randomness and fuzziness, whose generating mechanism and physical meanings both differ from each other [26]. As a result, it is necessary to integrate the randomness of electricity demand with interval features of consumer demand elasticity changing in the research on power market stability.In this paper, random factors are taken into consideration on a basis of power market interval models, presenting a stochastic model for power markets with interval parameters, combining the thought of randomness and interval. The stochastic differential equation theory is nowadays applied in many fields like finance, systematic science, and engineering control [27–29]. Adopting theories including economy, interval dynamical system, and stochastic differential equation to analyze the model’s stability, theorems of interval stochastic stability are provided and proved as well. It turned out that using the theory’s stability criterion is able to determine the range of suppliers and consumers demand elasticity which makes system stable. Finally, the validity of results is verified by using exact power market examples and stimulation analysis in Matlab. The results generate conclusions about power market certainty models [8], interval models [24], and stochastic models [25].In this text,· means the Euclidean norm of a vector, λmax· and λmin·, respectively, denote the matrix’s maximum and minimum eigenvalue. I is unit matrix.


## 2. The Stochastic Model for Power Markets with Interval Parameters

Let the generator cost functions and consumer utility functions be quadratic withm-suppliers and n-consumers. When a supplier observes a power market price λ higher than its production cost λgi, the supplier will expand production until the marginal cost of production equals the price. The rate of expansion is proportional to the difference between the observed market price and the actual cost of production. Let the generation power output be Pgi from supplier i, and the speed it can respond with is supplier-dependent, denoted by a time constant τgi for supplier i. The following model is proposed by Alvarado to describe power market dynamic behavior under the hypothesis in [8]:(1)τgiP˙gi=λ-bgi-cgiPgii=1,2,…,m,where Pgi is the generation power output, τgi is the responding speed of power output, λ is the price at any given time, bgi+cgiPgi is the marginal cost of supplier i, cgi is the demand elasticity of supplier i, and bgi is the linear cost coefficient of supplier i.And for the consumers, the model is formed like(2)τdjP˙dj=-λ+bdj+cdjPdjj=1,2,…,n,where Pdj is a consumer’s power demand, τdj is the speed of consumption demand expansion, bdj+cdjPdj is the marginal benefit of consumer j, cdj is the demand elasticity of consumer j, and bdj is the linear cost coefficient of consumer j. In addition, the equation holds:(3)∑i=1mPgi=∑j=1nPdj.Considering the congestion of power market, using flow distribution factors, a single congested condition can be represented:(4)sg1Pg1+sg2Pg2+⋯+sgmPgm+sd1Pd1+sd2Pd2+⋯+sdnPdn=h1.Usually, a complete model withnS-congested condition, m-suppliers, and n-consumers is described as follows:(5)T000P~˙Λ~˙=CSTS0P~Λ~+0h,where T=diagτg1,τg2,…,τgm,τd1,τd2,…,τdn, τgi>0, τdj>0, C=diag-cg1,-cg2,…,-cgm,cd1,cd2,…,cdn, Λ~=λμ1⋯μnS, i=1,2,…,m, j=1,2,…,n, Λ~∈RnS+1, P~∈Rm+n, μk(k=1,2,…,ns) denotes congested Lagrange multiplier, and matrix S stands for sensitivities of the constraints(6)1⋯1-1⋯-1sg11⋯sg1msd11⋯sd1n⋮⋮⋮⋮sgnS1⋯sgnSmsdnS1⋯sdnSn;the first line of the matrix denotes the power balance conditions; b=-bg1-bg2⋯-bgm⋯bd1bd2⋯bdnΤ is a cost vector of linear coefficients; h=0h1h2⋯hnST is a constant vector; and hi is the value of the right-hand side in the constraint equations in the remaining positions for i=1,2,…,nS.Since the system in equation (5) has at least one equilibrium point, the initial system can be turned into(7)T000P˙Λ˙=CSTS0PΛ.The demand elasticity of both suppliers and consumers varies in some ranges, as the former is influenced by installed capacity ands growing speed of