Differential Transformation of a Motor Load Model for Time-Domain Simulation

Kai Sun; Yang Liu

Differential Transformation of a Motor Load Model for Time-Domain Simulation

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that the nonlinear current-injection equation of a motor load becomes formally linear after Differential Transformation.

desk verdict A concise, sound extension of the DT method to induction motor loads; the linearity result holds, though the letter should be explicit about denominator conditions and what 'formally linear' buys you. read the letter →

arxiv 1908.09801 v1 pith:6OPYGIRT submitted 2019-08-23 eess.SY cs.SYeess.SPmath.OC

Yang Liu , Kai Sun This is my paper · ORCID

classification eess.SYcs.SYeess.SPmath.OC

keywords differentialtransformationpowersystemsimulationmotorloadmodelalgebraicequationscurrentinjectionformallylineartime-domaindynamic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This letter proves that the nonlinear algebraic current-injection equation of a third-order motor load model becomes a formally linear equation after Differential Transformation. The authors transform the load's differential equations and its algebraic current-voltage relation into recurrence equations in transform coefficients, then show that at every order $k$ the unknown current coefficients $I_{\mathrm{re}}(k), I_{\mathrm{im}}(k)$ depend linearly on the unknown voltage coefficients $V_{\mathrm{re}}(k), V_{\mathrm{im}}(k)$, with all lower-order terms collected into a known coefficient. If the proof is correct, a motor load can be included in time-domain power-system simulation without numerical iteration at each time step, extending the same property previously shown for generators and ZIP loads.

What carries the argument

The Differential Transformation (DT) is a series method that replaces a time function $x(t)$ by coefficients $X(k)$ and turns products into convolutions $\sum_m X(m)Y(k-m)$. The load-bearing machinery of this letter is the DT product and reciprocal rules applied to the intermediate variables $z_0,z_1,u_0,u_1,u_2$ defined in (4)-(5). The reciprocal rule produces the denominators $Z_1(0)$ and $U_0(0)$, and after isolating the $k$-th-order voltage terms, the remaining convolutions are collected into $B_1$ and $B_2$; the fixed matrix $A_m$ in (15) then makes (10) linear in $V_{\mathrm{re}}(k),V_{\mathrm{im}}(k)$ at every order.

What would settle it

Take a standard third-order motor-load parameter set and a terminal-voltage trajectory whose Differential Transformation coefficients are known through order 3; compute the current coefficients $I_{\mathrm{re}}(k),I_{\mathrm{im}}(k)$ directly from the nonlinear algebraic relation (2) by repeated differentiation, and compare them with the values produced by the linear relation (10) using (15)-(16). A single mismatch at any order $k$ would refute the proposition.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Proposition (10): after applying the Differential Transformation to the 3rd-order motor load model (1)-(2), the transformed current-injection equation satisfies $I(k) = \sum_m A_m V(k-m) + B_m$, where $I(k)=(I_{\mathrm{re}}(k), I_{\mathrm{im}}(k))$, $V(k)=(V_{\mathrm{re}}(k), V_{\mathrm{im}}(k))$, $A_m$ is a fixed coefficient matrix given by (15), and $B_m$ depends only on transform coefficients of order lower than $k$. The proof rewrites the intermediate voltage expressions $U_1(k)$ and $U_2(k)$ to expose the terms containing $V_{\mathrm{re}}(k)$ and $V_{\mathrm{im}}(k)$, and absorbs every other term into $B_m$. Thus the nonlinear algebraic constraint becomes a set of linear equations at each transform order.

Load-bearing premise

The load-bearing premise is that the Differential Transformation reciprocal and product rules, assumed from the authors' earlier work, apply exactly to this motor-load equation and that the denominator coefficients $Z_1(0)$ and $U_0(0)$ are nonzero at the operating point; if either condition fails, the linear relation (10) does not follow.

