Convergence analysis

Here we determine values of the price adjustment factor K at which market clearing using (4.8)–(4.10) will converge. Let et = pt−p∗ be the price error at iteration t with reference to the equilibrium pricep∗. Then,

Proposition 4.5.1 The convergence rate of classical price tˆatonnement is given by ||et+1|| ≤ ||I+ KJ∗||.||et|| (4.17)

where J∗ is [like (4.11)] the Jacobian matrix of z with respect to the price variable p, evaluated atp∗.Iis the unit matrix of the same size asJ∗. Also||.||denotes the Euclidean norm of a vector or matrix.

Proof We sketch a proof for the convergence rate (4.17) of classical tˆatonnement. Rewrit- ing the price updating rule (4.8) in a vector form we obtainpt+1=pt+ K.z(pt). Then,

(a)et+1=et+ K.z(pt)

By a geometric theorem on tangent representation [104], the differentiable function

z(p) is given as follows:

(b)z(p) =z(p∗) +J∗(p−p∗) +R

whereJ∗ is a first-derivative Jacobian matrix evaluated atp∗,Ris a residual term repre- senting the difference between the surfacez(p) and its tangent planez(p∗) +J∗(p−p∗) at any pointp within the N-dimension Euclidean space. Such a difference becomes smaller whenp approachesp∗.

Since, in a DRX market, the price pt oscillates around its equilibrium point p∗, the corresponding residual termR can be assumed small. Then z(pt) ≈ J∗(pt−p∗). (Note thatz(p∗) = 0.) Substituting this equation into (a), we obtain ||et+1||=||(I+ KJ∗).et||. Consequently we assert (4.17).

Remark Equation (4.17) suggests that the price adjustment error changes linearly after each iteration. This linear trend implies that||et|| ≤ ||I+ KJ∗||t_.||e0_{||for any t >0. Then}

a sufficient condition for price convergence (i.e.,||et|| →0 as t→+∞) is given as follows,

||I+ KJ∗||<1 (4.18)

Condition (4.18) can be satisfied by choosing appropriate values for the price adjust- ment factor K. However, as similar to the Newton method, the main issue here is that the DRXO cannot determine the exact Jacobian matrixJ∗ due to the lack of private cost and benefit data from participating agents. Therefore, the DRXO uses historical data to estimate J∗ such that K is determined appropriately. Note also that rather than finding

Robustness evaluation

the exact values of K, the DRXO only needs to know the feasible range to satisfy condition (4.18). This simplifies the task of parameter estimation.

However, as suggested by economists, there are some special cases where price adjust- ment using classical tˆatonnement cannot converge [103]. This issue arises when||I+KJ∗|| ≥

1 for any chosen/estimated value of K, causing prices to diverge. In such cases, the Newton method must be used instead.

Newton tˆatonnement

Here we examine the conditions under which price adjustment using the inexact New- ton method converges. Let 4t _{be the error in estimating the inverse Jacobian matrix}

(Jt)−1 in the price updating rule (4.11). (Note that4t _{also has sizeN ×N.) Then,} Proposition 4.5.2 The convergence rate of the proposed Newton pricing method is

||et+1|| ≤(L||4t||+ M||et||).||et|| (4.19)

whereL =||J||U_{(this denotes the upper bound of ||J||);M =} 1 2||J

−1_||U_||H||U _withHbeing

a Hessian matrix comprised of all second-order partial derivatives of z(p).

Proof Here we derive condition (4.19) that represents the convergence rate of the proposed inexact Newton method. By adding the error4t_{of estimatingJ}−1_{to the price adjustment}

equation (4.11), we obtainpt+1 =pt− (Jt)−1+4t .z(pt). Then, (a)et+1_=et₋ (Jt)−1₊₄t .z(pt₎

Assuming thatz(p) is twice differentiable, then its value at the equilibrium point p∗

can be expressed in terms of a second-order Taylor series expansion around any arbitrary pointp [104]. In particular,

(b)z(p∗) =z(pt) +Jt(p∗−pt) +1₂(p∗−pt)THξ(p∗−pt)

where (p∗ −pt)T is the transpose of (p∗ −pt). Hξ is the Hessian matrix evaluated at somepξbetweenp andp∗. Sincez(p∗) = 0,z(pt) =Jtet−1₂(et)THξet. Substituting this equation into (a), we obtain

||et+1||=||4t_Jt_et₋1 2(et)T (Jt)−1+4t Hξet|| ≤ ||4t_Jt_et_||+||1 2(e t₎T (Jt)−1+4t Hξet|| ≤ ||4t_||.||Jt_||.||et_||+ 1 2||(J t₎−1₊₄t_||.||Hξ_||.||(et_)||2 ∴ ||et+1|| ≤(L||4t_{||+ M||e}t_||).||et_|| where L , ||Jt||U _{= ||J||}U_{, M ,} 1 2||(J t₎−1₊₄t_||U_.||Hξ_||U ₌ 1 2||J −1_||U_.||H||U_{, assuming}

Remark Equation (4.19) implies that the price adjustment erroretchanges quadratically in each iteration. This quadratic trend, in the case that 4t _{= 0 for every t, corresponds}

to an exact Newton method that converges rapidly to the equilibrium solution [104]. Now we consider the case of 4t _{6= 0. A sufficient condition for convergence is that}

the first factor on the right hand side of (4.19), (L||4t_{||+ M||e}t_{||), is less then 1 for any}

t. This condition ensures that the error ||et|| decreases after each iteration, consequently approaching zero ast→+∞[104].

For (L||4t_{||+ M||e}t_||)<1,||4t_||and||et_{||should be limited as both L and M are fixed}

constants. While||4t_{||can be reduced by estimating more accurately the Jacobian matrix} Jt using current market data, ||et|| depends on ||e0|| and the price p0 that is initiated by the DRXO. In general, if the initiation is sufficiently close to the equilibrium point, price adjustment in subsequent iterations will converge quickly to this point. Otherwise, the price adjustment may either not converge or converge at a slower rate.

We formalize the term “sufficiently close” as the following particular condition to be used for price initiation:

L.||4t||U+ M.||e0||<1 (4.20) or ||e0||< 1−L.||4

t_||U

M (4.21)

This condition alone is sufficient for the method to converge. If the condition holds, then||e1_||<||e0_{||by (4.19). This implies||e}1_{||also satisfies (4.21), then||e}2_||<||e1_{||, etc.}

Therefore,||et|| is a decreasing sequence, which approaches zero. In fact, condition (4.20) is a stricter version of the above mentioned condition, (L||4t_{||+ M||e}t_{||)<1, as the former}

can imply the latter.

Although (4.20) is a strict condition, it is easy to implement. Specifically, by using historical data, the DRXO can determine the upper bounds L, M, and ||4t_||U _offline.

From these pre-determined parameters,p0 can be initiated satisfying condition (4.21). Besides price initiation, care must be taken estimating the Jacobian matrix J. If the error 4t _{is larger, the initial value e}0 _{must be smaller for the method to converge by}

condition (4.21). Achieving such a initiation closer to the equilibrium is obviously more difficult, as the DRXO does not know exactly the position of such an equilibrium point. Hence, a reasonable estimation of the Jacobian matrix, resulting in a relatively small error

Numerical simulation

Figure 4.9:The test system