generating energy, and the latter is affected by growing speed of power consumption. Let the range be c_gi≤cgi≤c¯gi and c_dj≤cdj≤c¯dj, i=1,2,…,m, j=1,2,…,n.Then,(8)C_=diag-c¯g1,-c¯g2,…,-c¯gm,c_d1,c_d2,…,c_dn,C¯=diag-c_g1,-c_g2,…,-c_gm,c¯d1,c¯d2,…,c¯dn.IfC=C-+C_/2, C~=C--C_/2, CI=C+ΔC, then ΔC∈[-C~,C~]. Equation (7) can be described as the interval dynamic system model below:(9)T000P˙Λ˙=CISTS0PΛ,where CI=[C_,C¯] is an interval matrix, C∈CI.Usually, there existsm+n>nS+1. Let r(S)=nS+1 and S=(S1,S2), where S1 is a nonsingular (nS+1)×(nS+1)-order submatrix of S. Dividing T and CI into(10)T=T100T2,CI=C1I00C2I,where T1 and C1I are (nS+1)×(nS+1)-order diagonal matrices, T2 and C2I are q×q-order diagonal matrices, and q=m+n-(ns+1). Similarly, matrices C1I and C2I are interval matrices with the form like [C1_,C1¯] and [C2_,C2¯], where C1∈C1I, C2∈C2I.The power market interval model can be derived after variations:(11)T3dP2t=C2I+S3TC1IS3P2tdt.Besides the interval features of demand elasticity for suppliers and consumers, considering the stochasticity of electricity demand, a random excitation is added to the right side of equation (11) and then the model with interval parameters is written as follows:(12)T3dP2t=C2I+S3TC1IS3P2tdt+QdBt,where t∈[t0,+∞), the initial value P2(t0)=P0 is bounded, and B(t) is Wiener process and independent of P2(t), which is consumers electricity demand changed with t. Q is the strength of random excitation representing the maximum value possible. The random excitation can be deemed as power demand stochastic fluctuation caused by objective factors, and these fluctuation are presumed to be Gaussian Wiener process generally.Remark 1.
LetQ=0, C1I=C1, C2I=C2; Q=0; C1I=C1, C2I=C2 in model (12), it can be simplified to the deterministic dynamic model proposed by Alvarado in [8], the interval dynamic system model in [24] and the stochastic dynamic model for power market in [25], respectively. Therefore, the model for power market (12) considering both the randomness of power demand and interval features of demand elasticity for suppliers and consumers, is a generalization of deterministic model, interval model and stochastic model proposed by Alvarado, instead of only considering one of the interval or stochastic features.


## 3. The Mean Interval Stability

In this part, we consider the stability of power market uncertain stochastic model (12). The definition of interval matrices in differential equation interval dynamical system is as follows.For any1≤i≤m and 1≤j≤n, m×n matrices A_=(uij)m×n, and A_=(uij)m×n satisfy uij≤vij. Let m×n matrix be(13)A_,A¯=A=aijm×n:uij≤aij≤vij,1≤i≤m,1≤j≤n.Consider this interval dynamic system(14)dXt=AIXtdt.Definition 2 (see [30]).
For anyAI∈[A-,A_], if the zero solution of system (14) is asymptotically stable, then system (14) is interval stable.
With Definition2, referring to the definition of mean stability in [31], the definition of system mean interval stability is provided.Definition 3.
For anyAI∈[A-,A_], if the solution process of the stochastic model for power markets with interval parameters (12) is asymptotically stable and it satisfies limt→∞⁡EX(t)<c, where c is a constant value greater than zero, then system (14) is interval stable.
The differential operatorL is [19](15)L=∂∂t+∑i=1nfiX,t∂∂xi,where f(X,t)=AIX(t).Next, we give some preliminary lemmas, which play an important role in the proof of the stability theorem.Lemma 4 (see [32]).
IfA is a n×n real symmetric positive definite matrix and x is a n-dimensional column vector, then λmin(A)XTX≤XTAX≤λmax(A)XTX.Lemma 5 (see [33]).
For∀X0 and t≥t0, there must exist(16)EXt;t0,X0p≤c2c1X0pe-c3t-t0,if the conditions below are satisfied:(i)
Linear ordinary differential equationdX(t)=AX(t)dt has equilibrium solutions.(ii)
There exists a positive definite functionV(X,t) and positive constants c1, c2, c3, satisfying LV(X,t)≤-c3V(X,t) and c1Xp≤V(X,t)≤c2Xp when (X,t)∈Rn×[t0,∞).The inequality means that the equilibrium solutions of the system are p-moment exponential stable.Lemma 6 (see [32]).