Editorial extensions

If this is right

At each transform order $k$, the current coefficients can be obtained from the voltage coefficients by one linear solve, removing the Newton-iteration loop that the algebraic motor-load equations otherwise require.
A time-domain simulator that already uses the DT method for generators and ZIP loads can treat motor loads with the same fixed-coefficient linear algebra, rather than a separate nonlinear solver.
The recurrence with fixed $A_m$ and history-dependent $B_m$ can be marched forward order by order, so the computational cost of each step scales with the convolution length rather than with a nonlinear iteration count.
The result supports the letter's claim that the formally-linear-after-DT property is not peculiar to one load type but extends to other dynamic load models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that operating points with $U_0(0)=0$, i.e. $z_{\mathrm{re}}^2+z_{\mathrm{im}}^2=0$, are singular for this construction; a shifted or regularized reciprocal rule might restore linearity there, but that is not tested.
The same formal linearization likely applies to any load whose current-voltage relation is rational in the terminal voltage and state variables, since only the product and reciprocal rules are used; testing a deeper-rotor induction model would show whether the structure survives.
Because $A_m$ is fixed while $B_m$ accumulates lower-order convolutions, truncating the DT series at order $N$ yields a natural error measure by comparing the $N$-th and $(N+1)$-st partial sums; the paper does not discuss this, but the recurrence makes it feasible.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a small but real result. The paper takes the authors' earlier work on differential transformation (DT) for ZIP loads and extends it to a third-order induction motor model. The new piece is the derivation of the DT of the motor's algebraic current-injection equation and the proof that, at each transform order k, the unknown voltage coefficient V(k) enters linearly, with everything else depending only on past coefficients. That is a useful property: it means the algebraic equations can be solved without Newton iteration at each time step. The derivation is algebraic and I found no hidden circularity or invented assumptions. The stress-test note matches my own reading: the key step is a quotient rule expansion of U1 and U2, and the claimed linear form follows from collecting the k-th order terms. The proof is not fully written out in the text, but what is shown is enough to reconstruct it.

What the paper does well: it states a crisp proposition and gives a direct algebraic proof. It also connects the result to the next step in the authors' research program without overselling it as a revolution. The heavy reliance on the authors' previous papers is not a flaw here, because the DT framework and quotient rules are genuinely established there. Self-citation in this context is legitimate.

Soft spots, in proportion: first, the DT of the differential equation (3) is given with details omitted. That is fine for a letter, but the authors would do well to say explicitly that the omitted derivation is not needed for the linearity proposition, or include it in a footnote. Second, denominators U0(0) and Z1(0) appear without explicit nondegeneracy conditions. Physically these correspond to a nonzero initial squared magnitude of the equivalent impedance and a nonzero initial value of the transformed denominator, so any reasonable motor operating point will satisfy them, but the assumption should be stated. Third, calling (10) a 'formally linear equation' is a bit generous: it is linear in the unknown current coefficient combined with the unknown V(k) at that order, but the B_m term depends on past values, so it is affine rather than globally linear. The proposition itself is stated precisely, so this is a wording issue, not a correctness issue. Finally, there is no numerical validation in the letter. That is a scope choice; the proof establishes the property. Still, a single small example would make the result easier to absorb.

For whom: anyone working on DT-based power system simulation, or on semi-analytical time-domain methods that rely on polynomial/Taylor representations of DAE solutions. For a broader power systems audience, it is a narrow incremental extension, not a must-read.

My recommendation: the paper deserves peer review and, after minor revision to state the denominator assumptions and adjust the wording around 'linear', it is acceptable.

Referee Report

0 major / 6 minor

Summary. This letter extends the authors' previous differential-transformation (DT) framework to a third-order induction motor load model. The algebraic current-injection equations of the motor are transformed term-by-term using DT convolution rules; the paper states a proposition that the transformed current injections I_re(k), I_im(k) are a "formally linear" function of the transformed bus voltages V_x(k), V_y(k) plus terms involving lower-order coefficients. The proof expands the numerator functions U_1, U_2, isolates the order-k voltage terms, and defines a coefficient matrix and a history vector. The paper concludes that the nonlinear motor-load current injection equation becomes linear after DT, enabling non-iterative solution within the DT simulation algorithm.

Significance. If correct, the result is a useful extension of the authors' earlier ZIP-load result: it shows that the algebraic part of a motor-load DAE also enjoys the "linear after DT" property, which is the key to avoiding Newton iterations in the DT time-domain solver. The algebraic proof is explicit and verifiable; the main steps—writing U_1 and U_2 as convolutions and applying the quotient rule—are correct under the standard DT rules. The contribution is an algebraic identity, not a numerical demonstration; no code or simulation is provided. The result is conditional on the DT rules of [1]-[2] and on nonvanishing initial values Z_0(0) and U_0(0), conditions that the manuscript currently leaves implicit.

minor comments (6)