IfA is a n×n real symmetric positive definite matrix, then there must exist an invertible symmetric matrix B satisfying A=B2.Lemma 7 (see [32]).
If there is an×n real symmetric positive definite matrix Z, which makes the stochastic model for power markets with interval parameters (12) satisfying these two conditions(17)h1T33-1P22≤T33-1P2TZT33-1P2≤h2T33-1P22;λmaxZ1/2T33TC3T33Z-1/2+Z-1/2T33TC3T33Z1/2+2C~2+C~1·S32·T332·Zh1<0,then there exists EeT33T(C2I+S3TC1IS3)T33(t-t0)X02≤βX02e-α(t-t0), where α is a positive constant, β=h2/h1.Proof.
SinceT3 is reversible, there must be a reversible matrix T33, satisfying T3=(T33-1)TT33-1. By substituting it into (12), we have(18)T33-1TT33-1dP2t=C2I+S3TC1IS3P2tdt+QdBt.
Both sides of the equation premultiplyT33T and let X(t)=T33-1P2(t); then, dX(t)=T33-1dP2(t) and (12) can be written as(19)dXt=T33TC2I+S3TC1IS3T33Xtdt+T33TQdBt.
The explicit expression of its solution [34] is(20)Xt=eT33TC2I+S3TC1IS3T33t-t0X0+∫t0teT33TC2I+S3TC1IS3T33t-sT33TQdBs.
LetQ=0; system (12) can be simplified as an interval model like this(21)T3dP2t=C2I+S3TC1IS3P2tdt,and the explicit expression of its solution is(22)Xt=eT33TC2I+S3TC1IS3T33t-t0X0.
Structure a Lyapunov functionV(X,t)=XΤZX. Since h1T33-1P22≤(T33-1P2)ΤZT33-1P2≤h2T33-1P22, we have h1X2≤XΤZX≤h2X2. It is obvious that V(X,t) has an infinitesimal upper bound and infinite lower bound.
LetC3=C2+S3TC1S3. Since ∂(XΤZX)/∂t=0, we obtain(23)LV=∑i=1kfiX,t∂∂xiXΤZX=2XΤZfX,t=2XΤZT33TC3T33X+2XΤZT33TΔC2+S3TΔC1S3T33X.
By Lemma4, the first item of the right-hand side can be turned into(24)2XΤZT33TC3T33X=XΤZ1/2Z1/2T33TC3T33Z-1/2+Z-1/2T33TC3T33Z1/2Z1/2X=Z1/2XΤZ1/2T33TC3T33Z-1/2+Z-1/2T33TC3T33Z1/2Z1/2X≤λmaxZ1/2T33TC3T33Z-1/2+Z-1/2T33TC3T33Z1/2XΤZX.
For anyε>0, there is 2ab≤εa2+b2/ε. Hence, the second item of the right-hand side of equation (23) can be formed as(25)2XΤZT33TΔC2+S3TΔC1S3T33X≤2XΤZ1/2Z1/2T33TΔC2+S3TΔC1S3T33X≤εXΤZ1/22+1εZ1/2T33TΔC2+S3TΔC1S3T33X2≤εXΤZX+1εZ·T334·ΔC2+S3TΔC1S32·X2.
By the norm property, we have(26)ΔC2+S3TΔC1S3≤ΔC2+ΔC1·S32≤C~2+C~1·S32.
By the condition ofh1X2≤XΤZX we can obtain X2≤XΤZX/h1, and(27)2XΤZT33TΔC2+S3TΔC1S3T33X≤ε+Z·T334·C~2+C~1·S322ε·h1XΤZX.
Letε=T332·(C~2+C~1·S32)·Z/h1; then, we acquire(28)2XΤZT33TΔC2+S3TΔC1S3T33X≤2T332·C~2+C~1·S32·Zh1·XΤZX.
Substituting (24) and (28) into (23), we have(29)LV≤λmaxZ1/2T33TC3T33Z-1/2+Z-1/2T33TC3T33Z1/2+2T332·C~2+C~1·S32·Zh1XΤZX.
Owing toV(X,t)=XΤZX and other conditions, there must exist h3>0 such that LV(X,t)≤-h3V(X,t). Since h1X2≤XΤZX≤h2X2, we confirm the system satisfies conditions in Lemma 5, so for any AI∈[A-,A_], EX(t;t0,X0)2≤h2X02e-h3(t-t0)/h1. Let h2/h1=β and h3=α; then, α>0 and β>0. Owing to equation (22), we have(30)EeT33TC2I+S3TC1IS3T33t-t0X02≤βX02e-αt-t0.
Considering any given initial times∈[t0,t), let the initial time of these varieties of stationary curve be X(s)=T33TQ; according to Lemma 5, we know that any solution curve satisfies(31)EXt;s,T33TQ2=EeT33TC2I+S3TC1IS3T33t-sT33TQ2≤βT33TQ2e-αt-s.Theorem 8.
If there exists a positive definite matrixZ, which makes(32)h1T33-1P22≤T33-1P2TZT33-1P2≤h2T33-1P22,λmaxZ1/2T33TC3T33Z-1/2+Z-1/2T33TC3T33Z1/2+2C~2+C~1·S32·T332·Zh1<0,where h1 and h2 are both positive constant, then model (12) is mean interval stable.Proof.
According to Cauchy-Schwarz inequality, we have(33)EXt2≤EXt2=EXTtXt=EeT33TC2I+S3TC1IS3T33t-t0X0TeT33TC2I+S3TC1IS3T33t-t0X0+EeT33TC2I+S3TC1IS3T33t-t0X0T∫t0teT33TC2I+S3TC1IS3T33t-sT33TQdBs+E∫t0teT33TC2I+S3TC1IS3T33t-sT33TQdBsTeT33TC2I+S3TC1IS3T33t-t0X0+E∫t0teT33TC2I+S3TC1IS3T33t-sT33TQdBsT∫t0teT33TC2I+S3TC1IS3T33t-sT33TQdBs.
Then, we analyze the right-side items of this inequality separately. For the first item, owing to the definition of vector norm and inequality (30) in Lemma 7, we obtain(34)EeT33TC2I+S3TC1IS3T33t-t0X0TeT33TC2I+S3TC1IS3T33t-t0X0=EeT33TC2I+S3TC1IS3T33t-t0X02≤βX02e-αt-t0.
For the second and third items, by the features of Wiener process, we have(35)EeT33TC2I+S3TC1IS3T33t-t0X0T∫t0teT33TC2I+S3TC1IS3T33t-sT33TQdBs=0,E∫t0teT33TC2I+S3TC1IS3T33t-sT33TQdBsTeT33TC2I+S3TC1IS3T33t-t0X0=0.
For the fourth item, using Ito isometry formula and Lemma7, we have(36)E∫t0teT33TC2I+S3TC1IS3T33t-sT33TQdBsT∫t0teT33TC2I+S3TC1IS3T33t-sT33TQdBs=E∫t0teT33TC2I+S3TC1IS3T33t-sT33TQdBs2=∫t0tEeT33TC2I+S3TC1IS3T33t-sT33TQ2ds≤∫t0tβT33TQ2e-αt-sds=βT33TQ21-e-αt-t0α.
Overall, it is proposed that(37)EXt2≤EXTtXt≤βX02e-αt-t0+βT33TQ21-e-αt-t0α.
Considering thatα>0 and β>0, take a limit on both sides and we have(38)limt→∞EXt2≤βT33TQ2α.
Letγ>βT33TQ2/α. If γ=βT33TQ2/α+1, then limt→∞⁡EX(t)<γ. By Definition 3, model (12) for power markets with interval parameters is proved to be mean interval stable.In particular, letZ=I and h1=h2=1; then, we have inferences below.Corollary 9.
For model (12) for power markets with interval parameters, if λmax(T33TC3T33)+(C~2+C~1·S32)T332<0, then the system is mean interval stable.A concise criterion on the stability of the stochastic model for power markets with interval parameters is given in the corollary. It suggests that using this criterion can structure a stable interval matrix of the system and find out the value range of suppliers and consumers demand elasticity, which makes the system stable.Remark 10.
According to Theorem8 and Corollary 9 above, we find the corollary strongly effective and practical. When C~1=C~2=0 in Corollary 9, C is a certain matrix and the consumers demand elasticity is certain. Then, the stability determine condition of model (12) can be simplified as λmax(T33TC3T33)<0; that is to say, all eigenvalues of C3 are negative, which is consistent with Theorem 2 in [10] and theorem in [25]. If Q=0, then the model gets rid of random factors and becomes a deterministic model, which agrees with the theorem in [24]. So we can see that the mean interval stable Theorem 8 and Corollary 9 are a generalization of [24, 25].