[II-B, Eq. (10)] The notation in Eq. (10) is inconsistent with the proof: the derivation establishes I(k) = A V(k) + B(k) with A a constant 2x2 matrix and B(k) depending on k (and on the current-order state coefficients through Z(k)); the subscripted A_m and B_m in (10) and (15) suggest a convolution over m and should be corrected to match the actual algebra.
[II-A, Eqs. (6) and (9)] The proof divides by Z_0(0) in (6) and by U_0(0) in (9) without stating that these quantities are nonzero; please add the explicit nondegeneracy condition or a statement that it holds at any physical motor operating point.
[II-A, Eq. (3)] The DT of the differential equation is given with "details omitted"; since this letter relies on [1]-[2] for the DT rules, cite the exact rule or include a short derivation so that the motor-load-specific steps are self-contained and checkable.
[II-A, Eqs. (1)-(2)] The parameters r, x, x', H, and the remaining parameters in (1) are not all defined; a reader cannot verify the model or the dimensions without a complete parameter list.
[General] The paper should state explicitly that the linearity at order k is meant after the state coefficient S(k) has been computed from the differential equation in the sequential DT solution; otherwise B(k) in (16) appears to depend on an unknown at the same order.
[General] The template placeholder "REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER" remains in the text and must be removed in the final submission.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the linear form is an algebraic rearrangement of the DT quotient rule.

full rationale

The paper proves a mathematical identity: after applying differential transformation (DT), the algebraic current-injection equation (2) of a third-order motor load is rearranged into (10), where the unknown V(k) at order k appears linearly with a coefficient matrix A_m given in (15), while B_m in (16) contains only earlier-order coefficients V(0..k-1) and I(0..k-1). This is not a prediction from fitted data and not a definition of the target result in terms of itself. Equations (13)-(14) explicitly isolate the V(k) terms from U1(k) and U2(k) using the DT quotient rule, and (15)-(16) define the coefficient matrix and the history term; that is a proof by construction, not circularity. The only inherited ingredients are the standard DT product and quotient rules, cited from the authors' prior work [1]-[2]. Those rules are parameter-free mathematical identities that do not assume the motor-load linearity result, so the self-citations are not load-bearing in the circularity sense. No parameter is fitted to data, no uniqueness theorem is imported, and no empirical claim is renamed. The formal nature of the linearity, namely that B_m contains previous I values, is explicitly stated and does not make the result circular. The derivation does assume the nondegeneracy condition U0(0) != 0, which is a standard physical condition equivalent to a nonzero initial equivalent impedance magnitude; this is an assumption about correctness, not circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It relies on the motor load model from [3]-[5] and on DT rules from the authors' prior work [1]-[2]. The main implicit assumptions are the validity of the DT reciprocal/convolution rules and the non-vanishing of the denominators U0(0) and Z1(0).

assumptions (3)

domain assumption The 3rd-order motor load model in (1)-(2) is an adequate representation of an induction motor load.
Taken from refs [3]-[5]; the central claim is a property of this specific model equation.
standard math The DT rules for products, reciprocals, and other nonlinear functions from the authors' prior work [1]-[2] apply to the functions in (2).
The convolution property of DT is standard, but the specific reciprocal rule is inherited without proof here.
domain assumption The power-series (DT) representation is used formally and assumed to converge or to be applicable for solving the DAE system.
No convergence analysis is given; the calculation treats DT coefficients as algebraic objects.

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Cite this review

Pith. "Pith review of Differential Transformation of a Motor Load Model for Time-Domain Simulation." pith.science (2026). https://pith.science/paper/6OPYGIRT

@misc{pith2026190809801,
  author       = {Pith},
  title        = {Pith review of: Differential Transformation of a Motor Load Model for Time-Domain Simulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6OPYGIRT}},
  note         = {Machine review of arXiv:1908.09801}
}

read the original abstract

The Differential Transformation (DT) method has demonstrated its potential in speeding up power system time-domain simulation by our previous work. This letter further derives DTs about a motor load model and proves that the nonlinear current injection equation about a motor load can be transformed into a linear equation by means of DT.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

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work page 2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Power system time domain simulation using a differential transformation method,

Y. Liu, K. Sun, R. Yao, B. Wang, "Power system time domain simulation using a differential transformation method," IEEE Trans. Power Syst., in press

work page

[2] [2]

Solving Power System Differential Algebraic Equations Using Differential Transformation

Y. Liu, K. Sun, " Solving Power System Differential Algebraic Equations Using Differential Transformation," arXiv:1903.00935, 2019

work page Pith review arXiv 1903

[3] [3]

Kundur, N

P. Kundur, N. J. Balu, M. G. Lauby, Power system stability and control. McGraw-hill New York, 1994

work page 1994

[4] [4]

Milano, Power system modelli ng and scripting

F. Milano, Power system modelli ng and scripting . Springer Science & Business Media, 2010

work page 2010

[5] [5]

A Time -Power Series Based Semi-Analytical Approach for Power System Simulation,

B. Wang, N . Duan, K . Sun, “A Time -Power Series Based Semi-Analytical Approach for Power System Simulation,” IEEE Trans. Power Syst., vol. 34, No. 2, pp. 841-851, March 2019

work page 2019