## 4. The Numerical Examples

In this part, we begin to use Theorem8 and Corollary 9 above to analyze the stochastic model for power markets with interval parameters. Since the linear interval stochastic model (19) is an equivalent transformation of model (12), we present the numerical examples of power market model (19) in order to analyze numbers and simulate numbers. These data are derived from Table 1 in [8], corresponding to a determinacy electricity market dynamic model proposed by Alvarado.In the power market stochastic model (12), consider the set of differential/algebraic equations (DAE) corresponding to the case of three suppliers and two consumers. The demands elasticity of suppliers is 0.3, 0.5, and 0.2, and response speeds of generation power output are 0.1, 0.3, and 0.2; the demands elasticity of consumers is −0.5 and −0.6, and expansion speeds of consumer demands are 0.2 and 0.25, respectively. The steady-state values of the power demand are 7.68 and 8.05; when the disturbed values are 0.5 and 0.25, and stochastic excitations are 0.03 and 0.06, respectively:(39)S1=1110.1-0.100.200.3,S2=-1-10.1-0.1-0.1-0.1.The corresponding data of model (19) are represented as follows:(40)C1=diag-0.3,-0.5,-0.2,C2=diag-0.5,-0.6,S3=0.25,-1.25;-0.75,-0.25;-0.5,0.5,T33≈-1.435,0.550;-0.594,-1.326,Q=0.030.06T,P2t=Pd1t,Pd2tT.We compute(41)λmaxT33TC3T33+C~2+C~1·S32T332=-2+2.412C~2+4C~1<0.IfC~2+4C~1<0.8292, then we have the conditions of Corollary 9 fulfilled. Let ΔC1=diag0.23,0.21,0.23 and ΔC2=diag0.35,0.361; then,(42)ΔC2+4ΔC1=0.3419<0.8292,λmaxT33TC3T33+ΔC2+ΔC1·S32T332=-0.0113<0,which meets the conditions of Corollary 9, suggesting the range is valid. We obtain(43)C_1=diag-0.53,-0.71,-0.43,C_2=diag-0.85,-0.961,C¯1=diag-0.07,-0.29,0.03,C¯2=diag-0.15,-0.239.We can also obtain the specific changing range of both sides of demand elasticity 0.07≤cg1≤0.53, 0.29≤cg2≤0.71, -0.03≤cg3≤0.43, -0.85≤cd1≤-0.15, and -0.961≤cd2≤-0.239.Known from Theorem8, system (19) is stable if the respective parameter is in the range above. Next, we begin the numerical stimulation to prove our theoretical analysis effective. Select several typical sets of data who have the greatest change of demand elasticity: the suppliers demand elasticity is 0.53, 0.71, and 0.43 and the consumers’ is −0.15 and −0.239. When t0=1s, set the electricity demand’s disturbed value of consumer 1 and consumer 2 to be, respectively, 0.5 and 0.25 and the excitation of electricity demand to be 0.03 and 0.06. Compare electricity demand of consumers with and without the stochastic excitation.Based on statistics theory, if one solution process is regarded as a sample from population, the average value of many solution processes is the mean value of samples. Equations below are used to calculate sample’s expectation and deviation percentage:(44)EPd=1N∑k=1NPdk,εPd=EPd-EPd0EPd0×100%,where Pd(k) represents the samples and N is the total number of samples. EPd0 are calculated by E-M numerical methods.Adding stochastic excitations and disturbed values whent0=1s, we can calculate sample means of consumer 1 and consumer 2 electricity demand and the deviation between the electricity demands with and without random excitations. Results are presented in Tables 2 and 3.It turns out that the sample’s expected value of electricity demand gradually comes to a steady-state value and the error decreases to zero with the increase of the time. It suggests that the changing demand elasticity has no effects on the system’s stability while demand elasticity of suppliers and consumers is in the given range; and small Gauss type random excitation causes only bounded stochastic fluctuation to the power system, instead of affecting its stability.In order to directly measure the changing trend of consumer 1 and consumer 2 electricity demand, Tables1 and 2, respectively, illustrate their steady-state value and the solution processes over 500 paths of sample’s expected values, while t0=1s and the random excitation term as well as disturbed value are added to the system; path 1 and path 2 are two of the 500 paths. We can figure out throughout the trend that random excitation terms and given demand elasticity do not play a role in the system’s stability. The sample mean of system’s solution processes is asymptotically stable, then the system is mean interval stable.Table 1
A three-supplier two-consumer case with increasing degrees of congestion.
Suppliers Consumers Stable solutions n S τ g c g b g τ d c d b d P g P d 0 0.1 0.3 1 0.20 −0.50 10 7.54 13.48 0.3 0.5 2 0.25 −0.60 8 2.52 7.90 0.2 0.2 1    11.31  1 0.1 0.3 1 0.20 −0.50 10 0.40 8.53 0.3 0.5 2 0.25 −0.60 8 7.47 11.47 0.2 0.2 1    12.13  2 0.1 0.3 1 0.20 −0.50 10 1.89 7.68 0.3 0.5 2 0.25 −0.60 8 11.52 8.05 0.2 0.2 1    2.31  3 0.1 0.3 1 0.20 −0.50 10 2.30 7.56 0.3 0.5 2 0.25 −0.60 8 11.51 8.35 0.2 0.2 1    2.10Table 2
Expected values and relative errors of solution process samples of consumer 1 in different time.
t   ( s ) Expected value Relative error (%) 2 7.883 1.986 3 7.765 1.039 4 7.715 0.443 5 7.693 0.171 6 7.684 0.058 7 7.681 0.017 8 7.681 0.013 9 7.681 0.009 10 7.680 0.005Table 3
Expected values and relative errors of solution process samples of consumer 2 in different time.
t   ( s ) Expected value Relative error (%) 2 8.181 1.430 3 8.118 0.840 4 8.087 0.456 5 8.071 0.261 6 8.059 0.114 7 8.053 0.036 8 8.054 0.044 9 8.053 0.031 10 8.053 0.034Conversely, letΔC^1=ΔC1 and ΔC^2=diag0.8,-0.361; then,(45)ΔC2+4ΔC1=0.8516>0.8292,λmaxT33TC3T33+ΔC2+ΔC1·S32T332=2.9531>0,which does not satisfy the conditions of Theorem 8 and Corollary 9, suggesting the range is not valid. Take the consumers demand elasticity as 0.3 and −0.239 with other data unchanged; Figures 3 to 4 correspond to Figures 1 to 2, illustrating that the system is not mean stable, which indicates that the system may not be mean stable if Theorem 8 and Corollary 9 are not satisfied.Figure 1
Average consumption demand of consumer 1 over 500 paths when we haveΔC1=diag0.23,0.21,0.23 and ΔC2=diag0.35,0.361.Figure 2
Average consumption demand of consumer 2 over 500 paths when we haveΔC1=diag0.23,0.21,0.23 and ΔC2=diag0.35,0.361.Figure 3
Average consumption demand of consumer 1 over 500 paths whenΔC^1=ΔC1 and ΔC^2=diag0.8,-0.361.Figure 4
Average consumption demand of consumer 2 over 500 paths whenΔC^1=ΔC1 and ΔC^2=diag0.8,-0.361.Remark 11.
Applying the mean interval stable Theorem8 and Corollary 9 in this article, demand elasticity C1 and C2 in [10, 20] can be generalized to intervals [C_1,C¯1] and [C_2,C¯2], that is to say, the demand elasticity varies in a certain range, and the system’s mean interval stability remains. Comparing to [24], we discuss the stochastic features of electricity demand and objective factors affecting system stability, which makes the conclusion more practical.


## 5. Conclusions

(1)
Based on the power market dynamic action model proposed by Alvarado, a stochastic model for power markets with interval parameters is presented considering electricity demand’s stochastic features caused by economic growth, industrial structure, and energy consumption pattern adjusting, as well as the interval features of consumers changing demand elasticity. (2)
Using the theory of economics, interval dynamic system, stochastic differential equations stability, and so on, proper Lyapunov functions are structured to analyze the system’s stability; then, the stability criterion for this model is provided. (3)
The results of numerical examples are consistent with the stability analysis. The conditions are proved to be effective. The research method of this paper is expected to analyze the stability of nonlinear model for power market in future.



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